(**************************************************************************)
include "ground_2/lib/star.ma".
-include "basic_2/notation/relations/predsubtystar_8.ma".
+include "basic_2/notation/relations/predsubtystar_7.ma".
include "basic_2/rt_transition/fpbq.ma".
(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
-definition fpbs: ∀h. sd h → tri_relation genv lenv term ≝
- λh,o. tri_TC … (fpbq h o).
+definition fpbs: ∀h. tri_relation genv lenv term ≝
+ λh. tri_TC … (fpbq h).
interpretation "parallel rst-computation (closure)"
- 'PRedSubTyStar h o G1 L1 T1 G2 L2 T2 = (fpbs h o G1 L1 T1 G2 L2 T2).
+ 'PRedSubTyStar h G1 L1 T1 G2 L2 T2 = (fpbs h G1 L1 T1 G2 L2 T2).
(* Basic eliminators ********************************************************)
-lemma fpbs_ind: ∀h,o,G1,L1,T1. ∀R:relation3 genv lenv term. R G1 L1 T1 →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+lemma fpbs_ind: ∀h,G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 →
+ (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
/3 width=8 by tri_TC_star_ind/ qed-.
-lemma fpbs_ind_dx: ∀h,o,G2,L2,T2. ∀R:relation3 genv lenv term. R G2 L2 T2 →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → R G1 L1 T1.
+lemma fpbs_ind_dx: ∀h,G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 →
+ (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G1 L1 T1.
/3 width=8 by tri_TC_star_ind_dx/ qed-.
(* Basic properties *********************************************************)
-lemma fpbs_refl: ∀h,o. tri_reflexive … (fpbs h o).
+lemma fpbs_refl: ∀h. tri_reflexive … (fpbs h).
/2 width=1 by tri_inj/ qed.
-lemma fpbq_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/2 width=1 by tri_inj/ qed.
-lemma fpbs_strap1: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbs_strap1: ∀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⦄.
/2 width=5 by tri_step/ qed-.
-lemma fpbs_strap2: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbs_strap2: ∀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⦄.
/2 width=5 by tri_TC_strap/ qed-.
(* Basic_2A1: uses: lleq_fpbs fleq_fpbs *)
-lemma ffdeq_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=1 by fpbq_fpbs, fpbq_ffdeq/ qed.
+lemma fdeq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+/3 width=1 by fpbq_fpbs, fpbq_fdeq/ qed.
(* Basic_2A1: uses: fpbs_lleq_trans *)
-lemma fpbs_ffdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=9 by fpbs_strap1, fpbq_ffdeq/ qed-.
+lemma fpbs_fdeq_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=9 by fpbs_strap1, fpbq_fdeq/ qed-.
(* Basic_2A1: uses: lleq_fpbs_trans *)
-lemma ffdeq_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_strap2, fpbq_ffdeq/ qed-.
+lemma fdeq_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+/3 width=5 by fpbs_strap2, fpbq_fdeq/ qed-.
+
+lemma tdeq_rdeq_lpx_fpbs: ∀h,T1,T2. T1 ≛ T2 → ∀L1,L0. L1 ≛[T2] L0 →
+ ∀G,L2. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃G,L1,T1⦄ ≥[h] ⦃G,L2,T2⦄.
+/4 width=5 by fdeq_fpbs, fpbs_strap1, fpbq_lpx, fdeq_intro_dx/ qed.
(* Basic_2A1: removed theorems 3:
fpb_fpbsa_trans fpbs_fpbsa fpbsa_inv_fpbs