details

property	value
status	complete
benchmark	kop11cai1.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n147.star.cs.uiowa.edu
space	Kop_13

run statistics

property	value
solver	Wanda 2.2a
configuration	default
runtime (wallclock)	0.770529 seconds
cpu usage	0.770917
user time	0.725782
system time	0.045135
max virtual memory	135464.0
max residence set size	20336.0

stage attributes

key	value
starexec-result	YES

output

YES We consider the system theBenchmark. Alphabet: 0 : [] --> nat cons : [nat * list] --> list map : [nat -> nat * list] --> list nil : [] --> list op : [nat -> nat * nat -> nat] --> nat -> nat pow : [nat -> nat * nat] --> nat -> nat s : [nat] --> nat Rules: map(f, nil) => nil map(f, cons(x, y)) => cons(f x, map(f, y)) pow(f, 0) => /\x.x pow(f, s(x)) => op(f, pow(f, x)) op(f, g) x => f (g x) /\x.f x => f 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 map : [nat -> nat * list] --> list nil : [] --> list op : [nat -> nat * nat -> nat] --> nat -> nat pow : [nat -> nat * nat] --> nat -> nat s : [nat] --> nat ~AP1 : [nat -> nat * nat] --> nat ~L1 : [nat -> nat] --> nat -> nat Rules: map(F, nil) => nil map(F, cons(X, Y)) => cons(~AP1(F, X), map(F, Y)) pow(F, 0) => ~L1(/\x.x) pow(F, s(X)) => op(F, pow(F, X)) op(F, G) X => ~AP1(F, ~AP1(G, X)) ~L1(/\x.~AP1(F, x)) => F ~L1(/\x.op(F, G) x) => op(F, G) ~AP1(F, X) => F X ~L1(F) => F We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): map(F, nil) >? nil map(F, cons(X, Y)) >? cons(~AP1(F, X), map(F, Y)) pow(F, 0) >? ~L1(/\x.x) pow(F, s(X)) >? op(F, pow(F, X)) op(F, G) X >? ~AP1(F, ~AP1(G, X)) ~L1(/\x.~AP1(F, x)) >? F ~L1(/\x.op(F, G) x) >? op(F, G) ~AP1(F, X) >? F X ~L1(F) >? F We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[nil]] = _|_ We choose Lex = {} and Mul = {0, @_{o -> o}, cons, map, op, pow, s, ~AP1, ~L1}, and the following precedence: 0 > pow > s > ~L1 > op > map = ~AP1 > @_{o -> o} > cons Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: map(F, _|_) >= _|_ map(F, cons(X, Y)) > cons(~AP1(F, X), map(F, Y)) pow(F, 0) > ~L1(/\x.x) pow(F, s(X)) >= op(F, pow(F, X)) @_{o -> o}(op(F, G), X) >= ~AP1(F, ~AP1(G, X)) ~L1(/\x.~AP1(F, x)) >= F ~L1(/\x.@_{o -> o}(op(F, G), x)) >= op(F, G) ~AP1(F, X) > @_{o -> o}(F, X) ~L1(F) >= F With these choices, we have: 1] map(F, _|_) >= _|_ by (Bot) 2] map(F, cons(X, Y)) > cons(~AP1(F, X), map(F, Y)) because [3], by definition 3] map*(F, cons(X, Y)) >= cons(~AP1(F, X), map(F, Y)) because map > cons, [4] and [9], by (Copy) 4] map*(F, cons(X, Y)) >= ~AP1(F, X) because map = ~AP1, map in Mul, [5] and [6], by (Stat) 5] F >= F by (Meta) 6] cons(X, Y) > X because [7], by definition 7] cons*(X, Y) >= X because [8], by (Select) 8] X >= X by (Meta) 9] map*(F, cons(X, Y)) >= map(F, Y) because map in Mul, [5] and [10], by (Stat) 10] cons(X, Y) > Y because [11], by definition 11] cons*(X, Y) >= Y because [12], by (Select) 12] Y >= Y by (Meta) popout

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all output return to Higher Order Rewriting Union Beta