(* Title: HOL/ex/Recdef.thy
ID: $Id$
Author: Konrad Slind and Lawrence C Paulson
Copyright 1996 University of Cambridge
Examples of recdef definitions. Most, but not all, are handled automatically.
*)
Recdefs = Main +
consts fact :: "nat => nat"
recdef fact "less_than"
"fact x = (if (x = 0) then 1 else x * fact (x - 1))"
consts Fact :: "nat => nat"
recdef Fact "less_than"
"Fact 0 = 1"
"Fact (Suc x) = (Fact x * Suc x)"
consts map2 :: "('a => 'b => 'c) * 'a list * 'b list => 'c list"
recdef map2 "measure(%(f,l1,l2).size l1)"
"map2(f, [], []) = []"
"map2(f, h#t, []) = []"
"map2(f, h1#t1, h2#t2) = f h1 h2 # map2 (f, t1, t2)"
consts finiteRchain :: "(['a,'a] => bool) * 'a list => bool"
recdef finiteRchain "measure (%(R,l).size l)"
"finiteRchain(R, []) = True"
"finiteRchain(R, [x]) = True"
"finiteRchain(R, x#y#rst) = (R x y & finiteRchain(R, y#rst))"
consts qsort ::"('a => 'a => bool) * 'a list => 'a list"
recdef qsort "measure (size o snd)"
simpset "simpset() addsimps [less_Suc_eq_le, length_filter]"
"qsort(ord, []) = []"
"qsort(ord, x#rst) = qsort(ord, filter(Not o ord x) rst)
@ [x] @
qsort(ord, filter(ord x) rst)"
(*Not handled automatically: too complicated.*)
consts variant :: "nat * nat list => nat"
recdef variant "measure(%(n::nat, ns). size(filter(%y. n <= y) ns))"
"variant(x, L) = (if (x mem L) then variant(Suc x, L) else x)"
consts gcd :: "nat * nat => nat"
recdef gcd "measure (%(x,y).x+y)"
simpset "simpset() addsimps [less_Suc_eq_le, le_add1, diff_le_self]"
"gcd (0,y) = y"
"gcd (Suc x, 0) = Suc x"
"gcd (Suc x, Suc y) = (if (y <= x) then gcd(x - y, Suc y)
else gcd(Suc x, y - x))"
(*Not handled automatically. In fact, g is the zero constant function.*)
consts g :: "nat => nat"
recdef g "less_than"
"g 0 = 0"
"g(Suc x) = g(g x)"
consts Div :: "nat * nat => nat * nat"
recdef Div "measure fst"
"Div(0,x) = (0,0)"
"Div(Suc x, y) =
(let (q,r) = Div(x,y)
in
if (y <= Suc r) then (Suc q,0) else (q, Suc r))"
(*Not handled automatically. Should be the predecessor function, but there
is an unnecessary "looping" recursive call in k(1) *)
consts k :: "nat => nat"
recdef k "less_than"
"k 0 = 0"
"k (Suc n) = (let x = k 1
in if (0=1) then k (Suc 1) else n)"
consts part :: "('a=>bool) * 'a list * 'a list * 'a list => 'a list * 'a list"
recdef part "measure (%(P,l,l1,l2).size l)"
"part(P, [], l1,l2) = (l1,l2)"
"part(P, h#rst, l1,l2) =
(if P h then part(P,rst, h#l1, l2)
else part(P,rst, l1, h#l2))"
consts fqsort :: "(['a,'a] => bool) * 'a list => 'a list"
recdef fqsort "measure (size o snd)"
"fqsort(ord,[]) = []"
"fqsort(ord, x#rst) =
(let (less,more) = part((%y. ord y x), rst, ([],[]))
in
fqsort(ord,less)@[x]@fqsort(ord,more))"
(* silly example which demonstrates the occasional need for additional
congruence rules (here: map_cong from List). If the congruence rule is
removed, an unprovable termination condition is generated!
Termination not proved automatically; see the ML file.
TFL requires (%x.mapf x) instead of mapf.
*)
consts mapf :: nat => nat list
recdef mapf "measure(%m. m)"
congs "[map_cong]"
"mapf 0 = []"
"mapf (Suc n) = concat(map (%x. mapf x) (replicate n n))"
end