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Higher Order Rewriting Union Beta pair #487094055
details
property
value
status
complete
benchmark
Applicative_first_order_05__#3.55.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n142.star.cs.uiowa.edu
space
Uncurried_Applicative_11
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
0.940409 seconds
cpu usage
0.904369
user time
0.819681
system time
0.084688
max virtual memory
138612.0
max residence set size
22896.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: 0 : [] --> b add : [b * c] --> c app : [c * c] --> c false : [] --> a filter : [b -> a * c] --> c filter2 : [a * b -> a * b * c] --> c high : [b * c] --> c if!fac6220high : [a * b * c] --> c if!fac6220low : [a * b * c] --> c le : [b * b] --> a low : [b * c] --> c map : [b -> b * c] --> c minus : [b * b] --> b nil : [] --> c quicksort : [c] --> c quot : [b * b] --> b s : [b] --> b true : [] --> a Rules: minus(x, 0) => x minus(s(x), s(y)) => minus(x, y) quot(0, s(x)) => 0 quot(s(x), s(y)) => s(quot(minus(x, y), s(y))) le(0, x) => true le(s(x), 0) => false le(s(x), s(y)) => le(x, y) app(nil, x) => x app(add(x, y), z) => add(x, app(y, z)) low(x, nil) => nil low(x, add(y, z)) => if!fac6220low(le(y, x), x, add(y, z)) if!fac6220low(true, x, add(y, z)) => add(y, low(x, z)) if!fac6220low(false, x, add(y, z)) => low(x, z) high(x, nil) => nil high(x, add(y, z)) => if!fac6220high(le(y, x), x, add(y, z)) if!fac6220high(true, x, add(y, z)) => high(x, z) if!fac6220high(false, x, add(y, z)) => add(y, high(x, z)) quicksort(nil) => nil quicksort(add(x, y)) => app(quicksort(low(x, y)), add(x, quicksort(high(x, y)))) map(f, nil) => nil map(f, add(x, y)) => add(f x, map(f, y)) filter(f, nil) => nil filter(f, add(x, y)) => filter2(f x, f, x, y) filter2(true, f, x, y) => add(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 observe that the rules contain a first-order subset: minus(X, 0) => X minus(s(X), s(Y)) => minus(X, Y) quot(0, s(X)) => 0 quot(s(X), s(Y)) => s(quot(minus(X, Y), s(Y))) le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) app(nil, X) => X app(add(X, Y), Z) => add(X, app(Y, Z)) low(X, nil) => nil low(X, add(Y, Z)) => if!fac6220low(le(Y, X), X, add(Y, Z)) if!fac6220low(true, X, add(Y, Z)) => add(Y, low(X, Z)) if!fac6220low(false, X, add(Y, Z)) => low(X, Z) high(X, nil) => nil high(X, add(Y, Z)) => if!fac6220high(le(Y, X), X, add(Y, Z)) if!fac6220high(true, X, add(Y, Z)) => high(X, Z) if!fac6220high(false, X, add(Y, Z)) => add(Y, high(X, Z)) quicksort(nil) => nil quicksort(add(X, Y)) => app(quicksort(low(X, Y)), add(X, quicksort(high(X, Y)))) Moreover, the system is orthogonal. Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is terminating when seen as a many-sorted first-order TRS. According to the external first-order termination prover, this system is indeed terminating: || Input TRS: || 1: minus(PeRCenTX,0()) -> PeRCenTX || 2: minus(s(PeRCenTX),s(PeRCenTY)) -> minus(PeRCenTX,PeRCenTY) || 3: quot(0(),s(PeRCenTX)) -> 0() || 4: quot(s(PeRCenTX),s(PeRCenTY)) -> s(quot(minus(PeRCenTX,PeRCenTY),s(PeRCenTY))) || 5: le(0(),PeRCenTX) -> true() || 6: le(s(PeRCenTX),0()) -> false() || 7: le(s(PeRCenTX),s(PeRCenTY)) -> le(PeRCenTX,PeRCenTY) || 8: app(nil(),PeRCenTX) -> PeRCenTX || 9: app(add(PeRCenTX,PeRCenTY),PeRCenTZ) -> add(PeRCenTX,app(PeRCenTY,PeRCenTZ)) || 10: low(PeRCenTX,nil()) -> nil() || 11: low(PeRCenTX,add(PeRCenTY,PeRCenTZ)) -> if!fac6220low(le(PeRCenTY,PeRCenTX),PeRCenTX,add(PeRCenTY,PeRCenTZ)) || 12: if!fac6220low(true(),PeRCenTX,add(PeRCenTY,PeRCenTZ)) -> add(PeRCenTY,low(PeRCenTX,PeRCenTZ)) || 13: if!fac6220low(false(),PeRCenTX,add(PeRCenTY,PeRCenTZ)) -> low(PeRCenTX,PeRCenTZ) || 14: high(PeRCenTX,nil()) -> nil() || 15: high(PeRCenTX,add(PeRCenTY,PeRCenTZ)) -> if!fac6220high(le(PeRCenTY,PeRCenTX),PeRCenTX,add(PeRCenTY,PeRCenTZ)) || 16: if!fac6220high(true(),PeRCenTX,add(PeRCenTY,PeRCenTZ)) -> high(PeRCenTX,PeRCenTZ) || 17: if!fac6220high(false(),PeRCenTX,add(PeRCenTY,PeRCenTZ)) -> add(PeRCenTY,high(PeRCenTX,PeRCenTZ)) || 18: quicksort(nil()) -> nil() || 19: quicksort(add(PeRCenTX,PeRCenTY)) -> app(quicksort(low(PeRCenTX,PeRCenTY)),add(PeRCenTX,quicksort(high(PeRCenTX,PeRCenTY))))
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