# Theory Nat

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

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, λx g. succ(g))"

definition mult :: "[i,i]⇒i"  (infixr "#*" 60)
where "a #* b == nrec(a, zero, λx g. b #+ g)"

definition sub :: "[i,i]⇒i"  (infixr "#-" 60)
where
"a #- b ==
letrec sub x y be ncase(y, x, λyy. ncase(x, zero, λxx. 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, λxx. ncase(y, false, λyy. 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), λx.
ncase(m,ack(x,succ(zero)), λy. 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 typechk
done

lemma multT: "⟦a:Nat; b:Nat⟧ ⟹ a #* b : Nat"
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
```