Spaces
Explore
Communities
Statistics
Reports
Cluster
Status
Help
Higher Order Rewriting Union Beta pair #487093643
details
property
value
status
complete
benchmark
h53.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n150.star.cs.uiowa.edu
space
Hamana_Kikuchi_18
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
11.666 seconds
cpu usage
11.6499
user time
11.4498
system time
0.20009
max virtual memory
380792.0
max residence set size
120644.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: cons : [a * alist] --> alist foldl : [a -> a -> a * a * alist] --> a nil : [] --> alist xap : [a -> a -> a * a] --> a -> a yap : [a -> a * a] --> a Rules: foldl(/\x./\y.yap(xap(f, x), y), z, nil) => z foldl(/\x./\y.yap(xap(f, x), y), z, cons(u, v)) => foldl(/\w./\x'.yap(xap(f, w), x'), yap(xap(f, z), u), v) 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: cons : [a * alist] --> alist foldl : [a -> a -> a * a * alist] --> a nil : [] --> alist yap : [a -> a * a] --> a Rules: foldl(/\x./\y.yap(F(x), y), X, nil) => X foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) => foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) 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]): foldl(/\x./\y.yap(F(x), y), X, nil) >? X foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >? foldl(/\z./\u.yap(F(z), u), yap(F(X), Y), Z) yap(F, X) >? F X We use a recursive path ordering as defined in [Kop12, Chapter 5]. Argument functions: [[foldl(x_1, x_2, x_3)]] = foldl(x_3, x_1, x_2) We choose Lex = {foldl} and Mul = {@_{o -> o}, cons, nil, yap}, and the following precedence: cons > nil > foldl > yap > @_{o -> o} Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows: foldl(/\x./\y.yap(F(x), y), X, nil) >= X foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) yap(F, X) > @_{o -> o}(F, X) With these choices, we have: 1] foldl(/\x./\y.yap(F(x), y), X, nil) >= X because [2], by (Star) 2] foldl*(/\x./\y.yap(F(x), y), X, nil) >= X because [3], by (Select) 3] X >= X by (Meta) 4] foldl(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) because [5], by (Star) 5] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= foldl(/\x./\y.yap(F(x), y), yap(F(X), Y), Z) because [6], [9], [19] and [29], by (Stat) 6] cons(Y, Z) > Z because [7], by definition 7] cons*(Y, Z) >= Z because [8], by (Select) 8] Z >= Z by (Meta) 9] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= /\x./\y.yap(F(x), y) because [10], by (F-Abs) 10] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z) >= /\x.yap(F(z), x) because [11], by (F-Abs) 11] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= yap(F(z), u) because foldl > yap, [12] and [17], by (Copy) 12] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= F(z) because [13], by (Select) 13] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z), z, u)), x) >= F(z) because [14], by (Eta)[Kop13:2] 14] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u)) >= F(z) because [15], by (Meta) 15] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= z because [16], by (Select) 16] z >= z by (Var) 17] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z), z, u) >= u because [18], by (Select) 18] u >= u by (Var) 19] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= yap(F(X), Y) because foldl > yap, [20] and [25], by (Copy) 20] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= F(X) because [21], by (Select) 21] /\x.yap(F(foldl*(/\y./\v.yap(F(y), v), X, cons(Y, Z))), x) >= F(X) because [22], by (Eta)[Kop13:2] 22] F(foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z))) >= F(X) because [23], by (Meta) 23] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= X because [24], by (Select) 24] X >= X by (Meta) 25] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Y because [26], by (Select) 26] cons(Y, Z) >= Y because [27], by (Star) 27] cons*(Y, Z) >= Y because [28], by (Select) 28] Y >= Y by (Meta) 29] foldl*(/\x./\y.yap(F(x), y), X, cons(Y, Z)) >= Z because [30], by (Select) 30] cons(Y, Z) >= Z because [7], by (Star) 31] yap(F, X) > @_{o -> o}(F, X) because [32], by definition 32] yap*(F, X) >= @_{o -> o}(F, X) because yap > @_{o -> o}, [33] and [35], by (Copy) 33] yap*(F, X) >= F because [34], by (Select) 34] F >= F by (Meta) 35] yap*(F, X) >= X because [36], by (Select) 36] X >= X by (Meta) We can thus remove the following rules:
popout
output may be truncated. 'popout' for the full output.
job log
popout
actions
all output
return to Higher Order Rewriting Union Beta