|
adamc@620
|
1 (* Copyright (c) 2009, Adam Chlipala
|
|
adamc@620
|
2 * All rights reserved.
|
|
adamc@620
|
3 *
|
|
adamc@620
|
4 * Redistribution and use in source and binary forms, with or without
|
|
adamc@620
|
5 * modification, are permitted provided that the following conditions are met:
|
|
adamc@620
|
6 *
|
|
adamc@620
|
7 * - Redistributions of source code must retain the above copyright notice,
|
|
adamc@620
|
8 * this list of conditions and the following disclaimer.
|
|
adamc@620
|
9 * - Redistributions in binary form must reproduce the above copyright notice,
|
|
adamc@620
|
10 * this list of conditions and the following disclaimer in the documentation
|
|
adamc@620
|
11 * and/or other materials provided with the distribution.
|
|
adamc@620
|
12 * - The names of contributors may not be used to endorse or promote products
|
|
adamc@620
|
13 * derived from this software without specific prior written permission.
|
|
adamc@620
|
14 *
|
|
adamc@620
|
15 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
|
|
adamc@620
|
16 * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
|
adamc@620
|
17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
|
adamc@620
|
18 * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
|
|
adamc@620
|
19 * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
|
|
adamc@620
|
20 * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
|
|
adamc@620
|
21 * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
|
|
adamc@620
|
22 * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
|
|
adamc@620
|
23 * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
|
|
adamc@620
|
24 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
|
|
adamc@620
|
25 * POSSIBILITY OF SUCH DAMAGE.
|
|
adamc@620
|
26 *)
|
|
adamc@620
|
27
|
|
adamc@620
|
28 Set Implicit Arguments.
|
|
adamc@620
|
29
|
|
adamc@620
|
30
|
|
adamc@620
|
31 Fixpoint name' (n : nat) : Type :=
|
|
adamc@620
|
32 match n with
|
|
adamc@620
|
33 | O => Empty_set
|
|
adamc@620
|
34 | S n' => option (name' n')
|
|
adamc@620
|
35 end.
|
|
adamc@620
|
36
|
|
adamc@620
|
37 Definition name'_eq_dec : forall n (x y : name' n), {x = y} + {x <> y}.
|
|
adamc@620
|
38 Hint Extern 1 (_ = _ -> False) => congruence.
|
|
adamc@620
|
39
|
|
adamc@620
|
40 induction n; simpl; intuition;
|
|
adamc@620
|
41 repeat match goal with
|
|
adamc@620
|
42 | [ x : Empty_set |- _ ] => destruct x
|
|
adamc@620
|
43 | [ x : option _ |- _ ] => destruct x
|
|
adamc@620
|
44 end; intuition;
|
|
adamc@620
|
45 match goal with
|
|
adamc@620
|
46 | [ IH : _, n1 : name' _, n2 : name' _ |- _ ] =>
|
|
adamc@620
|
47 destruct (IHn n1 n0); subst; intuition
|
|
adamc@620
|
48 end.
|
|
adamc@620
|
49 Qed.
|
|
adamc@620
|
50
|
|
adamc@620
|
51 Definition badName' n (P : name' n -> bool) :
|
|
adamc@620
|
52 {nm : _ | P nm = false} + {forall nm, P nm = true}.
|
|
adamc@620
|
53 Hint Constructors sig.
|
|
adamc@620
|
54 Hint Extern 1 (_ = true) =>
|
|
adamc@620
|
55 match goal with
|
|
adamc@620
|
56 | [ nm : option _ |- _ ] => destruct nm
|
|
adamc@620
|
57 end; auto.
|
|
adamc@620
|
58
|
|
adamc@620
|
59 induction n; simpl; intuition;
|
|
adamc@620
|
60 match goal with
|
|
adamc@620
|
61 | [ IH : forall P : _ -> _,_ |- _ ] =>
|
|
adamc@620
|
62 case_eq (P None);
|
|
adamc@620
|
63 destruct (IH (fun nm => P (Some nm))); firstorder eauto
|
|
adamc@620
|
64 end.
|
|
adamc@620
|
65 Qed.
|
|
adamc@620
|
66
|
|
adamc@620
|
67 Parameter numNames : nat.
|
|
adamc@620
|
68 Definition name := name' (S numNames).
|
|
adamc@620
|
69 Definition name_eq_dec : forall (x y : name), {x = y} + {x <> y} := @name'_eq_dec _.
|
|
adamc@620
|
70 Definition badName : forall P : name -> bool, {nm : _ | P nm = false} + {forall nm, P nm = true} := @badName' _.
|
|
adamc@620
|
71 Definition defaultName : name := None.
|
