(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_ext.ma".
-include "basic_2/computation/lpxs_ldrop.ma".
+include "basic_2/reduction/lpx_lleq.ma".
+include "basic_2/computation/cpxs_leq.ma".
+include "basic_2/computation/lpxs_drop.ma".
include "basic_2/computation/lpxs_cpxs.ma".
(* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
-(* Advanced properties ******************************************************)
-
-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 ⋕[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2.
+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-.
+
+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
elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
- #I1 #K1 #H #H2KL1 lapply (ldrop_inv_O2 … H) -H #H destruct
+ #K1 #H #H2KL1 lapply (drop_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)
| #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 (drop_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
+ #H2KL elim (lpxs_drop_trans_O1 … H1KL1 … HL1) -L1
+ #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
/3 width=4 by fqu_drop, ex3_intro/
- | lapply (ldrop_fwd_length_le2 … HL1) -L -T1 -g
+ | lapply (drop_fwd_length_le2 … HL1) -L -T1 -g
lapply (lleq_fwd_length … H2KL1) //
]
]
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
| * #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-.
projects / helm.git / blobdiff