(* *)
(**************************************************************************)
-include "basic_2/computation/lpxs_lleq.ma".
-include "basic_2/computation/fpbs_lift.ma".
-include "basic_2/computation/fpbg_fleq.ma".
+include "static_2/static/fdeq_fqup.ma".
+include "static_2/static/fdeq_fdeq.ma".
+include "basic_2/rt_transition/fpbq_fpb.ma".
+include "basic_2/rt_computation/fpbg.ma".
-(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+(* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
-(* Properties on "qrst" parallel reduction on closures **********************)
+(* Advanced forward lemmas **************************************************)
+
+lemma fpbg_fwd_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 fpbs_strap2, fpb_fpbq/
+qed-.
+
+(* Advanced properties with degree-based equivalence on closures ************)
+
+(* Basic_2A1: uses: fleq_fpbg_trans *)
+lemma fdeq_fpbg_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⦄.
+#h #o #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
+elim (fdeq_fpb_trans … H1 … H0) -G -L -T
+/4 width=9 by fpbs_strap2, fpbq_fdeq, ex2_3_intro/
+qed-.
+
+(* Properties with parallel proper rst-reduction on closures ****************)
lemma fpb_fpbg_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≻[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≛[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≛[h, o] ⦃G2, L2, T2⦄.
+ ⦃G1, L1, T1⦄ ≻[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ >[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
/3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.
+(* Properties with parallel rst-reduction on closures ***********************)
+
lemma fpbq_fpbg_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≛[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fpbq_ind_alt … H1) -H1
-/2 width=5 by fleq_fpbg_trans, fpb_fpbg_trans/
+ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ >[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
+elim (fpbq_inv_fpb … H1) -H1
+/2 width=5 by fdeq_fpbg_trans, fpb_fpbg_trans/
qed-.
-(* Properties on "qrst" parallel compuutation on closures *******************)
+(* Properties with parallel rst-compuutation on closures ********************)
lemma fpbs_fpbg_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⦄.
+ ∀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 #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
qed-.
-(* Note: this is used in the closure proof *)
-lemma fpbg_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⦄.
-#h #o #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
+(* Advanced inversion lemmas of parallel rst-computation on closures ********)
+
+(* Basic_2A1: was: fpbs_fpbg *)
+lemma fpbs_inv_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, 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 … H) -G2 -L2 -T2
+[ /2 width=1 by or_introl/
+| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1
+ elim (fpbq_inv_fpb … H2) -H2 #H2
+ [ /3 width=5 by fdeq_trans, or_introl/
+ | elim (fdeq_fpb_trans … H1 … H2) -G -L -T
+ /4 width=5 by ex2_3_intro, or_intror, fdeq_fpbs/
+ | /3 width=5 by fpbg_fdeq_trans, or_intror/
+ | /4 width=5 by fpbg_fpbq_trans, fpb_fpbq, or_intror/
+ ]
+]
qed-.
-(* Note: this is used in the closure proof *)
-lemma fqup_fpbg: ∀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 elim (fqup_inv_step_sn … H) -H
-/3 width=5 by fqus_fpbs, fpb_fqu, ex2_3_intro/
-qed.
-
-lemma cpxs_fpbg: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 →
- (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≛[h, o] ⦃G, L, T2⦄.
-#h #o #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
-/4 width=5 by cpxs_fpbs, fpb_cpx, ex2_3_intro/
-qed.
-
-lemma lstas_fpbg: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → (T1 = T2 → ⊥) →
- ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → ⦃G, L, T1⦄ >≛[h, o] ⦃G, L, T2⦄.
-/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
-
-lemma lpxs_fpbg: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
- (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≛[h, o] ⦃G, L2, T⦄.
-#h #o #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
-/4 width=5 by fpb_lpx, lpxs_lleq_fpbs, ex2_3_intro/
-qed.
+(* Advanced properties of parallel rst-computation on closures **************)
+
+lemma fpbs_fpb_trans: ∀h,o,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, o] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, o] ⦃G2, L2, U2⦄.
+#h #o #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_inv_fpbg … H) -H
+[ #H12 #G2 #L2 #U2 #H2 elim (fdeq_fpb_trans … H12 … H2) -F2 -K2 -T2
+ /3 width=5 by fdeq_fpbs, ex2_3_intro/
+| * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
+ @(ex2_3_intro … H4) -H4 /3 width=5 by fpbs_strap1, fpb_fpbq/
+]
+qed-.
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