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Higher Order Rewriting Union Beta pair #487094077
details
property
value
status
complete
benchmark
Applicative_first_order_05__08.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
Uncurried_Applicative_11
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.655012 seconds
cpu usage
0.655166
user time
0.613429
system time
0.041737
max virtual memory
129196.0
max residence set size
14344.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: !facminus : [a * a] --> a !facplus : [a * a] --> a !factimes : [a * a] --> a 0 : [] --> a 1 : [] --> a D : [a] --> a cons : [c * d] --> d constant : [] --> a false : [] --> b filter : [c -> b * d] --> d filter2 : [b * c -> b * c * d] --> d map : [c -> c * d] --> d nil : [] --> d t : [] --> a true : [] --> b Rules: D(t) => 1 D(constant) => 0 D(!facplus(x, y)) => !facplus(D(x), D(y)) D(!factimes(x, y)) => !facplus(!factimes(y, D(x)), !factimes(x, D(y))) D(!facminus(x, y)) => !facminus(D(x), D(y)) map(f, nil) => nil map(f, cons(x, y)) => cons(f x, map(f, y)) filter(f, nil) => nil filter(f, cons(x, y)) => filter2(f x, f, x, y) filter2(true, f, x, y) => cons(x, filter(f, y)) filter2(false, f, x, y) => filter(f, y) 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]): D(t) >? 1 D(constant) >? 0 D(!facplus(X, Y)) >? !facplus(D(X), D(Y)) D(!factimes(X, Y)) >? !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) D(!facminus(X, Y)) >? !facminus(D(X), D(Y)) map(F, nil) >? nil map(F, cons(X, Y)) >? cons(F X, map(F, Y)) filter(F, nil) >? nil filter(F, cons(X, Y)) >? filter2(F X, F, X, Y) filter2(true, F, X, Y) >? cons(X, filter(F, Y)) filter2(false, F, X, Y) >? filter(F, Y) We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[0]] = _|_ [[1]] = _|_ [[filter(x_1, x_2)]] = filter(x_2, x_1) [[filter2(x_1, x_2, x_3, x_4)]] = filter2(x_4, x_2, x_1, x_3) [[nil]] = _|_ We choose Lex = {filter, filter2} and Mul = {!facminus, !facplus, !factimes, @_{o -> o}, D, cons, constant, false, map, t, true}, and the following precedence: !factimes = D > !facminus > !facplus > constant > false > map > filter = filter2 > @_{o -> o} > cons > t > true Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: D(t) >= _|_ D(constant) >= _|_ D(!facplus(X, Y)) > !facplus(D(X), D(Y)) D(!factimes(X, Y)) > !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) D(!facminus(X, Y)) > !facminus(D(X), D(Y)) map(F, _|_) >= _|_ map(F, cons(X, Y)) >= cons(@_{o -> o}(F, X), map(F, Y)) filter(F, _|_) >= _|_ filter(F, cons(X, Y)) >= filter2(@_{o -> o}(F, X), F, X, Y) filter2(true, F, X, Y) >= cons(X, filter(F, Y)) filter2(false, F, X, Y) >= filter(F, Y) With these choices, we have: 1] D(t) >= _|_ by (Bot) 2] D(constant) >= _|_ by (Bot) 3] D(!facplus(X, Y)) > !facplus(D(X), D(Y)) because [4], by definition 4] D*(!facplus(X, Y)) >= !facplus(D(X), D(Y)) because D > !facplus, [5] and [9], by (Copy) 5] D*(!facplus(X, Y)) >= D(X) because D in Mul and [6], by (Stat) 6] !facplus(X, Y) > X because [7], by definition 7] !facplus*(X, Y) >= X because [8], by (Select) 8] X >= X by (Meta) 9] D*(!facplus(X, Y)) >= D(Y) because D in Mul and [10], by (Stat) 10] !facplus(X, Y) > Y because [11], by definition 11] !facplus*(X, Y) >= Y because [12], by (Select) 12] Y >= Y by (Meta) 13] D(!factimes(X, Y)) > !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) because [14], by definition 14] D*(!factimes(X, Y)) >= !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) because D > !facplus, [15] and [22], by (Copy) 15] D*(!factimes(X, Y)) >= !factimes(Y, D(X)) because D = !factimes, D in Mul, [16] and [19], by (Stat) 16] !factimes(X, Y) > Y because [17], by definition
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