details

property	value
status	complete
benchmark	h48.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	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	4.215 seconds
cpu usage	4.21533
user time	4.09511
system time	0.12022
max virtual memory	171752.0
max residence set size	56772.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: 0 : [] --> nat add : [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(/\x./\y.yap(xap(f, x), y), z, 0) => z rec(/\x./\y.yap(xap(f, x), y), z, s(u)) => yap(xap(f, u), rec(/\v./\w.yap(xap(f, v), w), z, u)) succ x y => s(y) add(x, y) => rec(/\z./\u.yap(xap(succ, z), u), x, y) add(x, 0) => x add(x, s(y)) => s(add(x, y)) 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 add : [nat * nat] --> nat rec : [nat -> nat -> nat * nat * nat] --> nat s : [nat] --> nat succ : [nat] --> nat -> nat yap : [nat -> nat * nat] --> nat Rules: rec(/\x./\y.yap(F(x), y), X, 0) => X rec(/\x./\y.yap(F(x), y), X, s(Y)) => yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) succ(X) Y => s(Y) add(X, Y) => rec(/\x./\y.yap(succ(x), y), X, Y) add(X, 0) => X add(X, s(Y)) => s(add(X, Y)) 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(/\x./\y.yap(F(x), y), X, 0) >? X rec(/\x./\y.yap(F(x), y), X, s(Y)) >? yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) succ(X) Y >? s(Y) add(X, Y) >? rec(/\x./\y.yap(succ(x), y), X, Y) add(X, 0) >? X add(X, s(Y)) >? s(add(X, Y)) yap(F, X) >? F X We use a recursive path ordering as defined in [Kop12, Chapter 5]. We choose Lex = {} and Mul = {0, @_{o -> o}, add, rec, s, succ, yap}, and the following precedence: 0 > add > succ > s > rec > yap > @_{o -> o} With these choices, we have: 1] rec(/\x./\y.yap(F(x), y), X, 0) > X because [2], by definition 2] rec*(/\x./\y.yap(F(x), y), X, 0) >= X because [3], by (Select) 3] X >= X by (Meta) 4] rec(/\x./\y.yap(F(x), y), X, s(Y)) > yap(F(Y), rec(/\x./\y.yap(F(x), y), X, Y)) because [5], by definition 5] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(Y), rec(/\x./\y.yap(F(x), y), X, Y)) because rec > yap, [6] and [13], by (Copy) 6] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= F(Y) because [7], by (Select) 7] /\x.yap(F(rec*(/\y./\z.yap(F(y), z), X, s(Y))), x) >= F(Y) because [8], by (Eta)[Kop13:2] 8] F(rec*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(Y) because [9], by (Meta) 9] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= Y because [10], by (Select) 10] s(Y) >= Y because [11], by (Star) 11] s*(Y) >= Y because [12], by (Select) 12] Y >= Y by (Meta) 13] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y) because rec in Mul, [14], [20] and [21], by (Stat) 14] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z) because [15], by (Abs) 15] /\z.yap(F(y), z) >= /\z.yap(F(y), z) because [16], by (Abs) 16] yap(F(y), x) >= yap(F(y), x) because yap in Mul, [17] and [19], by (Fun) 17] F(y) >= F(y) because [18], by (Meta) 18] y >= y by (Var) 19] x >= x by (Var) 20] X >= X by (Meta) 21] s(Y) > Y because [22], by definition 22] s*(Y) >= Y because [12], by (Select) 23] @_{o -> o}(succ(X), Y) > s(Y) because [24], by definition 24] @_{o -> o}*(succ(X), Y) >= s(Y) because [25], by (Select) 25] succ(X) @_{o -> o}*(succ(X), Y) >= s(Y) because [26] 26] succ*(X, @_{o -> o}*(succ(X), Y)) >= s(Y) because succ > s and [27], by (Copy) 27] succ*(X, @_{o -> o}*(succ(X), Y)) >= Y because [28], by (Select) 28] @_{o -> o}*(succ(X), Y) >= Y because [29], by (Select) 29] Y >= Y by (Meta) 30] add(X, Y) >= rec(/\x./\y.yap(succ(x), y), X, Y) because [31], by (Star) 31] add*(X, Y) >= rec(/\x./\y.yap(succ(x), y), X, Y) because add > rec, [32], [40] and [42], by (Copy) 32] add*(X, Y) >= /\y./\z.yap(succ(y), z) because [33], by (F-Abs) 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