(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predsubtystarproper_8.ma".
+include "ground_2/xoa/ex_2_3.ma".
+include "basic_2/notation/relations/predsubtystarproper_7.ma".
include "basic_2/rt_transition/fpb.ma".
include "basic_2/rt_computation/fpbs.ma".
(* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
-definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
- λh,o,G1,L1,T1,G2,L2,T2.
- â\88\83â\88\83G,L,T. â¦\83G1, L1, T1â¦\84 â\89»[h, o] â¦\83G, L, Tâ¦\84 & â¦\83G, L, Tâ¦\84 â\89¥[h, o] â¦\83G2, L2, T2â¦\84.
+definition fpbg: ∀h. tri_relation genv lenv term ≝
+ λh,G1,L1,T1,G2,L2,T2.
+ â\88\83â\88\83G,L,T. â\9dªG1,L1,T1â\9d« â\89»[h] â\9dªG,L,Tâ\9d« & â\9dªG,L,Tâ\9d« â\89¥[h] â\9dªG2,L2,T2â\9d«.
interpretation "proper parallel rst-computation (closure)"
- 'PRedSubTyStarProper h o G1 L1 T1 G2 L2 T2 = (fpbg h o G1 L1 T1 G2 L2 T2).
+ 'PRedSubTyStarProper h G1 L1 T1 G2 L2 T2 = (fpbg h G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fpb_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- â¦\83G1, L1, T1â¦\84 >[h, o] â¦\83G2, L2, T2â¦\84.
+lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ →
+ â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d«.
/2 width=5 by ex2_3_intro/ qed.
-lemma fpbg_fpbq_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
- â¦\83G1, L1, T1â¦\84 >[h, o] â¦\83G, L, Tâ¦\84 â\86\92 â¦\83G, L, Tâ¦\84 â\89½[h, o] â¦\83G2, L2, T2â¦\84 →
- â¦\83G1, L1, T1â¦\84 >[h, o] â¦\83G2, L2, T2â¦\84.
-#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
+lemma fpbg_fpbq_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
+ â\9dªG1,L1,T1â\9d« >[h] â\9dªG,L,Tâ\9d« â\86\92 â\9dªG,L,Tâ\9d« â\89½[h] â\9dªG2,L2,T2â\9d« →
+ â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d«.
+#h #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
+/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,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- â\88\80G1,L1,T1. â¦\83G1, L1, T1â¦\84 >[h, o] â¦\83G, L, Tâ¦\84 â\86\92 â¦\83G1, L1, T1â¦\84 >[h, o] â¦\83G2, L2, T2â¦\84.
-#h #o #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: ∀h,G,G2,L,L2,T,T2. ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫ →
+ â\88\80G1,L1,T1. â\9dªG1,L1,T1â\9d« >[h] â\9dªG,L,Tâ\9d« â\86\92 â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d«.
+#h #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_fdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >[h, o] ⦃G, L, T⦄ →
- â\88\80G2,L2,T2. â¦\83G, L, Tâ¦\84 â\89\9b[h, o] â¦\83G2, L2, T2â¦\84 â\86\92 â¦\83G1, L1, T1â¦\84 >[h, o] â¦\83G2, L2, T2â¦\84.
-/3 width=5 by fpbg_fpbq_trans, fpbq_fdeq/ qed-.
+lemma fpbg_feqx_trans: ∀h,G1,G,L1,L,T1,T. ❪G1,L1,T1❫ >[h] ❪G,L,T❫ →
+ â\88\80G2,L2,T2. â\9dªG,L,Tâ\9d« â\89\9b â\9dªG2,L2,T2â\9d« â\86\92 â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d«.
+/3 width=5 by fpbg_fpbq_trans, fpbq_feqx/ qed-.
(* Properties with t-bound rt-transition for terms **************************)
-lemma cpm_tdneq_cpm_fpbg (h) (o) (G) (L):
- ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛[h,o] T → ⊥) →
- ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2,h] T2 →
- ⦃G, L, T1⦄ >[h,o] ⦃G, L, T2⦄.
-/4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.
+lemma cpm_tneqx_cpm_fpbg (h) (G) (L):
+ ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) →
+ ∀n2,T2. ❪G,L❫ ⊢ T ➡[n2,h] T2 → ❪G,L,T1❫ >[h] ❪G,L,T2❫.
+/4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.