# Theory Nat

theory Nat
imports Wfd
```(*  Title:      CCL/ex/Nat.thy
Author:     Martin Coen, Cambridge University Computer Laboratory
*)

header {* 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)"

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 (tactic {* typechk_tac @{context} [] 1 *})
done

lemma multT: "[| a:Nat;  b:Nat |] ==> a #* b : Nat"
apply (tactic {* typechk_tac @{context} [] 1 *})
done

(* Defined to return zero if a<b *)
lemma subT: "[| a:Nat;  b:Nat |] ==> a #- b : Nat"
apply (unfold sub_def)
apply (tactic {* typechk_tac @{context} [] 1 *})
apply (tactic {* clean_ccs_tac @{context} *})
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 (tactic {* typechk_tac @{context} [] 1 *})
apply (tactic {* clean_ccs_tac @{context} *})
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 (tactic {* typechk_tac @{context} @{thms leT} 1 *})
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 (tactic {* gen_ccs_tac @{context} [] 1 *})
apply (erule NatPRI [THEN lexI1 [THEN relI]] NatPRI [THEN lexI2 [THEN relI]])+
done

end
```