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/xoa/ex_6_8.ma".
16 include "ground/arith/nat_plus.ma".
17 include "ground/counters/rtc.ma".
19 (* ADDITION FOR RT-TRANSITION COUNTERS **************************************)
21 definition rtc_plus (c1:rtc) (c2:rtc): rtc ≝
23 [ mk_rtc ri1 rs1 ti1 ts1 ⇒
25 [ mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1+ri2,rs1+rs2,ti1+ti2,ts1+ts2〉
31 'plus c1 c2 = (rtc_plus c1 c2).
33 (* Basic constructions ******************************************************)
35 lemma rtc_plus_rew (ri1) (ri2) (rs1) (rs2) (ti1) (ti2) (ts1) (ts2):
36 〈ri1+ri2,rs1+rs2,ti1+ti2,ts1+ts2〉 = 〈ri1,rs1,ti1,ts1〉+〈ri2,rs2,ti2,ts2〉.
39 lemma rtc_plus_zz_dx (c): c = c + 𝟘𝟘.
40 * #ri #rs #ti #ts <rtc_plus_rew //
43 (* Basic inversions *********************************************************)
45 lemma rtc_plus_inv_dx (ri) (rs) (ti) (ts) (c1) (c2):
46 〈ri,rs,ti,ts〉 = c1 + c2 →
47 ∃∃ri1,rs1,ti1,ts1,ri2,rs2,ti2,ts2.
48 ri1+ri2 = ri & rs1+rs2 = rs & ti1+ti2 = ti & ts1+ts2 = ts &
49 〈ri1,rs1,ti1,ts1〉 = c1 & 〈ri2,rs2,ti2,ts2〉 = c2.
50 #ri #rs #ti #ts * #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2
51 <rtc_plus_rew #H destruct /2 width=14 by ex6_8_intro/
54 (* Main constructions *******************************************************)
56 theorem rtc_plus_assoc: associative … rtc_plus.
57 * #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2 * #ri3 #rs3 #ti3 #ts3