(* *)
(**************************************************************************)
-include "basic_2/relocation/lleq_lleq.ma".
+include "basic_2/reduction/lpx_lleq.ma".
+include "basic_2/computation/cpxs_leq.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 **************)
+(* Properties on lazy equivalence for local environments ********************)
-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 *)
- ]
-]
+lemma lleq_lpxs_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 →
+ ∀L1,T,d. L1 ≡[T, d] L2 →
+ ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ≡[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-.
-(* 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.
+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 ≡[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
]
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[0, T1] 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[0, T2] 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[0, T1] 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[0, T2] 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[0, T1] 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[0, T2] 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
| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
]
qed-.
+
+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 &
+ (∀T. L0 ≡[T, d] L2 ↔ L1 ≡[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/
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
+| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
+ elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+ lapply (ysucc_inv_Y_dx … He) -He #He
+ elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+ @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H
+ elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+ elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+ @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_leq_repl … H) -H /3 width=1 by leq_sym, leq_succ/
+]
+qed-.
+
+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 &
+ (∀T. L0 ≡[T, d] L2 ↔ L1 ≡[T, d] L).
+/2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.
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