:: deftheorem defines is_primitive-recursively_expressed_by COMPUT_1:def 9 :
for g, f1, f2 being NAT * -defined to-naturals homogeneous Function
for i being Element of NAT holds
( g is_primitive-recursively_expressed_by f1,f2,i iff ex n being Element of NAT st
( dom g c= n -tuples_on NAT & i >= 1 & i <= n & (arity f1) + 1 = n & n + 1 = arity f2 & ( for p being FinSequence of NAT st len p = n holds
( ( p +* (i,0) in dom g implies Del (p,i) in dom f1 ) & ( Del (p,i) in dom f1 implies p +* (i,0) in dom g ) & ( p +* (i,0) in dom g implies g . (p +* (i,0)) = f1 . (Del (p,i)) ) & ( for n being Nat holds
( ( p +* (i,(n + 1)) in dom g implies ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 ) ) & ( p +* (i,n) in dom g & (p +* (i,n)) ^ <*(g . (p +* (i,n)))*> in dom f2 implies p +* (i,(n + 1)) in dom g ) & ( p +* (i,(n + 1)) in dom g implies g . (p +* (i,(n + 1))) = f2 . ((p +* (i,n)) ^ <*(g . (p +* (i,n)))*>) ) ) ) ) ) ) );