(* *)
(**************************************************************************)
-include "basic_2/substitution/fsups_fsups.ma".
+include "basic_2/substitution/fqus_fqus.ma".
include "basic_2/unfold/lsstas_lift.ma".
include "basic_2/reduction/cpx_lift.ma".
include "basic_2/computation/cpxs.ma".
(* Properties on supclosure *************************************************)
-lemma fsupq_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+lemma fquq_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+ ∀T1. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
#h #g #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2
-[ /3 width=3 by fsupq_fsups, ex2_intro/
+[ /3 width=3 by fquq_fqus, ex2_intro/
| #T #T2 #HT2 #_ #IHTU2 #T1 #HT1
- elim (fsupq_cpx_trans … HT1 … HT2) -T #T #HT1 #HT2
+ elim (fquq_cpx_trans … HT1 … HT2) -T #T #HT1 #HT2
elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
]
qed-.
-lemma fsupq_lsstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀U2,l1. ⦃G2, L2⦄ ⊢ T2 •*[h, g, l1] U2 →
- ∀l2. ⦃G2, L2⦄ ⊢ T2 ▪ [h, g] l2 → l1 ≤ l2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
-/3 width=5 by fsupq_cpxs_trans, lsstas_cpxs/ qed-.
+lemma fquq_lsstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀U2,l1. ⦃G2, L2⦄ ⊢ T2 •*[h, g, l1] U2 →
+ ∀l2. ⦃G2, L2⦄ ⊢ T2 ▪ [h, g] l2 → l1 ≤ l2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+/3 width=5 by fquq_cpxs_trans, lsstas_cpxs/ qed-.
-lemma fsups_cpxs_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fsups_ind … H) -G2 -L2 -T2
+lemma fqus_cpxs_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
[ /2 width=3 by ex2_intro/
| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
- elim (fsupq_cpxs_trans … HTU2 … HT2) -T2 #T2 #HT2 #HTU2
- elim (IHT1 … HT2) -T /3 width=7 by fsups_trans, ex2_intro/
+ elim (fquq_cpxs_trans … HTU2 … HT2) -T2 #T2 #HT2 #HTU2
+ elim (IHT1 … HT2) -T /3 width=7 by fqus_trans, ex2_intro/
]
qed-.
-lemma fsups_lsstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
- ∀U2,l1. ⦃G2, L2⦄ ⊢ T2 •*[h, g, l1] U2 →
- ∀l2. ⦃G2, L2⦄ ⊢ T2 ▪ [h, g] l2 → l1 ≤ l2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
-/3 width=7 by fsups_cpxs_trans, lsstas_cpxs/ qed-.
+lemma fqus_lsstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+ ∀U2,l1. ⦃G2, L2⦄ ⊢ T2 •*[h, g, l1] U2 →
+ ∀l2. ⦃G2, L2⦄ ⊢ T2 ▪ [h, g] l2 → l1 ≤ l2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+/3 width=7 by fqus_cpxs_trans, lsstas_cpxs/ qed-.
-lemma fsups_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fsups_ind … H) -G2 -L2 -T2
+lemma fqus_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
[ /2 width=3 by ex2_intro/
| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
- elim (fsupq_cpx_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
- elim (IHT1 … HT2) -T /3 width=7 by fsups_strap1, ex2_intro/
+ elim (fquq_cpx_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
+ elim (IHT1 … HT2) -T /3 width=7 by fqus_strap1, ex2_intro/
]
-qed-.
\ No newline at end of file
+qed-.