details

property	value
status	complete
benchmark	plode.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n139.star.cs.uiowa.edu
space	Mixed_HO_10

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.462039 seconds
cpu usage	0.46253
user time	0.422289
system time	0.040241
max virtual memory	113188.0
max residence set size	14308.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: cons : [nat * list] --> list explode : [list * nat -> nat * nat] --> nat implode : [list * nat -> nat * nat] --> nat nil : [] --> list op : [nat -> nat * nat -> nat] --> nat -> nat Rules: op(f, g) x => f (g x) implode(nil, f, x) => x implode(cons(x, y), f, z) => implode(y, f, f z) explode(nil, f, x) => x explode(cons(x, y), f, z) => explode(y, op(f, f), f z) 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]): op(F, G) X >? F (G X) implode(nil, F, X) >? X implode(cons(X, Y), F, Z) >? implode(Y, F, F Z) explode(nil, F, X) >? X explode(cons(X, Y), F, Z) >? explode(Y, op(F, F), F Z) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[explode(x_1, x_2, x_3)]] = explode(x_1, x_3, x_2) We choose Lex = {explode, implode} and Mul = {@_{o -> o}, cons, nil, op}, and the following precedence: cons > explode > nil > implode > op > @_{o -> o} Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: @_{o -> o}(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X)) implode(nil, F, X) >= X implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z)) explode(nil, F, X) > X explode(cons(X, Y), F, Z) > explode(Y, op(F, F), @_{o -> o}(F, Z)) With these choices, we have: 1] @_{o -> o}(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X)) because [2], by (Star) 2] @_{o -> o}*(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X)) because [3], by (Select) 3] op(F, G) @_{o -> o}*(op(F, G), X) >= @_{o -> o}(F, @_{o -> o}(G, X)) because [4] 4] op*(F, G, @_{o -> o}*(op(F, G), X)) >= @_{o -> o}(F, @_{o -> o}(G, X)) because op > @_{o -> o}, [5] and [7], by (Copy) 5] op*(F, G, @_{o -> o}*(op(F, G), X)) >= F because [6], by (Select) 6] F >= F by (Meta) 7] op*(F, G, @_{o -> o}*(op(F, G), X)) >= @_{o -> o}(G, X) because op > @_{o -> o}, [8] and [10], by (Copy) 8] op*(F, G, @_{o -> o}*(op(F, G), X)) >= G because [9], by (Select) 9] G >= G by (Meta) 10] op*(F, G, @_{o -> o}*(op(F, G), X)) >= X because [11], by (Select) 11] @_{o -> o}*(op(F, G), X) >= X because [12], by (Select) 12] op(F, G) @_{o -> o}*(op(F, G), X) >= X because [13] 13] op*(F, G, @_{o -> o}*(op(F, G), X)) >= X because [14], by (Select) 14] @_{o -> o}*(op(F, G), X) >= X because [15], by (Select) 15] X >= X by (Meta) 16] implode(nil, F, X) >= X because [17], by (Star) 17] implode*(nil, F, X) >= X because [18], by (Select) 18] X >= X by (Meta) 19] implode(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z)) because [20], by (Star) 20] implode*(cons(X, Y), F, Z) >= implode(Y, F, @_{o -> o}(F, Z)) because [21], [24], [26] and [28], by (Stat) 21] cons(X, Y) > Y because [22], by definition 22] cons*(X, Y) >= Y because [23], by (Select) 23] Y >= Y by (Meta) 24] implode*(cons(X, Y), F, Z) >= Y because [25], by (Select) 25] cons(X, Y) >= Y because [22], by (Star) 26] implode*(cons(X, Y), F, Z) >= F because [27], by (Select) 27] F >= F by (Meta) 28] implode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z) because implode > @_{o -> o}, [26] and [29], by (Copy) 29] implode*(cons(X, Y), F, Z) >= Z because [30], by (Select) 30] Z >= Z by (Meta) 31] explode(nil, F, X) > X because [32], by definition 32] explode*(nil, F, X) >= X because [33], by (Select) 33] X >= X by (Meta) 34] explode(cons(X, Y), F, Z) > explode(Y, op(F, F), @_{o -> o}(F, Z)) because [35], by definition 35] explode*(cons(X, Y), F, Z) >= explode(Y, op(F, F), @_{o -> o}(F, Z)) because [36], [39], [41] and [44], by (Stat) 36] cons(X, Y) > Y because [37], by definition 37] cons*(X, Y) >= Y because [38], by (Select) 38] Y >= Y by (Meta) 39] explode*(cons(X, Y), F, Z) >= Y because [40], by (Select) 40] cons(X, Y) >= Y because [37], by (Star) 41] explode*(cons(X, Y), F, Z) >= op(F, F) because explode > op, [42] and [42], by (Copy) 42] explode*(cons(X, Y), F, Z) >= F because [43], by (Select) 43] F >= F by (Meta) 44] explode*(cons(X, Y), F, Z) >= @_{o -> o}(F, Z) because explode > @_{o -> o}, [42] and [45], by (Copy) 45] explode*(cons(X, Y), F, Z) >= Z because [46], by (Select) 46] Z >= Z by (Meta) popout

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actions

all output return to Higher Order Rewriting Union Beta