details

property	value
status	complete
benchmark	sort.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n146.star.cs.uiowa.edu
space	Mixed_HO_10

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	3.25007 seconds
cpu usage	3.25005
user time	3.11927
system time	0.130782
max virtual memory	184800.0
max residence set size	68348.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: 0 : [] --> nat ascending!6220sort : [list] --> list cons : [nat * list] --> list descending!6220sort : [list] --> list insert : [nat * list * nat -> nat -> nat * nat -> nat -> nat] --> list max : [nat * nat] --> nat min : [nat * nat] --> nat nil : [] --> list s : [nat] --> nat sort : [list * nat -> nat -> nat * nat -> nat -> nat] --> list Rules: max(0, x) => x max(x, 0) => x max(s(x), s(y)) => s(max(x, y)) min(0, x) => 0 min(x, 0) => 0 min(s(x), s(y)) => s(min(x, y)) insert(x, nil, f, g) => cons(x, nil) insert(x, cons(y, z), f, g) => cons(f x y, insert(g x y, z, f, g)) sort(nil, f, g) => nil sort(cons(x, y), f, g) => insert(x, sort(y, f, g), f, g) ascending!6220sort(x) => sort(x, /\y./\z.min(y, z), /\u./\v.max(u, v)) descending!6220sort(x) => sort(x, /\y./\z.max(y, z), /\u./\v.min(u, v)) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): max(0, X) >? X max(X, 0) >? X max(s(X), s(Y)) >? s(max(X, Y)) min(0, X) >? 0 min(X, 0) >? 0 min(s(X), s(Y)) >? s(min(X, Y)) insert(X, nil, F, G) >? cons(X, nil) insert(X, cons(Y, Z), F, G) >? cons(F X Y, insert(G X Y, Z, F, G)) sort(nil, F, G) >? nil sort(cons(X, Y), F, G) >? insert(X, sort(Y, F, G), F, G) ascending!6220sort(X) >? sort(X, /\x./\y.min(x, y), /\z./\u.max(z, u)) descending!6220sort(X) >? sort(X, /\x./\y.max(x, y), /\z./\u.min(z, u)) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ [[insert(x_1, x_2, x_3, x_4)]] = insert(x_2, x_4, x_1, x_3) [[nil]] = _|_ We choose Lex = {insert} and Mul = {@_{o -> o -> o}, @_{o -> o}, ascending!6220sort, cons, descending!6220sort, max, min, s, sort}, and the following precedence: ascending!6220sort > descending!6220sort > max > min > s > sort > insert > @_{o -> o -> o} > @_{o -> o} > cons Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: max(_|_, X) > X max(X, _|_) >= X max(s(X), s(Y)) >= s(max(X, Y)) min(_|_, X) >= _|_ min(X, _|_) >= _|_ min(s(X), s(Y)) >= s(min(X, Y)) insert(X, _|_, F, G) >= cons(X, _|_) insert(X, cons(Y, Z), F, G) >= cons(@_{o -> o}(@_{o -> o -> o}(F, X), Y), insert(@_{o -> o}(@_{o -> o -> o}(G, X), Y), Z, F, G)) sort(_|_, F, G) >= _|_ sort(cons(X, Y), F, G) >= insert(X, sort(Y, F, G), F, G) ascending!6220sort(X) >= sort(X, /\x./\y.min(x, y), /\z./\u.max(z, u)) descending!6220sort(X) >= sort(X, /\x./\y.max(x, y), /\z./\u.min(z, u)) With these choices, we have: 1] max(_|_, X) > X because [2], by definition 2] max*(_|_, X) >= X because [3], by (Select) 3] X >= X by (Meta) 4] max(X, _|_) >= X because [5], by (Star) 5] max*(X, _|_) >= X because [6], by (Select) 6] X >= X by (Meta) 7] max(s(X), s(Y)) >= s(max(X, Y)) because [8], by (Star) 8] max*(s(X), s(Y)) >= s(max(X, Y)) because max > s and [9], by (Copy) 9] max*(s(X), s(Y)) >= max(X, Y) because max in Mul, [10] and [13], by (Stat) 10] s(X) >= X because [11], by (Star) 11] s*(X) >= X because [12], by (Select) 12] X >= X by (Meta) 13] s(Y) > Y because [14], by definition 14] s*(Y) >= Y because [15], by (Select) 15] Y >= Y by (Meta) 16] min(_|_, X) >= _|_ by (Bot) 17] min(X, _|_) >= _|_ by (Bot) 18] min(s(X), s(Y)) >= s(min(X, Y)) because [19], by (Star) popout

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actions

all output return to Higher Order Rewriting Union Beta