--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/cpxs_lift.ma".
+include "basic_2/computation/fpbs.ma".
+
+(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lstas_fpbs: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
+ ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
+/3 width=5 by cpxs_fpbs, lstas_cpxs/ qed.
+
+lemma sta_fpbs: ∀h,o,G,L,T,U,d.
+ ⦃G, L⦄ ⊢ T ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T •*[h, 1] U →
+ ⦃G, L, T⦄ ≥[h, o] ⦃G, L, U⦄.
+/2 width=5 by lstas_fpbs/ qed.
+
+(* Note: this is used in the closure proof *)
+lemma cpr_lpr_sta_fpbs: ∀h,o,G,L1,L2,T1,T2,U2,d2.
+ ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L2⦄ ⊢ T2 ▪[h, o] d2+1 → ⦃G, L2⦄ ⊢ T2 •*[h, 1] U2 →
+ ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, U2⦄.
+/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, sta_cpx, fpbq_cpx/ qed.
-(* Properties on supclosure *************************************************)
-
-lemma fqu_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, o] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, o] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma fqu_lpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpx_pair, ex3_2_intro/
[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpx_inv_pair1 … H) -H
#K2 #W2 #HLK2 #HVW2 #H destruct
qed-.
lemma fquq_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- â\88\80K2. â¦\83G2, L2â¦\84 â\8a¢ â\9e¡[h, o] K2 →
- â\88\83â\88\83K1,T. â¦\83G1, L1â¦\84 â\8a¢ â\9e¡[h, o] K1 & â¦\83G1, L1â¦\84 â\8a¢ T1 â\9e¡[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
+ â\88\80K2. â¦\83G2, L2â¦\84 â\8a¢ â¬\88[h, o] K2 →
+ â\88\83â\88\83K1,T. â¦\83G1, L1â¦\84 â\8a¢ â¬\88[h, o] K1 & â¦\83G1, L1â¦\84 â\8a¢ T1 â¬\88[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
[ #HT12 elim (fqu_lpx_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-
-lemma lpx_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpx_pair, ex3_2_intro/
-[ #I #G1 #L1 #V1 #X #H elim (lpx_inv_pair2 … H) -H
- #K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1)
- /4 width=7 by cpx_delta, fqu_drop, drop_drop, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #L0 #HL01
- elim (lpx_drop_trans_O1 … HL01 … HLK1) -L1
- /3 width=5 by fqu_drop, ex3_2_intro/
-]
-qed-.
-
-lemma lpx_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
-[ #HT12 elim (lpx_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
(* Properties on supclosure *************************************************)
-lemma lpx_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
- /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
-| #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
- #L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fqu_trans … H2 … HL0) -L
- #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
- /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
-]
-qed-.
-
-lemma lpx_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 [ /2 width=5 by ex3_2_intro/ ]
-#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
-#L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fquq_trans … H2 … HL0) -L
-#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
-/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
-qed-.
lemma lpxs_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 →
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≥ ≥ [ break term 46 h, break term 46 o ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'BTPRedStarAlt $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ffdeq.ma".
+include "basic_2/rt_computation/cpxs_lfdeq.ma".
+
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+
+(* Properties with degree-based equivalence for closures ********************)
+
+lemma ffdeq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T⦄ →
+ ∀T2. ⦃G2, L2⦄ ⊢ T ⬈*[h] T2 →
+ ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T0 & ⦃G1, L1, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
+elim (ffdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
+elim (lfdeq_cpxs_trans … HT2 … HL12) #T0 #HT0 #HT02
+lapply (cpxs_lfdeq_conf_dx … HT2 … HL12) -HL12 #HL12
+elim (tdeq_cpxs_trans … HT1 … HT0) -T #T #HT1 #HT0
+/4 width=5 by ffdeq_intro_dx, tdeq_trans, ex2_intro/
+qed-.
qed-.
(* Basic_2A1: was just: cpxs_lleq_conf *)
-lemma cpxs_lfdeq_trans: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
- ∀L2. L0 ≛[h, o, T0] L2 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
+lemma cpxs_lfdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
+ ∀L2. L0 ≛[h, o, T0] L2 →
+ ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
/3 width=3 by lfdeq_cpxs_trans, lfdeq_sym/ qed-.
(* Basic_2A1: was just: cpxs_lleq_conf_dx *)
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/lfpx_fsle.ma".
+include "basic_2/rt_transition/cpx_fqus.ma".
+include "basic_2/rt_transition/lfpx_fquq.ma".
include "basic_2/rt_computation/cpxs_drops.ma".
include "basic_2/rt_computation/cpxs_cpxs.ma".
∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
/4 width=5 by lfpx_cpxs_trans, cpxs_bind_dx, lfpx_pair_refl/ qed.
+
+(* Properties with plus-iterated structural successor for closures **********)
+
+(* Basic_2A1: uses: lpx_fqup_trans *)
+lemma lfpx_fqup_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐+[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lfpx_fqu_trans … H12 … HKL1) -L1
+ /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
+| #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+ #L0 #T0 #HT10 #HT0 #HL0 elim (lfpx_fqu_trans … H2 … HL0) -L
+ #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
+ /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
+]
+qed-.
+
+(* Properties with star-iterated structural successor for closures **********)
+
+(* Basic_2A1: uses: lpx_fqus_trans *)
+lemma lfpx_fqus_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐*[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fqus_inv_fqup … H) -H
+[ #H12 elim (lfpx_fqup_trans … H12 … HKL1) -L1 /3 width=5 by fqup_fqus, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
(* Properties with degree-based equivalence for terms ***********************)
lemma tdeq_cpxs_trans: ∀h,o,U1,T1. U1 ≛[h, o] T1 → ∀G,L,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- ∃∃U2. ⦃G, L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≛[h, o] T2.
+ ∃∃U2. ⦃G, L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≛[h, o] T2.
#h #o #U1 #T1 #HUT1 #G #L #T2 #HT12 @(cpxs_ind … HT12) -T2 /2 width=3 by ex2_intro/
#T #T2 #_ #HT2 * #U #HU1 #HUT elim (tdeq_cpx_trans … HUT … HT2) -T -T1
/3 width=3 by ex2_intro, cpxs_strap1/
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/btpredstar_8.ma".
-include "basic_2/multiple/fqus.ma".
-include "basic_2/reduction/fpbq.ma".
-include "basic_2/computation/cpxs.ma".
-include "basic_2/computation/lpxs.ma".
+include "ground_2/lib/star.ma".
+include "basic_2/notation/relations/predsubtystar_8.ma".
+include "basic_2/rt_transition/fpbq.ma".
-(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
+(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
definition fpbs: ∀h. sd h → tri_relation genv lenv term ≝
λh,o. tri_TC … (fpbq h o).
-interpretation "'qrst' parallel computation (closure)"
- 'BTPRedStar h o G1 L1 T1 G2 L2 T2 = (fpbs h o G1 L1 T1 G2 L2 T2).
+interpretation "parallel rst-computation (closure)"
+ 'PRedSubTyStar h o G1 L1 T1 G2 L2 T2 = (fpbs h o G1 L1 T1 G2 L2 T2).
(* Basic eliminators ********************************************************)
⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
/2 width=5 by tri_TC_strap/ qed-.
-lemma fqup_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
-qed.
+(* Basic_2A1: uses: lleq_fpbs *)
+lemma ffdeq_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=1 by fpbq_fpbs, fpbq_ffdeq/ qed.
-lemma fqus_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
-/3 width=5 by fpbq_fquq, tri_step/
-qed.
-
-lemma cpxs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-#h #o #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpbq_cpx, fpbs_strap1/
-qed.
-
-lemma lpxs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
-#h #o #G #L1 #L2 #T #H @(lpxs_ind … H) -L2
-/3 width=5 by fpbq_lpx, fpbs_strap1/
-qed.
-
-lemma lleq_fpbs: ∀h,o,G,L1,L2,T. L1 ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
-/3 width=1 by fpbq_fpbs, fpbq_lleq/ qed.
-
-lemma cprs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-/3 width=1 by cprs_cpxs, cpxs_fpbs/ qed.
-
-lemma lprs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
-/3 width=1 by lprs_lpxs, lpxs_fpbs/ qed.
-
-lemma fpbs_fqus_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind … H) -G2 -L2 -T2
-/3 width=5 by fpbs_strap1, fpbq_fquq/
-qed-.
-
-lemma fpbs_fqup_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
-
-lemma fpbs_cpxs_trans: ∀h,o,G1,G,L1,L,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-#h #o #G1 #G #L1 #L #T1 #T #T2 #H1 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpbs_strap1, fpbq_cpx/
-qed-.
-
-lemma fpbs_lpxs_trans: ∀h,o,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L⦄ ⊢ ➡*[h, o] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L2, T⦄.
-#h #o #G1 #G #L1 #L #L2 #T1 #T #H1 #H @(lpxs_ind … H) -L2
-/3 width=5 by fpbs_strap1, fpbq_lpx/
-qed-.
-
-lemma fpbs_lleq_trans: ∀h,o,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- L ≡[T, 0] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L2, T⦄.
-/3 width=5 by fpbs_strap1, fpbq_lleq/ qed-.
-
-lemma fqus_fpbs_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
-/3 width=5 by fpbs_strap2, fpbq_fquq/
-qed-.
-
-lemma cpxs_fpbs_trans: ∀h,o,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T #T2 #H1 #H @(cpxs_ind_dx … H) -T1
-/3 width=5 by fpbs_strap2, fpbq_cpx/
-qed-.
-
-lemma lpxs_fpbs_trans: ∀h,o,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ➡*[h, o] L → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L #L2 #T1 #T2 #H1 #H @(lpxs_ind_dx … H) -L1
-/3 width=5 by fpbs_strap2, fpbq_lpx/
-qed-.
-
-lemma lleq_fpbs_trans: ∀h,o,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- L1 ≡[T1, 0] L → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_strap2, fpbq_lleq/ qed-.
-
-lemma cpxs_fqus_fpbs: ∀h,o,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T →
- ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
-
-lemma cpxs_fqup_fpbs: ∀h,o,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T →
- ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
-
-lemma fqus_lpxs_fpbs: ∀h,o,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
- ⦃G2, L⦄ ⊢ ➡*[h, o] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
-
-lemma cpxs_fqus_lpxs_fpbs: ∀h,o,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T →
- ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, o] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
-
-lemma lpxs_lleq_fpbs: ∀h,o,G,L1,L,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, o] L →
- L ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
-/3 width=3 by lpxs_fpbs_trans, lleq_fpbs/ qed.
-
-(* Note: this is used in the closure proof *)
-lemma cpr_lpr_fpbs: ∀h,o,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, T2⦄.
-/4 width=5 by fpbs_strap1, fpbq_fpbs, lpr_fpbq, cpr_fpbq/
-qed.
+(* Basic_2A1: uses: fpbs_lleq_trans *)
+lemma fpbs_ffdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=9 by fpbs_strap1, fpbq_ffdeq/ qed-.
+
+(* Basic_2A1: uses: lleq_fpbs_trans *)
+lemma ffdeq_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_strap2, fpbq_ffdeq/ qed-.
+
+(* Basic_2A1: removed theorems 3:
+ fpb_fpbsa_trans fpbs_fpbsa fpbsa_inv_fpbs
+*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/btpredstaralt_8.ma".
-include "basic_2/multiple/lleq_fqus.ma".
-include "basic_2/computation/cpxs_lleq.ma".
-include "basic_2/computation/lpxs_lleq.ma".
-include "basic_2/computation/fpbs.ma".
-
-(* "QREST" PARALLEL COMPUTATION FOR CLOSURES ********************************)
-
-(* Note: alternative definition of fpbs *)
-definition fpbsa: ∀h. sd h → tri_relation genv lenv term ≝
- λh,o,G1,L1,T1,G2,L2,T2.
- ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T &
- ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ &
- ⦃G2, L0⦄ ⊢ ➡*[h, o] L & L ≡[T2, 0] L2.
-
-interpretation "'big tree' parallel computation (closure) alternative"
- 'BTPRedStarAlt h o G1 L1 T1 G2 L2 T2 = (fpbsa h o G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fpb_fpbsa_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≥≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #L1 #L #T1 #T * -G -L -T [ #G #L #T #HG1 | #T #HT1 | #L #HL1 | #L #HL1 ]
-#G2 #L2 #T2 * #L00 #L0 #T0 #HT0 #HG2 #HL00 #HL02
-[ elim (fquq_cpxs_trans … HT0 … HG1) -T
- /3 width=7 by fqus_strap2, ex4_3_intro/
-| /3 width=7 by cpxs_strap2, ex4_3_intro/
-| lapply (lpx_cpxs_trans … HT0 … HL1) -HT0 #HT10
- elim (lpx_fqus_trans … HG2 … HL1) -L
- /3 width=7 by lpxs_strap2, cpxs_trans, ex4_3_intro/
-| lapply (lleq_cpxs_trans … HT0 … HL1) -HT0 #HT0
- lapply (cpxs_lleq_conf_sn … HT0 … HL1) -HL1 #HL1
- elim (lleq_fqus_trans … HG2 … HL1) -L #K00 #HG12 #HKL00
- elim (lleq_lpxs_trans … HL00 … HKL00) -L00
- /3 width=9 by lleq_trans, ex4_3_intro/
-]
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem fpbs_fpbsa: ∀h,o,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
-/2 width=7 by fpb_fpbsa_trans, ex4_3_intro/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem fpbsa_inv_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ≥≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 *
-/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_lleq/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma fpbs_intro_alt: ∀h,o,G1,G2,L1,L0,L,L2,T1,T,T2.
- ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T → ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ →
- ⦃G2, L0⦄ ⊢ ➡*[h, o] L → L ≡[T2, 0] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ .
-/3 width=7 by fpbsa_inv_fpbs, ex4_3_intro/ qed.
-
-(* Advanced inversion lemmas *************************************************)
-
-lemma fpbs_inv_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T &
- ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ &
- ⦃G2, L0⦄ ⊢ ➡*[h, o] L & L ≡[T2, 0] L2.
-/2 width=1 by fpbs_fpbsa/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpxs.ma".
+include "basic_2/rt_computation/fpbs_fqup.ma".
+
+(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
+
+(* Properties with uncounted context-sensitive parallel rt-computation ******)
+
+lemma cpxs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
+#h #o #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=5 by fpbq_cpx, fpbs_strap1/
+qed.
+
+lemma fpbs_cpxs_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+ ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
+#h #o #G1 #G #L1 #L #T1 #T #H1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=5 by fpbs_strap1, fpbq_cpx/
+qed-.
+
+lemma cpxs_fpbs_trans: ∀h,o,G1,G2,L1,L2,T,T2. ⦃G1, L1, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+ ∀T1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T #T2 #H1 #T1 #H @(cpxs_ind_dx … H) -T1
+/3 width=5 by fpbs_strap2, fpbq_cpx/
+qed-.
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma cpxs_fqus_fpbs: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L2,T2. ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
+
+(* Properties with plus-iterated structural successor for closures **********)
+
+lemma cpxs_fqup_fpbs: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L2,T2. ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/s_computation/fqus_fqup.ma".
+include "basic_2/rt_computation/fpbs_fqus.ma".
+
+(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
+
+(* Properties with plus-iterated structural successor for closures **********)
+
+lemma fqup_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
+qed.
+
+lemma fpbs_fqup_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
+
+lemma fqup_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by fqus_fpbs_trans, fqup_fqus/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/s_computation/fqus.ma".
+include "basic_2/rt_computation/fpbs.ma".
+
+(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma fqus_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
+/3 width=5 by fpbq_fquq, tri_step/
+qed.
+
+lemma fpbs_fqus_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
+/3 width=5 by fpbs_strap1, fpbq_fquq/
+qed-.
+
+lemma fqus_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G #G2 #L #L2 #T #T2 #H1 #G1 #L1 #T1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
+/3 width=5 by fpbs_strap2, fpbq_fquq/
+qed-.
--- /dev/null
+(*
+lemma cprs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
+/3 width=1 by cprs_cpxs, cpxs_fpbs/ qed.
+
+lemma lprs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
+/3 width=1 by lprs_lpxs, lpxs_fpbs/ qed.
+
+(* Note: this is used in the closure proof *)
+lemma cpr_lpr_fpbs: ∀h,o,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, T2⦄.
+/4 width=5 by fpbs_strap1, fpbq_fpbs, lpr_fpbq, cpr_fpbq/
+qed.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ffdeq_fqup.ma".
+include "basic_2/static/ffdeq_fqus.ma".
+include "basic_2/static/ffdeq_ffdeq.ma".
+include "basic_2/rt_computation/cpxs_fqus.ma".
+include "basic_2/rt_computation/cpxs_ffdeq.ma".
+include "basic_2/rt_computation/cpxs_lfpx.ma".
+include "basic_2/rt_computation/lfpxs_ffdeq.ma".
+include "basic_2/rt_computation/fpbs_cpxs.ma".
+
+(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
+
+(* Properties with uncounted parallel rt-computation on referred entries ****)
+
+(* Basic_2A1: uses: lpxs_fpbs *)
+lemma lfpxs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
+#h #o #G #L1 #L2 #T #H @(lfpxs_ind_sn … H) -L2
+/3 width=5 by fpbq_lfpx, fpbs_strap1/
+qed.
+
+(* Basic_2A1: uses: fpbs_lpxs_trans *)
+lemma fpbs_lfpxs_trans: ∀h,o,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L, T2⦄ →
+ ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h, T2] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lfpxs_ind_sn … H) -L2
+/3 width=5 by fpbs_strap1, fpbq_lfpx/
+qed-.
+
+(* Basic_2A1: uses: lpxs_fpbs_trans *)
+lemma lfpxs_fpbs_trans: ∀h,o,G1,G2,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+ ∀L1. ⦃G1, L1⦄ ⊢ ⬈*[h, T1] L → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lfpxs_ind_dx … H) -L1
+/3 width=5 by fpbs_strap2, fpbq_lfpx/
+qed-.
+
+(* Basic_2A1: uses: lpxs_lleq_fpbs *)
+lemma lpxs_ffdeq_fpbs: ∀h,o,G1,L1,L,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, T1] L →
+ ∀G2,L2,T2. ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=3 by lfpxs_fpbs_trans, ffdeq_fpbs/ qed.
+
+(* Properties with star-iterated structural successor for closures **********)
+
+(* Basic_2A1: uses: fqus_lpxs_fpbs *)
+lemma fqus_lfpxs_fpbs: ∀h,o,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
+ ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h, T2] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=3 by fpbs_lfpxs_trans, fqus_fpbs/ qed.
+
+(* Properties with uncounted context-sensitive parallel rt-computation ******)
+
+(* Basic_2A1: uses: cpxs_fqus_lpxs_fpbs *)
+lemma cpxs_fqus_lfpxs_fpbs: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L,T2. ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ →
+ ∀L2.⦃G2, L⦄ ⊢ ⬈*[h, T2] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by cpxs_fqus_fpbs, fpbs_lfpxs_trans/ qed.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: fpbs_intro_alt *)
+lemma fpbs_intro_star: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G,L,T0. ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄ →
+ ∀L0. ⦃G, L⦄ ⊢ ⬈*[h, T0] L0 →
+ ∀G2,L2,T2. ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ .
+/3 width=5 by cpxs_fqus_lfpxs_fpbs, fpbs_strap1, fpbq_ffdeq/ qed.
+
+(* Advanced inversion lemmas *************************************************)
+
+(* Basic_2A1: uses: fpbs_inv_alt *)
+lemma fpbs_inv_star: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+ ∃∃G,L,L0,T,T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄
+ & ⦃G, L⦄ ⊢ ⬈*[h, T0] L0 & ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
+[ /2 width=9 by ex4_5_intro/
+| #G1 #G0 #L1 #L0 #T1 #T0 * -G0 -L0 -T0
+ [ #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
+ elim (fquq_cpxs_trans … HT03 … H10) -T0
+ /3 width=9 by fqus_strap2, ex4_5_intro/
+ | #T0 #HT10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
+ /3 width=9 by cpxs_strap2, ex4_5_intro/
+ | #L0 #HL10 #_ * #G3 #L3 #L4 #T3 #T4 #HT13 #H34 #HL34 #H42
+ lapply (lfpx_cpxs_trans … HT13 … HL10) -HT13 #HT13
+ lapply (lfpx_cpxs_conf … HT13 … HL10) -HL10 #HL10
+ elim (lfpx_fqus_trans … H34 … HL10) -L0
+ /3 width=9 by lfpxs_step_sn, cpxs_trans, ex4_5_intro/
+ | #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
+ elim (ffdeq_cpxs_trans … H10 … HT03) -T0 #T0 #HT10 #H03
+ elim (ffdeq_fqus_trans … H03 … H34) -G0 -L0 -T3 #G0 #L0 #T3 #H03 #H34
+ elim (ffdeq_lfpxs_trans … H34 … HL34) -L3 #L3 #HL03 #H34
+ /3 width=13 by ffdeq_trans, ex4_5_intro/
+ ]
+]
+qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/cpxs_lift.ma".
-include "basic_2/computation/fpbs.ma".
-
-(* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
-
-(* Advanced properties ******************************************************)
-
-lemma lstas_fpbs: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
- ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-/3 width=5 by cpxs_fpbs, lstas_cpxs/ qed.
-
-lemma sta_fpbs: ∀h,o,G,L,T,U,d.
- ⦃G, L⦄ ⊢ T ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T •*[h, 1] U →
- ⦃G, L, T⦄ ≥[h, o] ⦃G, L, U⦄.
-/2 width=5 by lstas_fpbs/ qed.
-
-(* Note: this is used in the closure proof *)
-lemma cpr_lpr_sta_fpbs: ∀h,o,G,L1,L2,T1,T2,U2,d2.
- ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
- ⦃G, L2⦄ ⊢ T2 ▪[h, o] d2+1 → ⦃G, L2⦄ ⊢ T2 •*[h, 1] U2 →
- ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, U2⦄.
-/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, sta_cpx, fpbq_cpx/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/ffdeq.ma".
+include "basic_2/rt_computation/lfpxs_lfdeq.ma".
+
+(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
+
+(* Properties with degree-based equivalence on closures *********************)
+
+lemma ffdeq_lfpxs_trans: ∀h,o,G1,G2,L1,L0,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L0, T2⦄ →
+ ∀L2. ⦃G2, L0⦄ ⊢⬈*[h, T2] L2 →
+ ∃∃L. ⦃G1, L1⦄ ⊢⬈*[h, T1] L & ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L0 #T1 #T2 #H1 #L2 #HL02
+elim (ffdeq_inv_gen_dx … H1) -H1 #HG #HL10 #HT12 destruct
+elim (lfdeq_lfpxs_trans … HL02 … HL10) -L0 #L0 #HL10 #HL02
+lapply (tdeq_lfpxs_trans … HT12 … HL10) -HL10 #HL10
+/3 width=3 by ffdeq_intro_dx, ex2_intro/
+qed-.
(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
+(* Properties with degree-based equivalence on terms ************************)
+
+lemma lfpxs_tdeq_trans: ∀h,o,G. s_r_confluent1 … (cdeq h o) (lfpxs h G).
+#h #o #G #L1 #T1 #T2 #HT12 #L2 #H @(lfpxs_ind_sn … H) -L2
+/3 width=6 by lfpxs_step_dx, lfpx_tdeq_conf/
+qed-.
+
+lemma tdeq_lfpxs_trans: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
+ ∀G,L1,L2. ⦃G, L1⦄ ⊢⬈*[h, T2] L2 → ⦃G, L1⦄ ⊢⬈*[h, T1] L2.
+/3 width=4 by lfpxs_tdeq_trans, tdeq_sym/ qed-.
+
(* Properties with degree-based equivalence on referred entries *************)
(* Basic_2A1: was: lleq_lpxs_trans *)
-cpxs.ma cpxs_tdeq.ma cpxs_theq.ma cpxs_theq_vector.ma cpxs_drops.ma cpxs_fqus.ma cpxs_lsubr.ma cpxs_lfdeq.ma cpxs_aaa.ma cpxs_lpx.ma cpxs_lfpx.ma cpxs_cnx.ma cpxs_cpxs.ma
+cpxs.ma cpxs_tdeq.ma cpxs_theq.ma cpxs_theq_vector.ma cpxs_drops.ma cpxs_fqus.ma cpxs_lsubr.ma cpxs_lfdeq.ma cpxs_ffdeq.ma cpxs_aaa.ma cpxs_lpx.ma cpxs_lfpx.ma cpxs_cnx.ma cpxs_cpxs.ma
cpxs_ext.ma
lpxs.ma lpxs_length.ma lpxs_lpx.ma lpxs_cpxs.ma
-lfpxs.ma lfpxs_length.ma lfpxs_drops.ma lfpxs_fqup.ma lfpxs_lfdeq.ma lfpxs_aaa.ma lfpxs_cpxs.ma lfpxs_lpxs.ma lfpxs_lfpxs.ma
+lfpxs.ma lfpxs_length.ma lfpxs_drops.ma lfpxs_fqup.ma lfpxs_lfdeq.ma lfpxs_ffdeq.ma lfpxs_aaa.ma lfpxs_cpxs.ma lfpxs_lpxs.ma lfpxs_lfpxs.ma
csx.ma csx_simple.ma csx_simple_theq.ma csx_drops.ma csx_fqus.ma csx_lsubr.ma csx_lfdeq.ma csx_aaa.ma csx_gcp.ma csx_gcr.ma csx_lfpx.ma csx_cnx.ma csx_cpxs.ma csx_lfpxs.ma csx_csx.ma
csx_vector.ma csx_cnx_vector.ma csx_csx_vector.ma
lfsx.ma lfsx_drops.ma lfsx_fqup.ma lfsx_lpx.ma lfsx_lpxs.ma lfsx_lfpxs.ma lfsx_csx.ma lfsx_lfsx.ma
lsubsx.ma lsubsx_lfsx.ma lsubsx_lsubsx.ma
+fpbs.ma fpbs_fqus.ma fpbs_fqup.ma fpbs_cpxs.ma fpbs_lfpxs.ma
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lfpx_drops.ma".
+include "basic_2/rt_transition/lfpx_fsle.ma".
+include "basic_2/s_transition/fquq.ma".
+
+(* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
+
+(* Properties with extended structural successor for closures ***************)
+
+(* Basic_2A1: uses: lpx_fqu_trans *)
+lemma lfpx_fqu_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #K #V #K1 #H
+ elim (lfpx_inv_zero_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
+ elim (lifts_total V (𝐔❴1❵)) #T #HVT
+ /3 width=7 by lfpx_cpx_conf, cpx_delta, fqu_drop, ex3_2_intro/
+| /3 width=7 by lfpx_fwd_pair_sn, cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
+| /3 width=6 by lfpx_fwd_bind_dx, cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
+| /3 width=8 by lfpx_fwd_bind_dx_void, cpx_pair_sn, fqu_clear, ex3_2_intro/
+| /3 width=7 by lfpx_fwd_flat_dx, cpx_pair_sn, fqu_flat_dx, ex3_2_intro/
+| #I #G #K #T #U #HTU #K1 #H
+ elim (lfpx_drops_trans … H (Ⓣ) … HTU) -H
+ [|*: /3 width=2 by drops_refl, drops_drop/ ] -I #K0 #HK10 #HK0
+ elim (drops_inv_succ … HK10) -HK10 #I #Y #HY #H destruct
+ lapply (drops_fwd_isid … HY ?) -HY // #H destruct
+ /3 width=5 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+(* Properties with extended optional structural successor for closures ******)
+
+(* Basic_2A1: uses: lpx_fquq_trans *)
+lemma lfpx_fquq_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h, T1] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h, T2] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
+[ #H12 elim (lfpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lfdeq_fqus.ma".
+include "basic_2/static/ffdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma ffdeq_fqus_trans: ∀h,o,b,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ∃∃G,L0,T0. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L0, T0⦄ & ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+#h #o #b #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H2
+elim(ffdeq_inv_gen_dx … H1) -H1 #HG #HL1 #HT1 destruct
+elim (lfdeq_fqus_trans … H2 … HL1) -L #L #T0 #H2 #HT02 #HL2
+elim (tdeq_fqus_trans … H2 … HT1) -T #L0 #T #H2 #HT0 #HL0
+lapply (tdeq_lfdeq_conf … HT02 … HL0) -HL0 #HL0
+/4 width=7 by ffdeq_intro_dx, lfdeq_trans, tdeq_trans, ex2_3_intro/
+qed-.
(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-(* Properties with supclosure ***********************************************)
+(* Properties with extended structural successor for closures ***************)
lemma fqu_tdeq_conf: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
∀U2. U1 ≛[h, o] U2 →
/3 width=5 by lfdeq_sym, tdeq_sym, ex3_2_intro/
qed-.
-(* Basic_2A1: was just: lleq_fqu_trans *)
+(* Basic_2A1: uses: lleq_fqu_trans *)
lemma lfdeq_fqu_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐[b] ⦃G2, K2, U⦄ →
∀L1. L1 ≛[h, o, T] L2 →
∃∃K1,U0. ⦃G1, L1, T⦄ ⊐[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
]
qed-.
+(* Properties with optional structural successor for closures ***************)
+
+lemma tdeq_fquq_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐⸮[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -H
+[ #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
+ /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
(* Basic_2A1: was just: lleq_fquq_trans *)
lemma lfdeq_fquq_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮[b] ⦃G2, K2, U⦄ →
∀L1. L1 ≛[h, o, T] L2 →
]
qed-.
+(* Properties with plus-iterated structural successor for closures **********)
+
(* Basic_2A1: was just: lleq_fqup_trans *)
lemma lfdeq_fqup_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, U⦄ →
∀L1. L1 ≛[h, o, T] L2 →
]
qed-.
+lemma tdeq_fqup_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐+[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1
+[ #G1 #L1 #U1 #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
+ /3 width=5 by fqu_fqup, ex3_2_intro/
+| #G1 #G #L1 #L #U1 #U #H #_ #IH #U2 #HU21
+ elim (tdeq_fqu_trans … H … HU21) -U1 #L0 #T #H1 #HTU #HL0
+ lapply (tdeq_lfdeq_div … HTU … HL0) -HL0 #HL0
+ elim (IH … HTU) -U #K2 #U1 #H2 #HUT1 #HKL2
+ elim (lfdeq_fqup_trans … H2 … HL0) -L #K #U #H2 #HU1 #HK2
+ lapply (tdeq_lfdeq_conf … HUT1 … HK2) -HK2 #HK2
+ /3 width=7 by lfdeq_trans, fqup_strap2, tdeq_trans, ex3_2_intro/
+]
+qed-.
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma tdeq_fqus_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐*[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H
+[ #H elim (tdeq_fqup_trans … H … HU21) -U1 /3 width=5 by fqup_fqus, ex3_2_intro/
+| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
(* Basic_2A1: was just: lleq_fqus_trans *)
lemma lfdeq_fqus_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐*[b] ⦃G2, K2, U⦄ →
∀L1. L1 ≛[h, o, T] L2 →
]
[ { "parallel qrst-computation" * } {
[ [ "" ] "fpbg ( ⦃?,?,?⦄ >≛[?,?] ⦃?,?,?⦄ )" "fpbg_lift" + "fpbg_fleq" + "fpbg_fpbs" + "fpbg_fpbg" * ]
- [ [ "" ] "fpbs ( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_alt ( ⦃?,?,?⦄ ≥≥[?,?] ⦃?,?,?⦄ )" "fpbs_lift" + "fpbs_aaa" + "fpbs_fpb" + "fpbs_fpbs" * ]
+ [ [ "" ] "fpbs_alt" + "fpbs_aaa" + "fpbs_fpb" + "fpbs_fpbs" * ]
}
]
[ { "decomposed rt-computation" * } {
}
]
*)
+ [ { "uncounted context-sensitive parallel rst-computation" * } {
+ [ [ "for closures" ] "fpbs" + "( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_fqup" + "fpbs_fqus" + "fpbs_cpxs" + "fpbs_lfpxs" * ]
+ }
+ ]
[ { "uncounted context-sensitive parallel rt-computation" * } {
[ [ "refinement for lenvs on selected entries" ] "lsubsx" + "( ? ⊢ ? ⊆ⓧ[?,?,?] ? )" "lsubsx_lfsx" + "lsubsx_lsubsx" * ]
[ [ "strongly normalizing for lenvs on referred entries" ] "lfsx" + "( ? ⊢ ⬈*[?,?,?] 𝐒⦃?⦄ )" "lfsx_drops" + "lfsx_fqup" + "lfsx_cpx" + "lfsx_cpxs" + "lfsx_lfpxs" + "lfsx_lfsx" * ]
[ [ "strongly normalizing for term vectors" ] "csx_vector" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_cnx_vector" + "csx_csx_vector" * ]
[ [ "strongly normalizing for terms" ] "csx" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_simple" + "csx_simple_theq" + "csx_drops" + "csx_fqus" + "csx_lsubr" + "csx_lfdeq" + "csx_aaa" + "csx_gcp" + "csx_gcr" + "csx_lfpx" + "csx_cnx" + "csx_cpxs" + "csx_lfpxs" + "csx_csx" * ]
- [ [ "for lenvs on referred entries" ] "lfpxs" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "lfpxs_length" + "lfpxs_drops" + "lfpxs_fqup" + "lfpxs_lfdeq" + "lfpxs_aaa" + "lfpxs_cpxs" + "lfpxs_lpxs" + "lfpxs_lfpxs" * ]
+ [ [ "for lenvs on referred entries" ] "lfpxs" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "lfpxs_length" + "lfpxs_drops" + "lfpxs_fqup" + "lfpxs_lfdeq" + "lfpxs_ffdeq" + "lfpxs_aaa" + "lfpxs_cpxs" + "lfpxs_lpxs" + "lfpxs_lfpxs" * ]
[ [ "for lenvs on all entries" ] "lpxs" + "( ⦃?,?⦄ ⊢ ⬈*[?] ? )" "lpxs_length" + "lpxs_lpx" + "lpxs_cpxs" * ]
[ [ "for binders" ] "cpxs_ext" + "( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" * ]
- [ [ "for terms" ] "cpxs" + "( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_theq" + "cpxs_theq_vector" + "cpxs_drops" + "cpxs_fqus" + "cpxs_lsubr" + "cpxs_lfdeq" + "cpxs_aaa" + "cpxs_lpx" + "cpxs_lfpx" + "cpxs_cnx" + "cpxs_cpxs" * ]
+ [ [ "for terms" ] "cpxs" + "( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_theq" + "cpxs_theq_vector" + "cpxs_drops" + "cpxs_fqus" + "cpxs_lsubr" + "cpxs_lfdeq" + "cpxs_ffdeq" + "cpxs_aaa" + "cpxs_lpx" + "cpxs_lfpx" + "cpxs_cnx" + "cpxs_cpxs" * ]
}
]
}
]
[ { "uncounted context-sensitive parallel rt-transition" * } {
[ [ "normal form for terms" ] "cnx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] 𝐍⦃?⦄ )" "cnx_simple" + "cnx_drops" + "cnx_cnx" * ]
- [ [ "for lenvs on referred entries" ] "lfpx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_drops" + "lfpx_fqup" + "lfpx_fsle" + "lfpx_lfdeq" + "lfpx_aaa" + "lfpx_lpx" + "lfpx_lfpx" * ]
+ [ [ "for lenvs on referred entries" ] "lfpx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_drops" + "lfpx_fquq" + "lfpx_fqup" + "lfpx_fsle" + "lfpx_lfdeq" + "lfpx_aaa" + "lfpx_lpx" + "lfpx_lfpx" * ]
[ [ "for lenvs on all entries" ] "lpx" + "( ⦃?,?⦄ ⊢ ⬈[?] ? )" * ]
[ [ "for binders" ] "cpx_ext" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" * ]
[ [ "for terms" ] "cpx" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_fqus" + "cpx_lsubr" + "cpx_lfeq" + "cpx_lfdeq" * ]
}
]
[ { "degree-based equivalence" * } {
- [ [ "for closures on referred entries" ] "ffdeq" + "( ⦃?,?,?⦄ ≛[?,?] ⦃?,?,?⦄ )" "ffdeq_fqup" + "ffdeq_ffdeq" * ]
+ [ [ "for closures on referred entries" ] "ffdeq" + "( ⦃?,?,?⦄ ≛[?,?] ⦃?,?,?⦄ )" "ffdeq_fqup" + "ffdeq_fqus" + "ffdeq_ffdeq" * ]
[ [ "for lenvs on referred entries" ] "lfdeq" + "( ? ≛[?,?,?] ? )" "lfdeq_length" + "lfdeq_drops" + "lfdeq_fqup" + "lfdeq_fqus" + "lfdeq_lfeq" + "lfdeq_lfdeq" * ]
}
]
projects / helm.git / commitdiff