(* *)
(**************************************************************************)
-include "ground_2/xoa/ex_2_3.ma".
-include "basic_2/notation/relations/predsubtystarproper_7.ma".
-include "basic_2/rt_transition/fpb.ma".
+include "ground/xoa/ex_3_6.ma".
+include "basic_2/notation/relations/predsubtystarproper_6.ma".
+include "basic_2/rt_transition/fpbc.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.
+ ∃∃G3,L3,T3,G4,L4,T4. ❨G1,L1,T1❩ ≥ ❨G3,L3,T3❩ & ❨G3,L3,T3❩ ≻ ❨G4,L4,T4❩ & ❨G4,L4,T4❩ ≥ ❨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).
+interpretation
+ "proper parallel rst-computation (closure)"
+ 'PRedSubTyStarProper G1 L1 T1 G2 L2 T2 = (fpbg G1 L1 T1 G2 L2 T2).
-(* Basic properties *********************************************************)
+(* Basic inversion lemmas ***************************************************)
-lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2.
- ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
-/2 width=5 by ex2_3_intro/ qed.
+lemma fpbg_inv_gen (G1) (G2) (L1) (L2) (T1) (T2):
+ ❨G1,L1,T1❩ > ❨G2,L2,T2❩ →
+ ∃∃G3,L3,T3,G4,L4,T4. ❨G1,L1,T1❩ ≥ ❨G3,L3,T3❩ & ❨G3,L3,T3❩ ≻ ❨G4,L4,T4❩ & ❨G4,L4,T4❩ ≥ ❨G2,L2,T2❩.
+// 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 *
-/3 width=9 by fpbs_strap1, ex2_3_intro/
-qed-.
+(* Basic properties *********************************************************)
-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/
+lemma fpbg_intro (G3) (G4) (L3) (L4) (T3) (T4):
+ ∀G1,L1,T1,G2,L2,T2.
+ ❨G1,L1,T1❩ ≥ ❨G3,L3,T3❩ → ❨G3,L3,T3❩ ≻ ❨G4,L4,T4❩ →
+ ❨G4,L4,T4❩ ≥ ❨G2,L2,T2❩ → ❨G1,L1,T1❩ > ❨G2,L2,T2❩.
+/2 width=9 by ex3_6_intro/ qed.
+
+(* Basic_2A1: was: fpbg_fpbq_trans *)
+lemma fpbg_fpb_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
+elim (fpbg_inv_gen … H1) -H1
+/3 width=13 by fpbs_strap1, fpbg_intro/
qed-.
-(* Note: this is used in the closure proof *)
-lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2.
- ❪G,L,T❫ ≥[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/
+(* Basic_2A1: was: fpbq_fpbg_trans *)
+lemma fpb_fpbg_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
+elim (fpbg_inv_gen … H2) -H2
+/3 width=13 by fpbs_strap2, fpbg_intro/
qed-.
-
-(* Basic_2A1: uses: fpbg_fleq_trans *)
-lemma fpbg_feqx_trans: ∀h,G1,G,L1,L,T1,T.
- ❪G1,L1,T1❫ >[h] ❪G,L,T❫ →
- ∀G2,L2,T2. ❪G,L,T❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ >[h] ❪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❫.
-/4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.