(* Title: Sequents/LK/Nat.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
*)
header {* Theory of the natural numbers: Peano's axioms, primitive recursion *}
theory Nat
imports LK
begin
typedecl nat
arities nat :: "term"
consts "0" :: nat ("0")
Suc :: "nat=>nat"
rec :: "[nat, 'a, [nat,'a]=>'a] => 'a"
"+" :: "[nat, nat] => nat" (infixl 60)
axioms
induct: "[| $H |- $E, P(0), $F;
!!x. $H, P(x) |- $E, P(Suc(x)), $F |] ==> $H |- $E, P(n), $F"
Suc_inject: "|- Suc(m)=Suc(n) --> m=n"
Suc_neq_0: "|- Suc(m) ~= 0"
rec_0: "|- rec(0,a,f) = a"
rec_Suc: "|- rec(Suc(m), a, f) = f(m, rec(m,a,f))"
add_def: "m+n == rec(m, n, %x y. Suc(y))"
ML {* use_legacy_bindings (the_context ()) *}
end