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