details

property	value
status	complete
benchmark	h57.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n151.star.cs.uiowa.edu
space	Hamana_Kikuchi_18

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.14134 seconds
cpu usage	0.141627
user time	0.121174
system time	0.020453
max virtual memory	113188.0
max residence set size	5756.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: 0 : [] --> nat cons : [] --> nat -> list -> list foldr : [] --> (nat -> nat -> nat) -> nat -> list -> nat length : [] --> list -> nat nil : [] --> list s : [] --> nat -> nat Rules: foldr (/\x./\y.f x y) z nil => z foldr (/\x./\y.f x y) z (cons u v) => f u (foldr (/\w./\x'.f w x') z v) length x => foldr (/\y./\z.s z) 0 x Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term. We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0). This leads to the following AFSM: Alphabet: 0 : [] --> nat cons : [nat * list] --> list foldr : [nat -> nat -> nat * nat * list] --> nat length : [list] --> nat nil : [] --> list s : [nat] --> nat ~AP1 : [nat -> nat -> nat * nat] --> nat -> nat Rules: foldr(/\x./\y.~AP1(F, x) y, X, nil) => X foldr(/\x./\y.~AP1(F, x) y, X, cons(Y, Z)) => ~AP1(F, Y) foldr(/\z./\u.~AP1(F, z) u, X, Z) length(X) => foldr(/\x./\y.s(y), 0, X) ~AP1(F, X) => F X Symbol ~AP1 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 cons : [nat * list] --> list foldr : [nat -> nat -> nat * nat * list] --> nat length : [list] --> nat nil : [] --> list s : [nat] --> nat Rules: foldr(/\x./\y.X(x, y), Y, nil) => Y foldr(/\x./\y.X(x, y), Y, cons(Z, U)) => X(Z, foldr(/\z./\u.X(z, u), Y, U)) length(X) => foldr(/\x./\y.s(y), 0, 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]): foldr(/\x./\y.X(x, y), Y, nil) >? Y foldr(/\x./\y.X(x, y), Y, cons(Z, U)) >? X(Z, foldr(/\z./\u.X(z, u), Y, U)) length(X) >? foldr(/\x./\y.s(y), 0, X) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ [[s(x_1)]] = x_1 We choose Lex = {} and Mul = {cons, foldr, length, nil}, and the following precedence: cons > length > foldr > nil Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: foldr(/\x./\y.X(x, y), Y, nil) >= Y foldr(/\x./\y.X(x, y), Y, cons(Z, U)) > X(Z, foldr(/\x./\y.X(x, y), Y, U)) length(X) >= foldr(/\x./\y.y, _|_, X) With these choices, we have: 1] foldr(/\x./\y.X(x, y), Y, nil) >= Y because [2], by (Star) 2] foldr*(/\x./\y.X(x, y), Y, nil) >= Y because [3], by (Select) 3] Y >= Y by (Meta) 4] foldr(/\x./\y.X(x, y), Y, cons(Z, U)) > X(Z, foldr(/\x./\y.X(x, y), Y, U)) because [5], by definition 5] foldr*(/\x./\y.X(x, y), Y, cons(Z, U)) >= X(Z, foldr(/\x./\y.X(x, y), Y, U)) because [6], by (Select) 6] X(foldr*(/\x./\y.X(x, y), Y, cons(Z, U)), foldr*(/\z./\u.X(z, u), Y, cons(Z, U))) >= X(Z, foldr(/\x./\y.X(x, y), Y, U)) because [7] and [11], by (Meta) 7] foldr*(/\x./\y.X(x, y), Y, cons(Z, U)) >= Z because [8], by (Select) 8] cons(Z, U) >= Z because [9], by (Star) 9] cons*(Z, U) >= Z because [10], by (Select) 10] Z >= Z by (Meta) 11] foldr*(/\x./\y.X(x, y), Y, cons(Z, U)) >= foldr(/\x./\y.X(x, y), Y, U) because foldr in Mul, [12], [17] and [18], by (Stat) 12] /\x./\z.X(x, z) >= /\x./\z.X(x, z) because [13], by (Abs) 13] /\z.X(y, z) >= /\z.X(y, z) because [14], by (Abs) 14] X(y, x) >= X(y, x) because [15] and [16], by (Meta) 15] y >= y by (Var) 16] x >= x by (Var) 17] Y >= Y by (Meta) 18] cons(Z, U) > U because [19], by definition popout

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all output return to Higher Order Rewriting Union Beta