(* Title: FOLP/ex/nat.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)
header {* Theory of the natural numbers: Peano's axioms, primitive recursion *}
theory Nat
imports FOLP
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)
(*Proof terms*)
nrec :: "[nat,p,[nat,p]=>p]=>p"
ninj :: "p=>p"
nneq :: "p=>p"
rec0 :: "p"
recSuc :: "p"
axioms
induct: "[| b:P(0); !!x u. u:P(x) ==> c(x,u):P(Suc(x))
|] ==> nrec(n,b,c):P(n)"
Suc_inject: "p:Suc(m)=Suc(n) ==> ninj(p) : m=n"
Suc_neq_0: "p:Suc(m)=0 ==> nneq(p) : R"
rec_0: "rec0 : rec(0,a,f) = a"
rec_Suc: "recSuc : rec(Suc(m), a, f) = f(m, rec(m,a,f))"
add_def: "m+n == rec(m, n, %x y. Suc(y))"
nrecB0: "b: A ==> nrec(0,b,c) = b : A"
nrecBSuc: "c(n,nrec(n,b,c)) : A ==> nrec(Suc(n),b,c) = c(n,nrec(n,b,c)) : A"
ML {* use_legacy_bindings (the_context ()) *}
end