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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground_2/ynat/ynat_plus.ma".
16 include "basic_2/grammar/lreq.ma".
18 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
20 (* Main properties **********************************************************)
22 theorem lreq_trans: ∀l,m. Transitive … (lreq l m).
23 #l #m #L1 #L2 #H elim H -L1 -L2 -l -m
24 [ #l #m #X #H lapply (lreq_inv_atom1 … H) -H
26 | #I1 #I #L1 #L #V1 #V #_ #IHL1 #X #H elim (lreq_inv_zero1 … H) -H
27 #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lreq_zero/
28 | #I #L1 #L #V #m #_ #IHL1 #X #H elim (lreq_inv_pair1 … H) -H //
29 #L2 #HL2 #H destruct /3 width=1 by lreq_pair/
30 | #I1 #I #L1 #L #V1 #V #l #m #_ #IHL1 #X #H elim (lreq_inv_succ1 … H) -H //
31 #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lreq_succ/
35 theorem lreq_canc_sn: ∀l,m,L,L1,L2. L ⩬[l, m] L1 → L ⩬[l, m] L2 → L1 ⩬[l, m] L2.
36 /3 width=3 by lreq_trans, lreq_sym/ qed-.
38 theorem lreq_canc_dx: ∀l,m,L,L1,L2. L1 ⩬[l, m] L → L2 ⩬[l, m] L → L1 ⩬[l, m] L2.
39 /3 width=3 by lreq_trans, lreq_sym/ qed-.
41 theorem lreq_join: ∀L1,L2,l,i. L1 ⩬[l, i] L2 →
42 ∀m. L1 ⩬[i+l, m] L2 → L1 ⩬[l, i+m] L2.
43 #L1 #L2 #l #i #H elim H -L1 -L2 -l -i //
44 [ #I #L1 #L2 #V #m #_ #IHL12 #m #H
45 lapply (lreq_inv_succ … H ?) -H // >ypred_succ /3 width=1 by lreq_pair/
46 | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #m #H
47 lapply (lreq_inv_succ … H ?) -H // >yplus_succ2 >ypred_succ /3 width=1 by lreq_succ/