(* *)
(**************************************************************************)
-include "basic_2/relocation/lleq_ldrop.ma".
+include "basic_2/multiple/lleq_drop.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
(* Main properties **********************************************************)
-theorem lleq_trans: ∀d,T. Transitive … (lleq d T).
+theorem lleq_trans: ∀l,T. Transitive … (lleq l T).
/2 width=3 by lleq_llpx_sn_trans/ qed-.
-theorem lleq_canc_sn: ∀L,L1,L2,T,d. L ⋕[d, T] L1→ L ⋕[d, T] L2 → L1 ⋕[d, T] L2.
+theorem lleq_canc_sn: ∀L,L1,L2,T,l. L ≡[l, T] L1→ L ≡[l, T] L2 → L1 ≡[l, T] L2.
/3 width=3 by lleq_trans, lleq_sym/ qed-.
-theorem lleq_canc_dx: ∀L1,L2,L,T,d. L1 ⋕[d, T] L → L2 ⋕[d, T] L → L1 ⋕[d, T] L2.
+theorem lleq_canc_dx: ∀L1,L2,L,T,l. L1 ≡[l, T] L → L2 ≡[l, T] L → L1 ≡[l, T] L2.
/3 width=3 by lleq_trans, lleq_sym/ qed-.
-(* Note: lleq_nlleq_trans: ∀d,T,L1,L. L1⋕[T, d] L →
- ∀L2. (L ⋕[T, d] L2 → ⊥) → (L1 ⋕[T, d] L2 → ⊥).
+(* Advanced properies on negated lazy equivalence *****************************)
+
+(* Note: for use in auto, works with /4 width=8/ so lleq_canc_sn is preferred *)
+lemma lleq_nlleq_trans: ∀l,T,L1,L. L1 ≡[T, l] L →
+ ∀L2. (L ≡[T, l] L2 → ⊥) → (L1 ≡[T, l] L2 → ⊥).
/3 width=3 by lleq_canc_sn/ qed-.
-works with /4 width=8/ so lleq_canc_sn is more convenient
-*)
+
+lemma nlleq_lleq_div: ∀l,T,L2,L. L2 ≡[T, l] L →
+ ∀L1. (L1 ≡[T, l] L → ⊥) → (L1 ≡[T, l] L2 → ⊥).
+/3 width=3 by lleq_trans/ qed-.