src/ZF/ex/Primrec.thy
author clasohm
Tue Feb 06 12:27:17 1996 +0100 (1996-02-06)
changeset 1478 2b8c2a7547ab
parent 1401 0c439768f45c
child 3840 e0baea4d485a
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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