(* *)
(**************************************************************************)
-include "static_2/static/feqx_fqus.ma".
-include "static_2/static/feqx_feqx.ma".
include "basic_2/rt_computation/cpxs_fqus.ma".
-include "basic_2/rt_computation/cpxs_feqx.ma".
-include "basic_2/rt_computation/lpxs_feqx.ma".
+include "basic_2/rt_computation/cpxs_reqg.ma".
+include "basic_2/rt_computation/lpxs_feqg.ma".
+include "basic_2/rt_computation/fpbs_lpx.ma".
include "basic_2/rt_computation/fpbs_cpxs.ma".
(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
-(* Properties with unbound rt-computation on full local environments *******)
+(* Properties with extended rt-computation on full local environments ******)
-lemma lpxs_fpbs: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → ⦃G,L1,T⦄ ≥[h] ⦃G,L2,T⦄.
-#h #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
-/3 width=5 by fpbq_lpx, fpbs_strap1/
+lemma lpxs_fpbs:
+ ∀G,L1,L2,T. ❪G,L1❫ ⊢ ⬈* L2 → ❪G,L1,T❫ ≥ ❪G,L2,T❫.
+#G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
+/3 width=5 by lpx_fpb, fpbs_strap1/
qed.
-lemma fpbs_lpxs_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L,T2⦄ →
- ∀L2. ⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
-#h #G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lpxs_ind_dx … H) -L2
-/3 width=5 by fpbs_strap1, fpbq_lpx/
+lemma fpbs_lpxs_trans:
+ ∀G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L,T2❫ →
+ ∀L2. ❪G2,L❫ ⊢ ⬈* L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+#G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lpxs_ind_dx … H) -L2
+/3 width=5 by fpbs_strap1, lpx_fpb/
qed-.
-lemma lpxs_fpbs_trans: ∀h,G1,G2,L,L2,T1,T2. ⦃G1,L,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
- ∀L1. ⦃G1,L1⦄ ⊢ ⬈*[h] L → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
-#h #G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lpxs_ind_sn … H) -L1
-/3 width=5 by fpbs_strap2, fpbq_lpx/
+lemma lpxs_fpbs_trans:
+ ∀G1,G2,L,L2,T1,T2. ❪G1,L,T1❫ ≥ ❪G2,L2,T2❫ →
+ ∀L1. ❪G1,L1❫ ⊢ ⬈* L → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+#G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lpxs_ind_sn … H) -L1
+/3 width=5 by fpbs_strap2, lpx_fpb/
qed-.
(* Basic_2A1: uses: lpxs_lleq_fpbs *)
-lemma lpxs_feqx_fpbs: ∀h,G1,L1,L,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] L →
- ∀G2,L2,T2. ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
-/3 width=3 by lpxs_fpbs_trans, feqx_fpbs/ qed.
-
-lemma fpbs_lpx_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L,T2⦄ →
- ∀L2. ⦃G2,L⦄ ⊢ ⬈[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
-/3 width=3 by fpbs_lpxs_trans, lpx_lpxs/ qed-.
+lemma lpxs_feqg_fpbs (S) (L):
+ reflexive … S → symmetric … S →
+ ∀G1,L1,T1. ❪G1,L1❫ ⊢ ⬈* L →
+ ∀G2,L2,T2. ❪G1,L,T1❫ ≛[S] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+/3 width=4 by lpxs_fpbs_trans, feqg_fpbs/ qed.
(* Properties with star-iterated structural successor for closures **********)
-lemma fqus_lpxs_fpbs: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ⬂* ⦃G2,L,T2⦄ →
- ∀L2. ⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+lemma fqus_lpxs_fpbs:
+ ∀G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ⬂* ❪G2,L,T2❫ →
+ ∀L2. ❪G2,L❫ ⊢ ⬈* L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
-(* Properties with unbound context-sensitive parallel rt-computation ********)
+(* Properties with extended context-sensitive parallel rt-computation *******)
-lemma cpxs_fqus_lpxs_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L,T2. ⦃G1,L1,T⦄ ⬂* ⦃G2,L,T2⦄ →
- ∀L2.⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+lemma cpxs_fqus_lpxs_fpbs:
+ ∀G1,L1,T1,T. ❪G1,L1❫ ⊢ T1 ⬈* T →
+ ∀G2,L,T2. ❪G1,L1,T❫ ⬂* ❪G2,L,T2❫ →
+ ∀L2.❪G2,L❫ ⊢ ⬈* L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
-lemma fpbs_cpxs_teqx_fqup_lpx_trans: ∀h,G1,G3,L1,L3,T1,T3. ⦃G1,L1,T1⦄ ≥ [h] ⦃G3,L3,T3⦄ →
- ∀T4. ⦃G3,L3⦄ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛ T5 →
- ∀G2,L4,T2. ⦃G3,L3,T5⦄ ⬂+ ⦃G2,L4,T2⦄ →
- ∀L2. ⦃G2,L4⦄ ⊢ ⬈[h] L2 → ⦃G1,L1,T1⦄ ≥ [h] ⦃G2,L2,T2⦄.
-#h #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
+lemma fpbs_cpxs_teqg_fqup_lpx_trans (S):
+ reflexive … S → symmetric … S →
+ ∀G1,G3,L1,L3,T1,T3. ❪G1,L1,T1❫ ≥ ❪G3,L3,T3❫ →
+ ∀T4. ❪G3,L3❫ ⊢ T3 ⬈* T4 → ∀T5. T4 ≛[S] T5 →
+ ∀G2,L4,T2. ❪G3,L3,T5❫ ⬂+ ❪G2,L4,T2❫ →
+ ∀L2. ❪G2,L4❫ ⊢ ⬈ L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+#S #H1S #H2S #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
@(fpbs_lpx_trans … HL42) -L2 (**) (* full auto too slow *)
@(fpbs_fqup_trans … H34) -G2 -L4 -T2
-/3 width=3 by fpbs_cpxs_trans, fpbs_teqx_trans/
+/3 width=6 by fpbs_cpxs_trans, fpbs_teqg_trans/
qed-.
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: fpbs_intro_alt *)
-lemma fpbs_intro_star: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G,L,T0. ⦃G1,L1,T⦄ ⬂* ⦃G,L,T0⦄ →
- ∀L0. ⦃G,L⦄ ⊢ ⬈*[h] L0 →
- ∀G2,L2,T2. ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ .
-/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_feqx/ qed.
+lemma fpbs_intro_star (S) (G) (T) (T0) (L) (L0):
+ reflexive … S → symmetric … S →
+ ∀G1,L1,T1. ❪G1,L1❫ ⊢ T1 ⬈* T →
+ ❪G1,L1,T❫ ⬂* ❪G,L,T0❫ → ❪G,L❫ ⊢ ⬈* L0 →
+ ∀G2,L2,T2. ❪G,L0,T0❫ ≛[S] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+/3 width=8 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, feqg_fpb/ qed.
(* Advanced inversion lemmas *************************************************)
(* Basic_2A1: uses: fpbs_inv_alt *)
-lemma fpbs_inv_star: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
- ∃∃G,L,L0,T,T0. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,L1,T⦄ ⬂* ⦃G,L,T0⦄
- & ⦃G,L⦄ ⊢ ⬈*[h] L0 & ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄.
-#h #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 (lpx_cpxs_trans … HT13 … HL10) -HT13 #HT13
- elim (lpx_fqus_trans … H34 … HL10) -L0
- /3 width=9 by lpxs_step_sn, cpxs_trans, ex4_5_intro/
- | #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
- elim (feqx_cpxs_trans … H10 … HT03) -T0 #T0 #HT10 #H03
- elim (feqx_fqus_trans … H03 … H34) -G0 -L0 -T3 #G0 #L0 #T3 #H03 #H34
- elim (feqx_lpxs_trans … H34 … HL34) -L3 #L3 #HL03 #H34
- /3 width=13 by feqx_trans, ex4_5_intro/
- ]
+lemma fpbs_inv_star (S):
+ reflexive … S → symmetric … S → Transitive … S →
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ →
+ ∃∃G,L,L0,T,T0. ❪G1,L1❫ ⊢ T1 ⬈* T & ❪G1,L1,T❫ ⬂* ❪G,L,T0❫ & ❪G,L❫ ⊢ ⬈* L0 & ❪G,L0,T0❫ ≛[S] ❪G2,L2,T2❫.
+#S #H1S #H2S #H3S #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
+[ /3 width=9 by feqg_refl, ex4_5_intro/
+| #G1 #G0 #L1 #L0 #T1 #T0 *
+ #L3 #T3 #H13 #HT30 #HL30 #_ *
+ #G4 #L4 #L5 #T4 #T5 #HT04 #H45 #HL45 #H52
+ lapply (rpx_cpx_conf_sn … HT30 … HL30) -HL30 #HL30
+ elim (fquq_cpx_trans … H13 … HT30) -T3 #T3 #HT13 #H30
+ elim (rpx_fwd_lpx_reqg S … HL30) -HL30 // #L #HL3 #HL0
+ lapply (reqg_cpxs_trans … HT04 … HL0) -HT04 // #HT04
+ lapply (cpxs_reqg_conf_sn … HT04 … HL0) -HL0 #HL0
+ lapply (lpx_cpxs_trans … HT04 … HL3) -HT04 #HT04
+ elim (fquq_cpxs_trans … HT04 … H30) -T0 #T0 #HT30 #H04
+ lapply (cpxs_strap2 … HT13 … HT30) -T3 #HT10
+ elim (reqg_fqus_trans … H45 … HL0) -L0 // #L0 #T3 #H43 #HT35 #HL04
+ lapply (feqg_intro_dx … G4 … HL04 … HT35) -HL04 -HT35 // #H35
+ elim (lpx_fqus_trans … H43 … HL3) -L #L #T #HT4 #H3 #HL0
+ elim (fquq_cpxs_trans … HT4 … H04) -T4 #T4 #HT04 #H4
+ lapply (cpxs_trans … HT10 … HT04) -T0 #HT14
+ lapply (fqus_strap2 … H4 … H3) -G0 -L3 -T #H43
+ elim (feqg_lpxs_trans … H35 … HL45) -L4 // #L4 #HL04 #H35
+ lapply (lpxs_step_sn … HL0 … HL04) -L0 #HL4
+ lapply (feqg_trans … H35 … H52) -L5 -T5 // #H32
+ /2 width=9 by ex4_5_intro/
]
qed-.
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