(* Title: CCL/ex/Nat.thy
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section {* Programs defined over the natural numbers *}
theory Nat
imports "../Wfd"
begin
definition not :: "i=>i"
where "not(b) == if b then false else true"
definition add :: "[i,i]=>i" (infixr "#+" 60)
where "a #+ b == nrec(a,b,%x g. succ(g))"
definition mult :: "[i,i]=>i" (infixr "#*" 60)
where "a #* b == nrec(a,zero,%x g. b #+ g)"
definition sub :: "[i,i]=>i" (infixr "#-" 60)
where
"a #- b ==
letrec sub x y be ncase(y,x,%yy. ncase(x,zero,%xx. sub(xx,yy)))
in sub(a,b)"
definition le :: "[i,i]=>i" (infixr "#<=" 60)
where
"a #<= b ==
letrec le x y be ncase(x,true,%xx. ncase(y,false,%yy. le(xx,yy)))
in le(a,b)"
definition lt :: "[i,i]=>i" (infixr "#<" 60)
where "a #< b == not(b #<= a)"
definition div :: "[i,i]=>i" (infixr "##" 60)
where
"a ## b ==
letrec div x y be if x #< y then zero else succ(div(x#-y,y))
in div(a,b)"
definition ackermann :: "[i,i]=>i"
where
"ackermann(a,b) ==
letrec ack n m be ncase(n,succ(m),%x.
ncase(m,ack(x,succ(zero)),%y. ack(x,ack(succ(x),y))))
in ack(a,b)"
lemmas nat_defs = not_def add_def mult_def sub_def le_def lt_def ackermann_def napply_def
lemma natBs [simp]:
"not(true) = false"
"not(false) = true"
"zero #+ n = n"
"succ(n) #+ m = succ(n #+ m)"
"zero #* n = zero"
"succ(n) #* m = m #+ (n #* m)"
"f^zero`a = a"
"f^succ(n)`a = f(f^n`a)"
by (simp_all add: nat_defs)
lemma napply_f: "n:Nat ==> f^n`f(a) = f^succ(n)`a"
apply (erule Nat_ind)
apply simp_all
done
lemma addT: "[| a:Nat; b:Nat |] ==> a #+ b : Nat"
apply (unfold add_def)
apply typechk
done
lemma multT: "[| a:Nat; b:Nat |] ==> a #* b : Nat"
apply (unfold add_def mult_def)
apply typechk
done
(* Defined to return zero if a<b *)
lemma subT: "[| a:Nat; b:Nat |] ==> a #- b : Nat"
apply (unfold sub_def)
apply typechk
apply clean_ccs
apply (erule NatPRI [THEN wfstI, THEN NatPR_wf [THEN wmap_wf, THEN wfI]])
done
lemma leT: "[| a:Nat; b:Nat |] ==> a #<= b : Bool"
apply (unfold le_def)
apply typechk
apply clean_ccs
apply (erule NatPRI [THEN wfstI, THEN NatPR_wf [THEN wmap_wf, THEN wfI]])
done
lemma ltT: "[| a:Nat; b:Nat |] ==> a #< b : Bool"
apply (unfold not_def lt_def)
apply (typechk leT)
done
subsection {* Termination Conditions for Ackermann's Function *}
lemmas relI = NatPR_wf [THEN NatPR_wf [THEN lex_wf, THEN wfI]]
lemma "[| a:Nat; b:Nat |] ==> ackermann(a,b) : Nat"
apply (unfold ackermann_def)
apply gen_ccs
apply (erule NatPRI [THEN lexI1 [THEN relI]] NatPRI [THEN lexI2 [THEN relI]])+
done
end