(* *)
(**************************************************************************)
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
include "basic_2/rt_transition/cpx_fqus.ma".
include "basic_2/rt_computation/cpxs_drops.ma".
#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
-(* Note: a proof based on fqu_cpx_trans_ntdeq might exist *)
-lemma fqu_cpxs_trans_ntdeq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h] U2 â\86\92 (T2 â\89¡[h, o] U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[h] U1 & T1 â\89¡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+(* Note: a proof based on fqu_cpx_trans_tdneq might exist *)
+lemma fqu_cpxs_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h] U2 â\86\92 (T2 â\89\9b[h, o] U2 → ⊥) →
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[h] U1 & T1 â\89\9b[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❴1❵)
#U2 #HVU2 @(ex3_intro … U2)
]
qed-.
-lemma fquq_cpxs_trans_ntdeq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h] U2 â\86\92 (T2 â\89¡[h, o] U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[h] U1 & T1 â\89¡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpxs_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h] U2 â\86\92 (T2 â\89\9b[h, o] U2 → ⊥) →
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[h] U1 & T1 â\89\9b[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
-[ #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_ntdeq … H12 … HTU2 H) -T2
+[ #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fquq, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
]
qed-.
-lemma fqup_cpxs_trans_ntdeq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h] U2 â\86\92 (T2 â\89¡[h, o] U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[h] U1 & T1 â\89¡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
+lemma fqup_cpxs_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
+ â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h] U2 â\86\92 (T2 â\89\9b[h, o] U2 → ⊥) →
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[h] U1 & T1 â\89\9b[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
-[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_ntdeq … H12 … HTU2 H) -T2
+[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fqup, ex3_intro/
| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
- #U1 #HTU1 #H #H12 elim (fqu_cpxs_trans_ntdeq … H1 … HTU1 H) -T1
+ #U1 #HTU1 #H #H12 elim (fqu_cpxs_trans_tdneq … H1 … HTU1 H) -T1
/3 width=8 by fqup_strap2, ex3_intro/
]
qed-.
-lemma fqus_cpxs_trans_ntdeq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h] U2 â\86\92 (T2 â\89¡[h, o] U2 → ⊥) →
- â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[h] U1 & T1 â\89¡[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
+lemma fqus_cpxs_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ â\88\80U2. â¦\83G2, L2â¦\84 â\8a¢ T2 â¬\88*[h] U2 â\86\92 (T2 â\89\9b[h, o] U2 → ⊥) →
+ â\88\83â\88\83U1. â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88*[h] U1 & T1 â\89\9b[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
-[ #H12 elim (fqup_cpxs_trans_ntdeq … H12 … HTU2 H) -T2
+[ #H12 elim (fqup_cpxs_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqup_fqus, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
]