⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
/2 width=5 by tri_TC_strap/ qed-.
-lemma fqup_fpbs: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83+ ⦃G2, L2, T2⦄ →
+lemma fqup_fpbs: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90+ ⦃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/
qed.
-lemma fqus_fpbs: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83* ⦃G2, L2, T2⦄ →
+lemma fqus_fpbs: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90* ⦃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 fpb_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.
+
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 lprs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
/3 width=1 by lprs_lpxs, lpxs_fpbs/ qed.
-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, lpr_fpb, cpr_fpb/ qed.
-
lemma fpbs_fqus_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
- â¦\83G, L, Tâ¦\84 â\8a\83* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+ â¦\83G, L, Tâ¦\84 â\8a\90* ⦃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/
qed-.
+lemma fpbs_fqup_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⦄.
+/3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
+
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_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-.
+
lemma fqus_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- â¦\83G1, L1, T1â¦\84 â\8a\83* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+ â¦\83G1, L1, T1â¦\84 â\8a\90* ⦃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/
qed-.
#h #g #G1 #G2 #L1 #L #L2 #T1 #T2 #H1 #H @(lpxs_ind_dx … H) -L1
/3 width=5 by fpbs_strap2, fpb_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-.
+
+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⦄.
+/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
+
+lemma cpxs_fqup_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⦄.
+/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
+
+lemma fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
+ ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
+
+lemma cpxs_fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+ ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
+
+(* 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/
+qed.