(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_leq.ma".
include "basic_2/reduction/lpx_lleq.ma".
include "basic_2/computation/cpxs_leq.ma".
include "basic_2/computation/lpxs_ldrop.ma".
(* Properties on lazy equivalence for local environments ********************)
lemma lleq_lpxs_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.
#h #g #G #L2 #K2 #H @(lpxs_ind … H) -K2 /2 width=3 by ex2_intro/
#K #K2 #_ #HK2 #IH #L1 #T #d #HT elim (IH … HT) -L2
#L #HL1 #HT elim (lleq_lpx_trans … HK2 … HT) -K
/3 width=3 by lpxs_strap1, ex2_intro/
qed-.
-lemma lpxs_lleq_fqu_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83 ⦃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\83 â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡*[h, g] L2 & K2 â\8b\95[T2, 0] L2.
+lemma lpxs_lleq_fqu_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90 ⦃G2, L2, T2⦄ →
+ 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 (lpxs_inv_pair2 … H1) -H1
#K0 #V0 #H1KL1 #_ #H destruct
| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
/2 width=4 by fqu_flat_dx, ex3_intro/
| #G1 #L1 #L #T1 #U1 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1
- elim (ldrop_O1_le (e+1) K1)
+ elim (ldrop_O1_le (Ⓕ) (e+1) K1)
[ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
#H2KL elim (lpxs_ldrop_trans_O1 … H1KL1 … HL1) -L1
#K0 #HK10 #H1KL lapply (ldrop_mono … HK10 … HK1) -HK10 #H destruct
]
qed-.
-lemma lpxs_lleq_fquq_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83⸮ ⦃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\83⸮ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡*[h, g] L2 & K2 â\8b\95[T2, 0] L2.
+lemma lpxs_lleq_fquq_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90⸮ ⦃G2, L2, T2⦄ →
+ 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 (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
]
qed-.
-lemma lpxs_lleq_fqup_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83+ ⦃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\83+ â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡*[h, g] L2 & K2 â\8b\95[T2, 0] L2.
+lemma lpxs_lleq_fqup_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90+ ⦃G2, L2, T2⦄ →
+ 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 (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
/3 width=4 by fqu_fqup, ex3_intro/
]
qed-.
-lemma lpxs_lleq_fqus_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\83* ⦃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\83* â¦\83G2, K2, T2â¦\84 & â¦\83G2, K2â¦\84 â\8a¢ â\9e¡*[h, g] L2 & K2 â\8b\95[T2, 0] L2.
+lemma lpxs_lleq_fqus_trans: â\88\80h,g,G1,G2,L1,L2,T1,T2. â¦\83G1, L1, T1â¦\84 â\8a\90* ⦃G2, L2, T2⦄ →
+ 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 (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
fact leq_lpxs_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 >(lpxs_inv_atom1 … H) -H
/3 width=5 by ex3_intro, conj/
lemma leq_lpxs_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_lpxs_trans_lleq_aux/ qed-.