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1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||         The HELM team.                                      *)
8 (*      ||A||         http://helm.cs.unibo.it                             *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 include "ground_2/notation/functions/tuple_4.ma".
16 include "ground_2/notation/functions/zerozero_0.ma".
17 include "ground_2/notation/functions/zeroone_0.ma".
18 include "ground_2/notation/functions/onezero_0.ma".
19 include "ground_2/lib/arith.ma".
20
21 (* RT-TRANSITION COUNTER ****************************************************)
22
23 record rtc: Type[0] ≝ {
24    ri: nat; (* Note: inner r-steps *)
25    rs: nat; (* Note: spine r-steps *)
26    ti: nat; (* Note: inner t-steps *)
27    ts: nat  (* Note: spine t-steps *)
28 }.
29
30 interpretation "constructor (rtc)"
31    'Tuple ri rs ti ts = (mk_rtc ri rs ti ts).
32
33 interpretation "one structural step (rtc)"
34    'ZeroZero = (mk_rtc O O O O).
35
36 interpretation "one r-step (rtc)"
37    'OneZero = (mk_rtc O (S O) O O).
38
39 interpretation "one t-step (rtc)"
40    'ZeroOne = (mk_rtc O O O (S O)).
41
42 definition eq_f: relation rtc ≝ λc1,c2. ⊤.
43
44 inductive eq_t: relation rtc ≝
45 | eq_t_intro: ∀ri1,ri2,rs1,rs2,ti,ts.
46               eq_t (〈ri1,rs1,ti,ts〉) (〈ri2,rs2,ti,ts〉)
47 .
48
49 (* Basic properties *********************************************************)
50
51 lemma eq_t_refl: reflexive …  eq_t.
52 * // qed.
53
54 (* Basic inversion lemmas ***************************************************)
55
56 fact eq_t_inv_dx_aux: ∀x,y. eq_t x y →
57                       ∀ri1,rs1,ti,ts. x = 〈ri1,rs1,ti,ts〉 →
58                       ∃∃ri2,rs2. y = 〈ri2,rs2,ti,ts〉.
59 #x #y * -x -y
60 #ri1 #ri #rs1 #rs #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #H destruct -ri2 -rs2
61 /2 width=3 by ex1_2_intro/
62 qed-.
63
64 lemma eq_t_inv_dx: ∀ri1,rs1,ti,ts,y. eq_t (〈ri1,rs1,ti,ts〉) y →
65                    ∃∃ri2,rs2. y = 〈ri2,rs2,ti,ts〉.
66 /2 width=5 by eq_t_inv_dx_aux/ qed-.