details

property	value
status	complete
benchmark	AotoYamada_05__014.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n147.star.cs.uiowa.edu
space	Uncurried_Applicative_11

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.666639 seconds
cpu usage	0.637339
user time	0.561627
system time	0.075712
max virtual memory	130448.0
max residence set size	16204.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: 0 : [] --> b cons : [b * a] --> a double : [] --> a -> a inc : [] --> a -> a map : [b -> b] --> a -> a nil : [] --> a plus : [b] --> b -> b s : [b] --> b times : [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 map(f) nil => nil map(f) cons(x, y) => cons(f x, map(f) y) inc => map(plus(s(0))) double => map(times(s(s(0)))) 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 map(F) nil >? nil map(F) cons(X, Y) >? cons(F X, map(F) Y) inc >? map(plus(s(0))) double >? map(times(s(s(0)))) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ [[nil]] = _|_ We choose Lex = {} and Mul = {@_{o -> o}, cons, double, inc, map, plus, s, times}, and the following precedence: double > inc > times > plus > @_{o -> o} > cons > map > s Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: @_{o -> o}(plus(_|_), X) >= X @_{o -> o}(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y)) @_{o -> o}(times(_|_), X) >= _|_ @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) @_{o -> o}(map(F), _|_) >= _|_ @_{o -> o}(map(F), cons(X, Y)) > cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y)) inc > map(plus(s(_|_))) double > map(times(s(s(_|_)))) With these choices, we have: 1] @_{o -> o}(plus(_|_), X) >= X because [2], by (Star) 2] @_{o -> o}*(plus(_|_), X) >= X because [3], by (Select) 3] X >= X by (Meta) 4] @_{o -> o}(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y)) because [5], by (Star) 5] @_{o -> o}*(plus(s(X)), Y) >= s(@_{o -> o}(plus(X), Y)) because @_{o -> o} > s and [6], by (Copy) 6] @_{o -> o}*(plus(s(X)), Y) >= @_{o -> o}(plus(X), Y) because [7], by (Select) 7] plus(s(X)) @_{o -> o}*(plus(s(X)), Y) >= @_{o -> o}(plus(X), Y) because [8] 8] plus*(s(X), @_{o -> o}*(plus(s(X)), Y)) >= @_{o -> o}(plus(X), Y) because plus > @_{o -> o}, [9] and [13], by (Copy) 9] plus*(s(X), @_{o -> o}*(plus(s(X)), Y)) >= plus(X) because plus in Mul and [10], by (Stat) 10] s(X) > X because [11], by definition 11] s*(X) >= X because [12], by (Select) 12] X >= X by (Meta) 13] plus*(s(X), @_{o -> o}*(plus(s(X)), Y)) >= Y because [14], by (Select) 14] @_{o -> o}*(plus(s(X)), Y) >= Y because [15], by (Select) 15] Y >= Y by (Meta) 16] @_{o -> o}(times(_|_), X) >= _|_ by (Bot) 17] @_{o -> o}(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) because [18], by (Star) 18] @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) because [19], by (Select) 19] times(s(X)) @_{o -> o}*(times(s(X)), Y) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) because [20] 20] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(plus(@_{o -> o}(times(X), Y)), Y) because times > @_{o -> o}, [21] and [27], by (Copy) 21] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= plus(@_{o -> o}(times(X), Y)) because times > plus and [22], by (Copy) 22] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= @_{o -> o}(times(X), Y) because times > @_{o -> o}, [23] and [27], by (Copy) 23] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= times(X) because times in Mul and [24], by (Stat) 24] s(X) > X because [25], by definition 25] s*(X) >= X because [26], by (Select) 26] X >= X by (Meta) 27] times*(s(X), @_{o -> o}*(times(s(X)), Y)) >= Y because [28], by (Select) 28] @_{o -> o}*(times(s(X)), Y) >= Y because [29], by (Select) 29] Y >= Y by (Meta) 30] @_{o -> o}(map(F), _|_) >= _|_ by (Bot) 31] @_{o -> o}(map(F), cons(X, Y)) > cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y)) because [32], by definition 32] @_{o -> o}*(map(F), cons(X, Y)) >= cons(@_{o -> o}(F, X), @_{o -> o}(map(F), Y)) because @_{o -> o} > cons, [33] and [43], by (Copy) popout

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actions

all output return to Higher Order Rewriting Union Beta