details

property	value
status	complete
benchmark	AotoYamada_05__020.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n151.star.cs.uiowa.edu
space	Uncurried_Applicative_11

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.173213 seconds
cpu usage	0.173483
user time	0.150155
system time	0.023328
max virtual memory	113188.0
max residence set size	6596.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: 0 : [] --> a comp : [b -> b * b -> b] --> b -> b plus : [a * a] --> a s : [a] --> a times : [a * a] --> a twice : [b -> b] --> b -> b Rules: plus(0, x) => x plus(s(x), y) => s(plus(x, y)) times(0, x) => 0 times(s(x), y) => plus(times(x, y), y) comp(f, g) x => f (g x) twice(f) => comp(f, f) 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]): plus(0, X) >? X plus(s(X), Y) >? s(plus(X, Y)) times(0, X) >? 0 times(s(X), Y) >? plus(times(X, Y), Y) comp(F, G) X >? F (G X) twice(F) >? comp(F, F) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ We choose Lex = {} and Mul = {@_{o -> o}, comp, plus, s, times, twice}, and the following precedence: times > plus > s > twice > comp > @_{o -> o} Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: plus(_|_, X) >= X plus(s(X), Y) >= s(plus(X, Y)) times(_|_, X) >= _|_ times(s(X), Y) > plus(times(X, Y), Y) @_{o -> o}(comp(F, G), X) > @_{o -> o}(F, @_{o -> o}(G, X)) twice(F) >= comp(F, F) With these choices, we have: 1] plus(_|_, X) >= X because [2], by (Star) 2] plus*(_|_, X) >= X because [3], by (Select) 3] X >= X by (Meta) 4] plus(s(X), Y) >= s(plus(X, Y)) because [5], by (Star) 5] plus*(s(X), Y) >= s(plus(X, Y)) because plus > s and [6], by (Copy) 6] plus*(s(X), Y) >= plus(X, Y) because plus in Mul, [7] and [10], by (Stat) 7] s(X) > X because [8], by definition 8] s*(X) >= X because [9], by (Select) 9] X >= X by (Meta) 10] Y >= Y by (Meta) 11] times(_|_, X) >= _|_ by (Bot) 12] times(s(X), Y) > plus(times(X, Y), Y) because [13], by definition 13] times*(s(X), Y) >= plus(times(X, Y), Y) because times > plus, [14] and [19], by (Copy) 14] times*(s(X), Y) >= times(X, Y) because times in Mul, [15] and [18], by (Stat) 15] s(X) > X because [16], by definition 16] s*(X) >= X because [17], by (Select) 17] X >= X by (Meta) 18] Y >= Y by (Meta) 19] times*(s(X), Y) >= Y because [18], by (Select) 20] @_{o -> o}(comp(F, G), X) > @_{o -> o}(F, @_{o -> o}(G, X)) because [21], by definition 21] @_{o -> o}*(comp(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X)) because [22], by (Select) 22] comp(F, G) @_{o -> o}*(comp(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X)) because [23] 23] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= @_{o -> o}(F, @_{o -> o}(G, X)) because comp > @_{o -> o}, [24] and [26], by (Copy) 24] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= F because [25], by (Select) 25] F >= F by (Meta) 26] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= @_{o -> o}(G, X) because comp > @_{o -> o}, [27] and [29], by (Copy) 27] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= G because [28], by (Select) 28] G >= G by (Meta) 29] comp*(F, G, @_{o -> o}*(comp(F, G), X)) >= X because [30], by (Select) 30] @_{o -> o}*(comp(F, G), X) >= X because [31], by (Select) 31] X >= X by (Meta) 32] twice(F) >= comp(F, F) because [33], by (Star) 33] twice*(F) >= comp(F, F) because twice > comp, [34] and [34], by (Copy) 34] twice*(F) >= F because [35], by (Select) 35] F >= F by (Meta) We can thus remove the following rules: times(s(X), Y) => plus(times(X, Y), Y) comp(F, G) X => F (G X) We use rule removal, following [Kop12, Theorem 2.23]. popout

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all output return to Higher Order Rewriting Union Beta