(* *)
(**************************************************************************)
-include "static_2/s_computation/fqus_fqup.ma".
-include "static_2/static/feqg_fqup.ma".
-include "basic_2/rt_computation/fpbs_fqus.ma".
+include "basic_2/rt_transition/fpb_fqup.ma".
+include "basic_2/rt_computation/fpbs.ma".
(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
-(* Advanced properties ******************************************************)
+(* Advanced eliminators *****************************************************)
+
+lemma fpbs_ind:
+ ∀G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 →
+ (∀G,G2,L,L2,T,T2. ❪G1,L1,T1❫ ≥ ❪G,L,T❫ → ❪G,L,T❫ ≽ ❪G2,L2,T2❫ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ → Q G2 L2 T2.
+/3 width=8 by tri_TC_star_ind/ qed-.
-lemma teqx_fpbs_trans:
- ∀T1,T. T1 ≅ T →
- ∀G1,G2,L1,L2,T2. ❪G1,L1,T❫ ≥ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
-/3 width=5 by feqx_fpbs_trans, teqg_feqg/ qed-.
+lemma fpbs_ind_dx:
+ ∀G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 →
+ (∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≽ ❪G,L,T❫ → ❪G,L,T❫ ≥ ❪G2,L2,T2❫ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ → Q G1 L1 T1.
+/3 width=8 by tri_TC_star_ind_dx/ qed-.
+
+(* Advanced properties ******************************************************)
-lemma fpbs_teqx_trans:
- ∀G1,G2,L1,L2,T1,T. ❪G1,L1,T1❫ ≥ ❪G2,L2,T❫ →
- ∀T2. T ≅ T2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
-/3 width=5 by fpbs_feqx_trans, teqg_feqg/ qed-.
+lemma fpbs_refl:
+ tri_reflexive … fpbs.
+/2 width=1 by tri_inj/ qed.
(* Properties with plus-iterated structural successor for closures **********)
lemma fqup_fpbs:
∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ⬂+ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
+/4 width=5 by fqu_fquq, fquq_fpb, tri_step/
qed.
lemma fpbs_fqup_trans:
∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≥ ❪G,L,T❫ →
∀G2,L2,T2. ❪G,L,T❫ ⬂+ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
-/3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
+#G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=5 by fpbs_strap1, fqu_fpb/
+qed-.
lemma fqup_fpbs_trans:
∀G,G2,L,L2,T,T2. ❪G,L,T❫ ≥ ❪G2,L2,T2❫ →
∀G1,L1,T1. ❪G1,L1,T1❫ ⬂+ ❪G,L,T❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
-/3 width=5 by fqus_fpbs_trans, fqup_fqus/ qed-.
+#G #G2 #L #L2 #T #T2 #H1 #G1 #L1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+/3 width=9 by fpbs_strap2, fqu_fpb/
+qed-.
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