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15 include "basic_2A/multiple/lleq_drop.ma".
17 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
19 (* Main properties **********************************************************)
21 theorem lleq_trans: ∀l,T. Transitive … (lleq l T).
22 /2 width=3 by lleq_llpx_sn_trans/ qed-.
24 theorem lleq_canc_sn: ∀L,L1,L2,T,l. L ≡[l, T] L1→ L ≡[l, T] L2 → L1 ≡[l, T] L2.
25 /3 width=3 by lleq_trans, lleq_sym/ qed-.
27 theorem lleq_canc_dx: ∀L1,L2,L,T,l. L1 ≡[l, T] L → L2 ≡[l, T] L → L1 ≡[l, T] L2.
28 /3 width=3 by lleq_trans, lleq_sym/ qed-.
30 (* Advanced properies on negated lazy equivalence *****************************)
32 (* Note: for use in auto, works with /4 width=8/ so lleq_canc_sn is preferred *)
33 lemma lleq_nlleq_trans: ∀l,T,L1,L. L1 ≡[T, l] L →
34 ∀L2. (L ≡[T, l] L2 → ⊥) → (L1 ≡[T, l] L2 → ⊥).
35 /3 width=3 by lleq_canc_sn/ qed-.
37 lemma nlleq_lleq_div: ∀l,T,L2,L. L2 ≡[T, l] L →
38 ∀L1. (L1 ≡[T, l] L → ⊥) → (L1 ≡[T, l] L2 → ⊥).
39 /3 width=3 by lleq_trans/ qed-.