(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predsubtyproper_6.ma".
-include "static_2/s_transition/fqu.ma".
-include "static_2/static/reqx.ma".
-include "basic_2/rt_transition/lpr_lpx.ma".
+include "basic_2/notation/relations/predsubty_6.ma".
+include "static_2/s_transition/fquq.ma".
+include "basic_2/rt_transition/rpx.ma".
-(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
+(* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
-inductive fpb (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpb_fqu: ∀G2,L2,T2. ❪G1,L1,T1❫ ⬂ ❪G2,L2,T2❫ → fpb G1 L1 T1 G2 L2 T2
-| fpb_cpx: ∀T2. ❪G1,L1❫ ⊢ T1 ⬈ T2 → (T1 ≅ T2 → ⊥) → fpb G1 L1 T1 G1 L1 T2
-| fpb_lpx: ∀L2. ❪G1,L1❫ ⊢ ⬈ L2 → (L1 ≅[T1] L2 → ⊥) → fpb G1 L1 T1 G1 L2 T1
-.
+(* Basic_2A1: uses: fpbq *)
+definition fpb (G1) (L1) (T1) (G2) (L2) (T2): Prop ≝
+ ∃∃L,T. ❨G1,L1,T1❩ ⬂⸮ ❨G2,L,T❩ & ❨G2,L❩ ⊢ T ⬈ T2 & ❨G2,L❩ ⊢ ⬈[T] L2.
interpretation
- "proper parallel rst-transition (closure)"
- 'PRedSubTyProper G1 L1 T1 G2 L2 T2 = (fpb G1 L1 T1 G2 L2 T2).
+ "parallel rst-transition (closure)"
+ 'PRedSubTy G1 L1 T1 G2 L2 T2 = (fpb G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-(* Basic_2A1: includes: cpr_fpb *)
-lemma cpm_fpb (h) (n) (G) (L):
- â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â\9e¡[h,n] T2 â\86\92 (T1 â\89\85 T2 â\86\92 â\8a¥) â\86\92 â\9dªG,L,T1â\9d« â\89» â\9dªG,L,T2â\9d«.
-/3 width=3 by fpb_cpx, cpm_fwd_cpx/ qed.
+lemma fpb_intro (G1) (L1) (T1) (G2) (L2) (T2):
+ ∀L,T. ❨G1,L1,T1❩ ⬂⸮ ❨G2,L,T❩ → ❨G2,L❩ ⊢ T ⬈ T2 →
+ â\9d¨G2,Lâ\9d© â\8a¢ â¬\88[T] L2 â\86\92 â\9d¨G1,L1,T1â\9d© â\89½ â\9d¨G2,L2,T2â\9d©.
+/2 width=5 by ex3_2_intro/ qed.
-lemma lpr_fpb (h) (G) (T):
- ∀L1,L2. ❪G,L1❫ ⊢ ➡[h,0] L2 → (L1 ≅[T] L2 → ⊥) → ❪G,L1,T❫ ≻ ❪G,L2,T❫.
-/3 width=2 by fpb_lpx, lpr_fwd_lpx/ qed.
+lemma rpx_fpb (G) (T):
+ ∀L1,L2. ❨G,L1❩ ⊢ ⬈[T] L2 → ❨G,L1,T❩ ≽ ❨G,L2,T❩.
+/2 width=5 by fpb_intro/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fpb_inv_gen (G1) (L1) (T1) (G2) (L2) (T2):
+ ❨G1,L1,T1❩ ≽ ❨G2,L2,T2❩ →
+ ∃∃L,T. ❨G1,L1,T1❩ ⬂⸮ ❨G2,L,T❩ & ❨G2,L❩ ⊢ T ⬈ T2 & ❨G2,L❩ ⊢ ⬈[T] L2.
+// qed-.
+
+(* Basic_2A1: removed theorems 2:
+ fpbq_fpbqa fpbqa_inv_fpbq
+*)