-lemma fqus_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
-/3 width=5 by fpbq_fquq, tri_step/
-qed.
-
-lemma cpxs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-#h #o #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpbq_cpx, fpbs_strap1/
-qed.
-
-lemma lpxs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
-#h #o #G #L1 #L2 #T #H @(lpxs_ind … H) -L2
-/3 width=5 by fpbq_lpx, fpbs_strap1/
-qed.
-
-lemma lleq_fpbs: ∀h,o,G,L1,L2,T. L1 ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
-/3 width=1 by fpbq_fpbs, fpbq_lleq/ qed.
-
-lemma cprs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-/3 width=1 by cprs_cpxs, cpxs_fpbs/ qed.
-
-lemma lprs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
-/3 width=1 by lprs_lpxs, lpxs_fpbs/ qed.
-
-lemma fpbs_fqus_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind … H) -G2 -L2 -T2
-/3 width=5 by fpbs_strap1, fpbq_fquq/
-qed-.
-
-lemma fpbs_fqup_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
-
-lemma fpbs_cpxs_trans: ∀h,o,G1,G,L1,L,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-#h #o #G1 #G #L1 #L #T1 #T #T2 #H1 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpbs_strap1, fpbq_cpx/
-qed-.
-
-lemma fpbs_lpxs_trans: ∀h,o,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L⦄ ⊢ ➡*[h, o] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L2, T⦄.
-#h #o #G1 #G #L1 #L #L2 #T1 #T #H1 #H @(lpxs_ind … H) -L2
-/3 width=5 by fpbs_strap1, fpbq_lpx/
-qed-.
-
-lemma fpbs_lleq_trans: ∀h,o,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- L ≡[T, 0] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L2, T⦄.
-/3 width=5 by fpbs_strap1, fpbq_lleq/ qed-.
-
-lemma fqus_fpbs_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
-/3 width=5 by fpbs_strap2, fpbq_fquq/
-qed-.
-
-lemma cpxs_fpbs_trans: ∀h,o,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T #T2 #H1 #H @(cpxs_ind_dx … H) -T1
-/3 width=5 by fpbs_strap2, fpbq_cpx/
-qed-.
-
-lemma lpxs_fpbs_trans: ∀h,o,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ➡*[h, o] L → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L #L2 #T1 #T2 #H1 #H @(lpxs_ind_dx … H) -L1
-/3 width=5 by fpbs_strap2, fpbq_lpx/
-qed-.
-
-lemma lleq_fpbs_trans: ∀h,o,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- L1 ≡[T1, 0] L → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_strap2, fpbq_lleq/ qed-.
-
-lemma cpxs_fqus_fpbs: ∀h,o,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T →
- ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
-
-lemma cpxs_fqup_fpbs: ∀h,o,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T →
- ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
-
-lemma fqus_lpxs_fpbs: ∀h,o,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
- ⦃G2, L⦄ ⊢ ➡*[h, o] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
-
-lemma cpxs_fqus_lpxs_fpbs: ∀h,o,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T →
- ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, o] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
-
-lemma lpxs_lleq_fpbs: ∀h,o,G,L1,L,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, o] L →
- L ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
-/3 width=3 by lpxs_fpbs_trans, lleq_fpbs/ qed.
-
-(* Note: this is used in the closure proof *)
-lemma cpr_lpr_fpbs: ∀h,o,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, T2⦄.
-/4 width=5 by fpbs_strap1, fpbq_fpbs, lpr_fpbq, cpr_fpbq/
-qed.