(*  Title:      CCL/ex/List.thy
    ID:         $Id: List.thy,v 1.6 2005/09/17 15:35:31 wenzelm Exp $
    Author:     Martin Coen, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge
*)

header {* Programs defined over lists *}

theory List
imports Nat
begin

consts
  map       :: "[i=>i,i]=>i"
  "o"       :: "[i=>i,i=>i]=>i=>i"             (infixr 55)
  "@"       :: "[i,i]=>i"             (infixr 55)
  mem       :: "[i,i]=>i"             (infixr 55)
  filter    :: "[i,i]=>i"
  flat      :: "i=>i"
  partition :: "[i,i]=>i"
  insert    :: "[i,i,i]=>i"
  isort     :: "i=>i"
  qsort     :: "i=>i"

axioms

  map_def:     "map(f,l)   == lrec(l,[],%x xs g. f(x)$g)"
  comp_def:    "f o g == (%x. f(g(x)))"
  append_def:  "l @ m == lrec(l,m,%x xs g. x$g)"
  mem_def:     "a mem l == lrec(l,false,%h t g. if eq(a,h) then true else g)"
  filter_def:  "filter(f,l) == lrec(l,[],%x xs g. if f`x then x$g else g)"
  flat_def:    "flat(l) == lrec(l,[],%h t g. h @ g)"

  insert_def:  "insert(f,a,l) == lrec(l,a$[],%h t g. if f`a`h then a$h$t else h$g)"
  isort_def:   "isort(f) == lam l. lrec(l,[],%h t g. insert(f,h,g))"

  partition_def:
  "partition(f,l) == letrec part l a b be lcase(l,<a,b>,%x xs.
                            if f`x then part(xs,x$a,b) else part(xs,a,x$b))
                    in part(l,[],[])"
  qsort_def:   "qsort(f) == lam l. letrec qsortx l be lcase(l,[],%h t.
                                   let p be partition(f`h,t)
                                   in split(p,%x y. qsortx(x) @ h$qsortx(y)))
                          in qsortx(l)"

ML {* use_legacy_bindings (the_context ()) *}

end