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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basic_2/relocation/lreq_lreq.ma".
16 include "basic_2/static/lfxs_lfxs.ma".
17 include "basic_2/static/lfeq_lreq.ma".
19 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
21 (* Main properties **********************************************************)
23 theorem lfeq_bind: ∀p,I,L1,L2,V1,V2,T.
24 L1 ≡[V1] L2 → L1.ⓑ{I}V1 ≡[T] L2.ⓑ{I}V2 →
26 /2 width=2 by lfxs_bind/ qed.
28 theorem lfeq_flat: ∀I,L1,L2,V,T. L1 ≡[V] L2 → L1 ≡[T] L2 →
30 /2 width=1 by lfxs_flat/ qed.
32 (* Note: /2 width=3 by lfeq_lfxs_trans/ *)
33 theorem lfeq_trans: ∀T. Transitive … (lfeq T).
34 #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
35 lapply (frees_lreq_conf … Hf1 … HL1) #H0
36 lapply (frees_mono … Hf2 … H0) -Hf2 -H0
37 /4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/
40 theorem lfeq_canc_sn: ∀T. left_cancellable … (lfeq T).
41 /3 width=3 by lfeq_trans, lfeq_sym/ qed-.
43 theorem lfeq_canc_dx: ∀T. right_cancellable … (lfeq T).
44 /3 width=3 by lfeq_trans, lfeq_sym/ qed-.
46 (* Advanced properies on negated lazy equivalence *****************************)
48 (* Note: for use in auto, works with /4 width=8/ so lfeq_canc_sn is preferred *)
49 lemma lfeq_nlfeq_trans: ∀T,L1,L. L1 ≡[T] L →
50 ∀L2. (L ≡[T] L2 → ⊥) → (L1 ≡[T] L2 → ⊥).
51 /3 width=3 by lfeq_canc_sn/ qed-.
53 lemma nlfeq_lfeq_div: ∀T,L2,L. L2 ≡[T] L →
54 ∀L1. (L1 ≡[T] L → ⊥) → (L1 ≡[T] L2 → ⊥).
55 /3 width=3 by lfeq_trans/ qed-.