details

property	value
status	complete
benchmark	Applicative_05__ReverseLastInit.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n144.star.cs.uiowa.edu
space	Uncurried_Applicative_11

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.443796 seconds
cpu usage	0.44167
user time	0.388852
system time	0.052818
max virtual memory	113188.0
max residence set size	15004.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: compose : [a -> a * a -> a] --> a -> a cons : [a * a] --> a hd : [] --> a -> a init : [] --> a -> a last : [] --> a -> a nil : [] --> a reverse : [] --> a -> a reverse2 : [a * a] --> a tl : [] --> a -> a Rules: compose(f, g) x => g (f x) reverse x => reverse2(x, nil) reverse2(nil, x) => x reverse2(cons(x, y), z) => reverse2(y, cons(x, z)) hd cons(x, y) => x tl cons(x, y) => y last => compose(hd, reverse) init => compose(reverse, compose(tl, reverse)) 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]): compose(F, G) X >? G (F X) reverse X >? reverse2(X, nil) reverse2(nil, X) >? X reverse2(cons(X, Y), Z) >? reverse2(Y, cons(X, Z)) hd cons(X, Y) >? X tl cons(X, Y) >? Y last >? compose(hd, reverse) init >? compose(reverse, compose(tl, reverse)) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[hd]] = _|_ [[nil]] = _|_ [[reverse]] = _|_ [[tl]] = _|_ We choose Lex = {reverse2} and Mul = {@_{o -> o}, compose, cons, init, last}, and the following precedence: init > last > compose > @_{o -> o} > reverse2 > cons Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: @_{o -> o}(compose(F, G), X) >= @_{o -> o}(G, @_{o -> o}(F, X)) @_{o -> o}(_|_, X) >= reverse2(X, _|_) reverse2(_|_, X) > X reverse2(cons(X, Y), Z) >= reverse2(Y, cons(X, Z)) @_{o -> o}(_|_, cons(X, Y)) > X @_{o -> o}(_|_, cons(X, Y)) > Y last >= compose(_|_, _|_) init > compose(_|_, compose(_|_, _|_)) With these choices, we have: 1] @_{o -> o}(compose(F, G), X) >= @_{o -> o}(G, @_{o -> o}(F, X)) because [2], by (Star) 2] @_{o -> o}*(compose(F, G), X) >= @_{o -> o}(G, @_{o -> o}(F, X)) because [3], by (Select) 3] compose(F, G) @_{o -> o}*(compose(F, G), X) >= @_{o -> o}(G, @_{o -> o}(F, X)) because [4] 4] compose*(F, G, @_{o -> o}*(compose(F, G), X)) >= @_{o -> o}(G, @_{o -> o}(F, X)) because compose > @_{o -> o}, [5] and [7], by (Copy) 5] compose*(F, G, @_{o -> o}*(compose(F, G), X)) >= G because [6], by (Select) 6] G >= G by (Meta) 7] compose*(F, G, @_{o -> o}*(compose(F, G), X)) >= @_{o -> o}(F, X) because compose > @_{o -> o}, [8] and [10], by (Copy) 8] compose*(F, G, @_{o -> o}*(compose(F, G), X)) >= F because [9], by (Select) 9] F >= F by (Meta) 10] compose*(F, G, @_{o -> o}*(compose(F, G), X)) >= X because [11], by (Select) 11] @_{o -> o}*(compose(F, G), X) >= X because [12], by (Select) 12] X >= X by (Meta) 13] @_{o -> o}(_|_, X) >= reverse2(X, _|_) because [14], by (Star) 14] @_{o -> o}*(_|_, X) >= reverse2(X, _|_) because @_{o -> o} > reverse2, [15] and [17], by (Copy) 15] @_{o -> o}*(_|_, X) >= X because [16], by (Select) 16] X >= X by (Meta) 17] @_{o -> o}*(_|_, X) >= _|_ by (Bot) 18] reverse2(_|_, X) > X because [19], by definition 19] reverse2*(_|_, X) >= X because [20], by (Select) 20] X >= X by (Meta) 21] reverse2(cons(X, Y), Z) >= reverse2(Y, cons(X, Z)) because [22], by (Star) 22] reverse2*(cons(X, Y), Z) >= reverse2(Y, cons(X, Z)) because [23], [26] and [28], by (Stat) 23] cons(X, Y) > Y because [24], by definition 24] cons*(X, Y) >= Y because [25], by (Select) 25] Y >= Y by (Meta) 26] reverse2*(cons(X, Y), Z) >= Y because [27], by (Select) 27] cons(X, Y) >= Y because [24], by (Star) 28] reverse2*(cons(X, Y), Z) >= cons(X, Z) because reverse2 > cons, [29] and [33], by (Copy) 29] reverse2*(cons(X, Y), Z) >= X because [30], by (Select) 30] cons(X, Y) >= X because [31], by (Star) 31] cons*(X, Y) >= X because [32], by (Select) 32] X >= X by (Meta) popout

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all output return to Higher Order Rewriting Union Beta