(**************************************************************************)
include "basic_2/notation/relations/dpconvstar_6.ma".
-include "basic_2/unfold/lsstas.ma".
+include "basic_2/static/da.ma".
+include "basic_2/unfold/lstas.ma".
include "basic_2/equivalence/cpcs.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, g, l2] T & ⦃G, L⦄ ⊢ T ⬌* T2.
+ ∃∃T,l1,l2. l2 ≤ l1 & ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 & ⦃G, L⦄ ⊢ T1 •*[h, l2] T & ⦃G, L⦄ ⊢ T ⬌* T2.
interpretation "decomposed extended parallel equivalence (term)"
'DPConvStar h g G L T1 T2 = (cpes h g G L T1 T2).
(* Basic properties *********************************************************)
-lemma ssta_cpcs_cpes: ∀h,g,G,L,T1,T,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T1 •[h, g] T →
- ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g] T2.
-/3 width=7/ qed.
+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 lsstas_cpes: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l → ⦃G, L⦄ ⊢ T1 •*[h, g, l] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g] T2.
-/2 width=7/ 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 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/ qed.
+/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/ qed.
+/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/
+#h #g #G #L #T1 #T #T2 * /3 width=9 by cpcs_strap1, ex4_3_intro/
qed.