(* *)
(**************************************************************************)
-include "basic_2/notation/relations/dpconvstar_6.ma".
-include "basic_2/static/da.ma".
-include "basic_2/unfold/lstas.ma".
-include "basic_2/equivalence/cpcs.ma".
+include "basic_2/notation/relations/dpconvstar_8.ma".
+include "basic_2/computation/cpds.ma".
(* DECOMPOSED EXTENDED PARALLEL EQUIVALENCE FOR TERMS ***********************)
-definition cpes: ∀h. sd h → relation4 genv lenv term term ≝
- λh,g,G,L,T1,T2.
- ∃∃T,l1,l2. l2 ≤ l1 & ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 & ⦃G, L⦄ ⊢ T1 •*[h, l2] T & ⦃G, L⦄ ⊢ T ⬌* T2.
+definition cpes: ∀h. sd h → nat → nat → relation4 genv lenv term term ≝
+ λh,g,l1,l2,G,L,T1,T2.
+ ∃∃T. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l1] T & ⦃G, L⦄ ⊢ T2 •*➡*[h, g, l2] T.
interpretation "decomposed extended parallel equivalence (term)"
- 'DPConvStar h g G L T1 T2 = (cpes h g G L T1 T2).
+ 'DPConvStar h g l1 l2 G L T1 T2 = (cpes h g l1 l2 G L T1 T2).
(* Basic properties *********************************************************)
-lemma sta_cpcs_cpes: ∀h,g,G,L,T1,T,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 •[h] T →
- ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g] T2.
-/3 width=7 by sta_lstas, ex4_3_intro/ qed.
+lemma cpds_div: ∀h,g,G,L,T1,T2,T,l1,l2.
+ ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l1] T → ⦃G, L⦄ ⊢ T2 •*➡*[h, g, l2] T →
+ ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l1, l2] T2.
+/2 width=3 by ex2_intro/ qed.
-lemma lstas_cpes: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l → ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g] T2.
-/2 width=7 by ex4_3_intro/ qed.
+lemma cpes_refl_O_O: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l →
+ ⦃G, L⦄ ⊢ T •*⬌*[h, g, 0, 0] T.
+/3 width=3 by cpds_refl, cpds_div/ qed.
-lemma cpcs_cpes: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l → ⦃G, L⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g] T2.
-/2 width=7 by lstar_O, ex4_3_intro/ qed.
-
-lemma cpes_refl: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ⦃G, L⦄ ⊢ T •*⬌*[h, g] T.
-/2 width=2 by cpcs_cpes/ qed.
-
-lemma cpes_strap1: ∀h,g,G,L,T1,T,T2.
- ⦃G, L⦄ ⊢ T1 •*⬌*[h, g] T → ⦃G, L⦄ ⊢ T ⬌ T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g] T2.
-#h #g #G #L #T1 #T #T2 * /3 width=9 by cpcs_strap1, ex4_3_intro/
-qed.
+lemma cpes_sym: ∀h,g,G,L,T1,T2,l1,l2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l1, l2] T2 →
+ ⦃G, L⦄ ⊢ T2 •*⬌*[h, g, l2, l1] T1.
+#h #g #G #L #T1 #T2 #L1 #l2 * /2 width=3 by cpds_div/
+qed-.