(* *)
(**************************************************************************)
-include "basic_2/relocation/lleq_lleq.ma".
+include "basic_2/substitution/lleq_ext.ma".
include "basic_2/computation/lpxs_ldrop.ma".
include "basic_2/computation/lpxs_cpxs.ma".
(* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
-(* Inversion lemmas on lazy equivalence for local environments **************)
+(* Advanced properties ******************************************************)
-lemma lpxs_nlleq_inv_step_sn: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[d, T] L2 → ⊥) →
- ∃∃L. ⦃G, L1⦄ ⊢ ➡*[h, g] L & L1 ⋕[d, T] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, g] L2.
-#h #g #G #L1 #L2 #T #d #H @(lpxs_ind_dx … H) -L1
-[ #H elim H -H //
-| #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L d) #H
- [ -H2 elim IH2 -IH2
- /4 width=4 by lpxs_strap2, lleq_canc_sn, lleq_trans, ex3_intro/
- | -IH2 -H12 /3 width=4 by lpx_lpxs, ex3_intro/ (**) (* auto fails without clear *)
- ]
-]
-qed-.
+axiom lleq_lpxs_trans: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ∀K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 →
+ ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ⋕[T, d] K2.
+(*
+#h #g #G #L1 #L2 #T #d #H @(lleq_ind_alt … H) -L1 -L2 -T -d
+[
+|
+|
+|
+|
+| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #K2 #HLK2
+ elim (IHV … HLK2) -IHV #KV #HLKV #HV
+ elim (IHT (K2.ⓑ{I}V)) -IHT /2 width=1 by lpxs_pair_refl/ -HLK2 #Y #H #HT
+ elim (lpxs_inv_pair1 … H) -H #KT #VT #HLKT #_ #H destruct
+
+#h #g #G #L1 #L2 #T #d * #HL12 #IH #K2 #HLK2
+*)
(* Properties on lazy equivalence for local environments ********************)
lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[0, T1] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[0, T2] L2.
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[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
elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
- #K1 #H #H2KL1 lapply (ldrop_inv_O2 … H) -H #H destruct
+ #I1 #K1 #H #H2KL1 lapply (ldrop_inv_O2 … H) -H #H destruct
/2 width=4 by fqu_lref_O, ex3_intro/
| * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
[ elim (lleq_inv_bind … H)
qed-.
lemma lpxs_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[0, T1] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[0, T2] L2.
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[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: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[0, T1] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[0, T2] L2.
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[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: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[0, T1] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[0, T2] L2.
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[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