include "basic_2/notation/relations/btpredstar_8.ma".
include "basic_2/multiple/fqus.ma".
-include "basic_2/reduction/fpb.ma".
+include "basic_2/reduction/fpbq.ma".
include "basic_2/computation/cpxs.ma".
include "basic_2/computation/lpxs.ma".
(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
definition fpbs: ∀h. sd h → tri_relation genv lenv term ≝
- λh,g. tri_TC … (fpb h g).
+ λh,g. tri_TC … (fpbq h g).
interpretation "'qrst' parallel computation (closure)"
'BTPRedStar h g G1 L1 T1 G2 L2 T2 = (fpbs h g G1 L1 T1 G2 L2 T2).
lemma fpbs_refl: ∀h,g. tri_reflexive … (fpbs h g).
/2 width=1 by tri_inj/ qed.
-lemma fpb_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma fpbq_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
/2 width=1 by tri_inj/ qed.
lemma fpbs_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
lemma fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/4 width=5 by fqu_fquq, fpb_fquq, tri_step/
+/4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
qed.
lemma fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
-/3 width=5 by fpb_fquq, tri_step/
+/3 width=5 by fpbq_fquq, tri_step/
qed.
lemma cpxs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpb_cpx, fpbs_strap1/
+/3 width=5 by fpbq_cpx, fpbs_strap1/
qed.
lemma lpxs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
#h #g #G #L1 #L2 #T #H @(lpxs_ind … H) -L2
-/3 width=5 by fpb_lpx, fpbs_strap1/
+/3 width=5 by fpbq_lpx, fpbs_strap1/
qed.
lemma lleq_fpbs: ∀h,g,G,L1,L2,T. L1 ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
-/3 width=1 by fpb_fpbs, fpb_lleq/ qed.
+/3 width=1 by fpbq_fpbs, fpbq_lleq/ qed.
lemma cprs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
/3 width=1 by cprs_cpxs, cpxs_fpbs/ qed.
lemma fpbs_fqus_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind … H) -G2 -L2 -T2
-/3 width=5 by fpbs_strap1, fpb_fquq/
+/3 width=5 by fpbs_strap1, fpbq_fquq/
qed-.
lemma fpbs_fqup_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
lemma fpbs_cpxs_trans: ∀h,g,G1,G,L1,L,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
#h #g #G1 #G #L1 #L #T1 #T #T2 #H1 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpbs_strap1, fpb_cpx/
+/3 width=5 by fpbs_strap1, fpbq_cpx/
qed-.
lemma fpbs_lpxs_trans: ∀h,g,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
⦃G, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L2, T⦄.
#h #g #G1 #G #L1 #L #L2 #T1 #T #H1 #H @(lpxs_ind … H) -L2
-/3 width=5 by fpbs_strap1, fpb_lpx/
+/3 width=5 by fpbs_strap1, fpbq_lpx/
qed-.
lemma fpbs_lleq_trans: ∀h,g,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
L ≡[T, 0] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L2, T⦄.
-/3 width=5 by fpbs_strap1, fpb_lleq/ qed-.
+/3 width=5 by fpbs_strap1, fpbq_lleq/ qed-.
lemma fqus_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
-/3 width=5 by fpbs_strap2, fpb_fquq/
+/3 width=5 by fpbs_strap2, fpbq_fquq/
qed-.
lemma cpxs_fpbs_trans: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T #T2 #H1 #H @(cpxs_ind_dx … H) -T1
-/3 width=5 by fpbs_strap2, fpb_cpx/
+/3 width=5 by fpbs_strap2, fpbq_cpx/
qed-.
lemma lpxs_fpbs_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
⦃G1, L1⦄ ⊢ ➡*[h, g] L → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L #L2 #T1 #T2 #H1 #H @(lpxs_ind_dx … H) -L1
-/3 width=5 by fpbs_strap2, fpb_lpx/
+/3 width=5 by fpbs_strap2, fpbq_lpx/
qed-.
lemma lleq_fpbs_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
L1 ≡[T1, 0] L → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_strap2, fpb_lleq/ qed-.
+/3 width=5 by fpbs_strap2, fpbq_lleq/ qed-.
lemma cpxs_fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
(* Note: this is used in the closure proof *)
lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
-/4 width=5 by fpbs_strap1, fpb_fpbs, lpr_fpb, cpr_fpb/
+/4 width=5 by fpbs_strap1, fpbq_fpbs, lpr_fpbq, cpr_fpbq/
qed.