17481
|
1 |
(* Title: Sequents/LK/Nat.thy
|
7091
|
2 |
ID: $Id$
|
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
|
7095
|
4 |
Copyright 1999 University of Cambridge
|
7091
|
5 |
*)
|
|
6 |
|
17481
|
7 |
header {* Theory of the natural numbers: Peano's axioms, primitive recursion *}
|
|
8 |
|
|
9 |
theory Nat
|
|
10 |
imports LK
|
|
11 |
begin
|
|
12 |
|
|
13 |
typedecl nat
|
|
14 |
arities nat :: "term"
|
7091
|
15 |
consts "0" :: nat ("0")
|
17481
|
16 |
Suc :: "nat=>nat"
|
|
17 |
rec :: "[nat, 'a, [nat,'a]=>'a] => 'a"
|
|
18 |
"+" :: "[nat, nat] => nat" (infixl 60)
|
7091
|
19 |
|
17481
|
20 |
axioms
|
|
21 |
induct: "[| $H |- $E, P(0), $F;
|
7123
|
22 |
!!x. $H, P(x) |- $E, P(Suc(x)), $F |] ==> $H |- $E, P(n), $F"
|
7095
|
23 |
|
17481
|
24 |
Suc_inject: "|- Suc(m)=Suc(n) --> m=n"
|
|
25 |
Suc_neq_0: "|- Suc(m) ~= 0"
|
|
26 |
rec_0: "|- rec(0,a,f) = a"
|
|
27 |
rec_Suc: "|- rec(Suc(m), a, f) = f(m, rec(m,a,f))"
|
|
28 |
add_def: "m+n == rec(m, n, %x y. Suc(y))"
|
|
29 |
|
|
30 |
ML {* use_legacy_bindings (the_context ()) *}
|
|
31 |
|
7091
|
32 |
end
|