(**************************************************************************)
include "ground/xoa/ex_2_3.ma".
-include "basic_2/notation/relations/predsubtystarproper_7.ma".
+include "basic_2/notation/relations/predsubtystarproper_6.ma".
include "basic_2/rt_transition/fpb.ma".
include "basic_2/rt_computation/fpbs.ma".
(* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
-definition fpbg: ∀h. tri_relation genv lenv term ≝
- λh,G1,L1,T1,G2,L2,T2.
- ∃∃G,L,T. ❪G1,L1,T1❫ ≻[h] ❪G,L,T❫ & ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫.
+definition fpbg: tri_relation genv lenv term ≝
+ λG1,L1,T1,G2,L2,T2.
+ ∃∃G,L,T. ❪G1,L1,T1❫ ≻ ❪G,L,T❫ & ❪G,L,T❫ ≥ ❪G2,L2,T2❫.
interpretation "proper parallel rst-computation (closure)"
- 'PRedSubTyStarProper h G1 L1 T1 G2 L2 T2 = (fpbg h G1 L1 T1 G2 L2 T2).
+ 'PRedSubTyStarProper G1 L1 T1 G2 L2 T2 = (fpbg G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2.
- ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
+lemma fpb_fpbg:
+ ∀G1,G2,L1,L2,T1,T2.
+ ❪G1,L1,T1❫ ≻ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
/2 width=5 by ex2_3_intro/ qed.
-lemma fpbg_fpbq_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
- ❪G1,L1,T1❫ >[h] ❪G,L,T❫ → ❪G,L,T❫ ≽[h] ❪G2,L2,T2❫ →
- ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
-#h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
+lemma fpbg_fpbq_trans:
+ ∀G1,G,G2,L1,L,L2,T1,T,T2.
+ ❪G1,L1,T1❫ > ❪G,L,T❫ → ❪G,L,T❫ ≽ ❪G2,L2,T2❫ →
+ ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
/3 width=9 by fpbs_strap1, ex2_3_intro/
qed-.
-lemma fpbg_fqu_trans (h): ∀G1,G,G2,L1,L,L2,T1,T,T2.
- ❪G1,L1,T1❫ >[h] ❪G,L,T❫ → ❪G,L,T❫ ⬂ ❪G2,L2,T2❫ →
- ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
-#h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
+lemma fpbg_fqu_trans:
+ ∀G1,G,G2,L1,L,L2,T1,T,T2.
+ ❪G1,L1,T1❫ > ❪G,L,T❫ → ❪G,L,T❫ ⬂ ❪G2,L2,T2❫ →
+ ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
/4 width=5 by fpbg_fpbq_trans, fpbq_fquq, fqu_fquq/
qed-.
(* Note: this is used in the closure proof *)
-lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2.
- â\9dªG,L,Tâ\9d« â\89¥[h] ❪G2,L2,T2❫ →
- ∀G1,L1,T1. ❪G1,L1,T1❫ >[h] ❪G,L,T❫ → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
-#h #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
+lemma fpbg_fpbs_trans:
+ â\88\80G,G2,L,L2,T,T2. â\9dªG,L,Tâ\9d« â\89¥ ❪G2,L2,T2❫ →
+ ∀G1,L1,T1. ❪G1,L1,T1❫ > ❪G,L,T❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
+#G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
qed-.
(* Basic_2A1: uses: fpbg_fleq_trans *)
-lemma fpbg_feqx_trans: ∀h,G1,G,L1,L,T1,T.
- â\9dªG1,L1,T1â\9d« >[h] ❪G,L,T❫ →
- ∀G2,L2,T2. ❪G,L,T❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
+lemma fpbg_feqx_trans:
+ â\88\80G1,G,L1,L,T1,T. â\9dªG1,L1,T1â\9d« > ❪G,L,T❫ →
+ ∀G2,L2,T2. ❪G,L,T❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
/3 width=5 by fpbg_fpbq_trans, fpbq_feqx/ qed-.
(* Properties with t-bound rt-transition for terms **************************)
lemma cpm_tneqx_cpm_fpbg (h) (G) (L):
∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡[h,n1] T → (T1 ≛ T → ⊥) →
- ∀n2,T2. ❪G,L❫ ⊢ T ➡[h,n2] T2 → ❪G,L,T1❫ >[h] ❪G,L,T2❫.
+ ∀n2,T2. ❪G,L❫ ⊢ T ➡[h,n2] T2 → ❪G,L,T1❫ > ❪G,L,T2❫.
/4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.