(* Properties on lazy equivalence for local environments ********************)
axiom lleq_lpx_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡[h, g] K2 →
- â\88\80L1,T,d. L1 â\8b\95[T, d] L2 →
- â\88\83â\88\83K1. â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, g] K1 & K1 â\8b\95[T, d] K2.
+ â\88\80L1,T,d. L1 â\89¡[T, d] L2 →
+ â\88\83â\88\83K1. â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, g] K1 & K1 â\89¡[T, d] K2.
lemma lpx_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\8b\95[T1, 0] L1 →
- â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90 â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\8b\95[T2, 0] L2.
+ â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\89¡[T1, 0] L1 →
+ â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90 â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\89¡[T2, 0] L2.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpx_inv_pair2 … H1) -H1
#K0 #V0 #H1KL1 #_ #H destruct
qed-.
lemma lpx_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\8b\95[T1, 0] L1 →
- â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90⸮ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\8b\95[T2, 0] L2.
+ â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\89¡[T1, 0] L1 →
+ â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90⸮ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\89¡[T2, 0] L2.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
elim (fquq_inv_gen … H) -H
[ #H elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
qed-.
lemma lpx_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\8b\95[T1, 0] L1 →
- â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90+ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\8b\95[T2, 0] L2.
+ â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\89¡[T1, 0] L1 →
+ â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90+ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\89¡[T2, 0] L2.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
/3 width=4 by fqu_fqup, ex3_intro/
qed-.
lemma lpx_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\8b\95[T1, 0] L1 →
- â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90* â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\8b\95[T2, 0] L2.
+ â\88\80K1. â¦\83G1, K1â¦\84 â\8a¢ â\9e¡[h, g] L1 â\86\92 K1 â\89¡[T1, 0] L1 →
+ â\88\83â\88\83K2. â¦\83G1, K1, T1â¦\84 â\8a\90* â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡[h, g] L2 & K2 â\89¡[T2, 0] L2.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
elim (fqus_inv_gen … H) -H
[ #H elim (lpx_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
fact leq_lpx_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ≃[d, e] L0 → e = ∞ →
∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
∃∃L. L ≃[d, e] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
- (â\88\80T. L0 â\8b\95[T, d] L2 â\86\94 L1 â\8b\95[T, d] L).
+ (â\88\80T. L0 â\89¡[T, d] L2 â\86\94 L1 â\89¡[T, d] L).
#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
[ #d #e #_ #L2 #H >(lpx_inv_atom1 … H) -H
/3 width=5 by ex3_intro, conj/
lemma leq_lpx_trans_lleq: ∀h,g,G,L1,L0,d. L1 ≃[d, ∞] L0 →
∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
∃∃L. L ≃[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
- (â\88\80T. L0 â\8b\95[T, d] L2 â\86\94 L1 â\8b\95[T, d] L).
+ (â\88\80T. L0 â\89¡[T, d] L2 â\86\94 L1 â\89¡[T, d] L).
/2 width=1 by leq_lpx_trans_lleq_aux/ qed-.