(* Title: ZF/ex/Primrec.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Primitive Recursive Functions
Proof adopted from
Nora Szasz,
A Machine Checked Proof that Ackermann's Function is not Primitive Recursive,
In: Huet & Plotkin, eds., Logical Environments (CUP, 1993), 317-338.
See also E. Mendelson, Introduction to Mathematical Logic.
(Van Nostrand, 1964), page 250, exercise 11.
*)
Primrec = List +
consts
primrec :: i
SC :: i
CONST :: i=>i
PROJ :: i=>i
COMP :: [i,i]=>i
PREC :: [i,i]=>i
ACK :: i=>i
ack :: [i,i]=>i
translations
"ack(x,y)" == "ACK(x) ` [y]"
defs
SC_def "SC == lam l:list(nat).list_case(0, %x xs.succ(x), l)"
CONST_def "CONST(k) == lam l:list(nat).k"
PROJ_def "PROJ(i) == lam l:list(nat). list_case(0, %x xs.x, drop(i,l))"
COMP_def "COMP(g,fs) == lam l:list(nat). g ` map(%f. f`l, fs)"
(*Note that g is applied first to PREC(f,g)`y and then to y!*)
PREC_def "PREC(f,g) ==
lam l:list(nat). list_case(0,
%x xs. rec(x, f`xs, %y r. g ` Cons(r, Cons(y, xs))), l)"
ACK_def "ACK(i) == rec(i, SC,
%z r. PREC (CONST (r`[1]), COMP(r,[PROJ(0)])))"
inductive
domains "primrec" <= "list(nat)->nat"
intrs
SC "SC : primrec"
CONST "k: nat ==> CONST(k) : primrec"
PROJ "i: nat ==> PROJ(i) : primrec"
COMP "[| g: primrec; fs: list(primrec) |] ==> COMP(g,fs): primrec"
PREC "[| f: primrec; g: primrec |] ==> PREC(f,g): primrec"
monos "[list_mono]"
con_defs "[SC_def,CONST_def,PROJ_def,COMP_def,PREC_def]"
type_intrs "nat_typechecks @ list.intrs @
[lam_type, list_case_type, drop_type, map_type,
apply_type, rec_type]"
end