(* Title: FOLP/ex/nat.thy ID: $Id: Nat.thy,v 1.5 2005/09/18 12:25:49 wenzelm Exp $ 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