1473

1 
(* Title: FOL/ex/nat.thy

0

2 
ID: $Id$

1473

3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory

0

4 
Copyright 1992 University of Cambridge


5 


6 
Examples for the manual "Introduction to Isabelle"


7 


8 
Theory of the natural numbers: Peano's axioms, primitive recursion


9 


10 
INCOMPATIBLE with nat2.thy, Nipkow's example


11 
*)


12 


13 
Nat = FOL +

352

14 
types nat


15 
arities nat :: term

1322

16 
consts "0" :: nat ("0")


17 
Suc :: nat=>nat


18 
rec :: [nat, 'a, [nat,'a]=>'a] => 'a


19 
"+" :: [nat, nat] => nat (infixl 60)

0

20 
rules induct "[ P(0); !!x. P(x) ==> P(Suc(x)) ] ==> P(n)"


21 
Suc_inject "Suc(m)=Suc(n) ==> m=n"


22 
Suc_neq_0 "Suc(m)=0 ==> R"


23 
rec_0 "rec(0,a,f) = a"


24 
rec_Suc "rec(Suc(m), a, f) = f(m, rec(m,a,f))"


25 
add_def "m+n == rec(m, n, %x y. Suc(y))"


26 
end
