1 (**************************************************************************)
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/lib/star.ma".
16 include "basic_2/syntax/ext2.ma".
18 (* EXTENSION TO BINDERS OF A RELATION FOR TERMS *****************************)
20 (* Properties with transitive closure ***************************************)
22 lemma ext2_tc_pair: ∀R,I,V1,V2. TC … R V1 V2 →
23 TC … (ext2 R) (BPair I V1) (BPair I V2).
24 #R #I #V1 #V2 #H elim H -H -V2
25 /3 width=3 by ext2_pair, step, inj/
28 lemma ext2_tc_inj: ∀R,I1,I2. ext2 R I1 I2 → ext2 (TC … R) I1 I2.
30 /3 width=1 by ext2_unit, ext2_pair, inj/
33 (* Main properties with transitive closure **********************************)
35 theorem ext2_tc_step: ∀R,I1,I. ext2 (TC … R) I1 I →
36 ∀I2. ext2 R I I2 → ext2 (TC … R) I1 I2.
38 [ #I #Z #H >(ext2_inv_unit_sn … H) -Z /2 width=1 by ext2_unit/
39 | #I #V1 #V #HV1 #Z #H
40 elim (ext2_inv_pair_sn … H) -H #V2 #HV2 #H destruct
41 /3 width=3 by ext2_pair, step/
45 (* Advanced properties with transitive closure ******************************)
47 lemma ext2_tc: ∀R,I1,I2. TC … (ext2 R) I1 I2 → ext2 (TC … R) I1 I2.
48 #R #I1 #I2 #H elim H -I2
49 /2 width=3 by ext2_tc_step, ext2_tc_inj/
52 (* Advanced inversion lemmas with transitive closure ************************)
54 lemma ext2_inv_tc: ∀R,I1,I2. ext2 (TC … R) I1 I2 → TC … (ext2 R) I1 I2.
56 /3 width=1 by ext2_tc_pair, ext2_unit, inj/