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 *)
13 (**************************************************************************)
15 include "ground/lib/star.ma".
16 include "ground/lib/ltc.ma".
18 (* LABELLED TRANSITIVE CLOSURE FOR RELATIONS ********************************)
20 alias symbol "subseteq" = "relation inclusion". (* * alias *)
22 (* Constructions with ctc ***************************************************)
24 lemma ltc_CTC (C) (A) (i) (f) (B) (R:relation4 C A B B):
26 ∀c. CTC … (λc. R c i) c ⊆ ltc … f … (R c) i.
27 #C #A #i #f #B #R #Hf #c #b1 #b2 #H elim H -b2 /2 width=1 by ltc_rc/
28 #b #b2 #_ #Hb2 #IH >(Hf i) -Hf /2 width=3 by ltc_dx/
31 (* Inversions with ctc ******************************************************)
33 lemma ltc_inv_CTC (C) (A) (i) (f) (B) (R:relation4 C A B B):
34 associative … f → annulment_2 … f i →
35 ∀c. ltc … f … (R c) i ⊆ CTC … (λc. R c i) c.
36 #C #A #i #f #B #R #H1f #H2f #c #b1 #b2
37 @(insert_eq_1 … i) #a #H
38 @(ltc_ind_dx A f B … H) -a -b2 /2 width=1 by inj/ -H1f
39 #a1 #a2 #b #b2 #_ #IH #Hb2 #H <H
40 elim (H2f … H) -H2f -H #H1 #H2 destruct