details

property	value
status	complete
benchmark	h50.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n137.star.cs.uiowa.edu
space	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	11.6399 seconds
cpu usage	11.6392
user time	11.3876
system time	0.2516
max virtual memory	388668.0
max residence set size	150408.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: 0 : [] --> nat plus : [nat * nat] --> nat rec : [nat * nat * nat -> nat -> nat] --> nat s : [nat] --> nat succ : [] --> nat -> nat -> nat xap : [nat -> nat -> nat * nat] --> nat -> nat yap : [nat -> nat * nat] --> nat Rules: rec(0, x, /\y./\z.yap(xap(f, y), z)) => x rec(s(x), y, /\z./\u.yap(xap(f, z), u)) => yap(xap(f, x), rec(x, y, /\v./\w.yap(xap(f, v), w))) succ x y => s(y) plus(x, y) => rec(x, y, succ) xap(f, x) => f x yap(f, x) => f x This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). Symbol xap is an encoding for application that is only used in innocuous ways. We can simplify the program (without losing non-termination) by removing it. This gives: Alphabet: 0 : [] --> nat plus : [nat * nat] --> nat rec : [nat * nat * nat -> nat -> nat] --> nat s : [nat] --> nat succ : [] --> nat -> nat -> nat yap : [nat -> nat * nat] --> nat Rules: rec(0, X, /\x./\y.yap(F(x), y)) => X rec(s(X), Y, /\x./\y.yap(F(x), y)) => yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) succ X Y => s(Y) plus(X, Y) => rec(X, Y, succ) yap(F, X) => F X We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): rec(0, X, /\x./\y.yap(F(x), y)) >? X rec(s(X), Y, /\x./\y.yap(F(x), y)) >? yap(F(X), rec(X, Y, /\z./\u.yap(F(z), u))) succ X Y >? s(Y) plus(X, Y) >? rec(X, Y, succ) yap(F, X) >? F X We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[succ]] = _|_ We choose Lex = {} and Mul = {0, @_{o -> o -> o}, @_{o -> o}, plus, rec, s, yap}, and the following precedence: 0 > @_{o -> o -> o} > s > plus > rec > yap > @_{o -> o} Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: rec(0, X, /\x./\y.yap(F(x), y)) >= X rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y))) @_{o -> o}(@_{o -> o -> o}(_|_, X), Y) >= s(Y) plus(X, Y) > rec(X, Y, _|_) yap(F, X) >= @_{o -> o}(F, X) With these choices, we have: 1] rec(0, X, /\x./\y.yap(F(x), y)) >= X because [2], by (Star) 2] rec*(0, X, /\x./\y.yap(F(x), y)) >= X because [3], by (Select) 3] X >= X by (Meta) 4] rec(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y))) because [5], by (Star) 5] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y))) because [6], by (Select) 6] yap(F(rec*(s(X), Y, /\x./\y.yap(F(x), y))), rec*(s(X), Y, /\z./\u.yap(F(z), u))) >= yap(F(X), rec(X, Y, /\x./\y.yap(F(x), y))) because yap in Mul, [7] and [12], by (Fun) 7] F(rec*(s(X), Y, /\x./\y.yap(F(x), y))) >= F(X) because [8], by (Meta) 8] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= X because [9], by (Select) 9] s(X) >= X because [10], by (Star) 10] s*(X) >= X because [11], by (Select) 11] X >= X by (Meta) 12] rec*(s(X), Y, /\x./\y.yap(F(x), y)) >= rec(X, Y, /\x./\y.yap(F(x), y)) because rec in Mul, [13], [15] and [16], by (Stat) 13] s(X) > X because [14], by definition 14] s*(X) >= X because [11], by (Select) 15] Y >= Y by (Meta) 16] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z) because [17], by (Abs) 17] /\z.yap(F(y), z) >= /\z.yap(F(y), z) because [18], by (Abs) 18] yap(F(y), x) >= yap(F(y), x) because yap in Mul, [19] and [21], by (Fun) 19] F(y) >= F(y) because [20], by (Meta) 20] y >= y by (Var) 21] x >= x by (Var) 22] @_{o -> o}(@_{o -> o -> o}(_|_, X), Y) >= s(Y) because [23], by (Star) 23] @_{o -> o}*(@_{o -> o -> o}(_|_, X), Y) >= s(Y) because [24], by (Select) 24] @_{o -> o -> o}(_|_, X) @_{o -> o}*(@_{o -> o -> o}(_|_, X), Y) >= s(Y) because [25] 25] @_{o -> o -> o}*(_|_, X, @_{o -> o}*(@_{o -> o -> o}(_|_, X), Y)) >= s(Y) because @_{o -> o -> o} > s and [26], by (Copy) 26] @_{o -> o -> o}*(_|_, X, @_{o -> o}*(@_{o -> o -> o}(_|_, X), Y)) >= Y because [27], by (Select) 27] @_{o -> o}*(@_{o -> o -> o}(_|_, X), Y) >= Y because [28], by (Select) popout

output may be truncated. 'popout' for the full output.

job log

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actions

all output return to Higher Order Rewriting Union Beta