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Higher Order Rewriting Union Beta pair #487094041
details
property
value
status
complete
benchmark
Applicative_AG01_innermost__#4.36.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n149.star.cs.uiowa.edu
space
Uncurried_Applicative_11
run statistics
property
value
solver
Wanda 2.2a
configuration
default
runtime (wallclock)
1.53873 seconds
cpu usage
0.995298
user time
0.919263
system time
0.076035
max virtual memory
139264.0
max residence set size
23376.0
stage attributes
key
value
starexec-result
YES
output
YES We consider the system theBenchmark. Alphabet: 0 : [] --> b cons : [b * c] --> c eq : [b * b] --> a false : [] --> a filter : [b -> a * c] --> c filter2 : [a * b -> a * b * c] --> c if!fac6220min : [a * c] --> b if!fac6220replace : [a * b * b * c] --> c le : [b * b] --> a map : [b -> b * c] --> c min : [c] --> b nil : [] --> c replace : [b * b * c] --> c s : [b] --> b sort : [c] --> c true : [] --> a Rules: eq(0, 0) => true eq(0, s(x)) => false eq(s(x), 0) => false eq(s(x), s(y)) => eq(x, y) le(0, x) => true le(s(x), 0) => false le(s(x), s(y)) => le(x, y) min(cons(0, nil)) => 0 min(cons(s(x), nil)) => s(x) min(cons(x, cons(y, z))) => if!fac6220min(le(x, y), cons(x, cons(y, z))) if!fac6220min(true, cons(x, cons(y, z))) => min(cons(x, z)) if!fac6220min(false, cons(x, cons(y, z))) => min(cons(y, z)) replace(x, y, nil) => nil replace(x, y, cons(z, u)) => if!fac6220replace(eq(x, z), x, y, cons(z, u)) if!fac6220replace(true, x, y, cons(z, u)) => cons(y, u) if!fac6220replace(false, x, y, cons(z, u)) => cons(z, replace(x, y, u)) sort(nil) => nil sort(cons(x, y)) => cons(min(cons(x, y)), sort(replace(min(cons(x, y)), x, 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 observe that the rules contain a first-order subset: eq(0, 0) => true eq(0, s(X)) => false eq(s(X), 0) => false eq(s(X), s(Y)) => eq(X, Y) le(0, X) => true le(s(X), 0) => false le(s(X), s(Y)) => le(X, Y) min(cons(0, nil)) => 0 min(cons(s(X), nil)) => s(X) min(cons(X, cons(Y, Z))) => if!fac6220min(le(X, Y), cons(X, cons(Y, Z))) if!fac6220min(true, cons(X, cons(Y, Z))) => min(cons(X, Z)) if!fac6220min(false, cons(X, cons(Y, Z))) => min(cons(Y, Z)) replace(X, Y, nil) => nil replace(X, Y, cons(Z, U)) => if!fac6220replace(eq(X, Z), X, Y, cons(Z, U)) if!fac6220replace(true, X, Y, cons(Z, U)) => cons(Y, U) if!fac6220replace(false, X, Y, cons(Z, U)) => cons(Z, replace(X, Y, U)) sort(nil) => nil sort(cons(X, Y)) => cons(min(cons(X, Y)), sort(replace(min(cons(X, Y)), 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: eq(0(),0()) -> true() || 2: eq(0(),s(PeRCenTX)) -> false() || 3: eq(s(PeRCenTX),0()) -> false() || 4: eq(s(PeRCenTX),s(PeRCenTY)) -> eq(PeRCenTX,PeRCenTY) || 5: le(0(),PeRCenTX) -> true() || 6: le(s(PeRCenTX),0()) -> false() || 7: le(s(PeRCenTX),s(PeRCenTY)) -> le(PeRCenTX,PeRCenTY) || 8: min(cons(0(),nil())) -> 0() || 9: min(cons(s(PeRCenTX),nil())) -> s(PeRCenTX) || 10: min(cons(PeRCenTX,cons(PeRCenTY,PeRCenTZ))) -> if!fac6220min(le(PeRCenTX,PeRCenTY),cons(PeRCenTX,cons(PeRCenTY,PeRCenTZ))) || 11: if!fac6220min(true(),cons(PeRCenTX,cons(PeRCenTY,PeRCenTZ))) -> min(cons(PeRCenTX,PeRCenTZ)) || 12: if!fac6220min(false(),cons(PeRCenTX,cons(PeRCenTY,PeRCenTZ))) -> min(cons(PeRCenTY,PeRCenTZ)) || 13: replace(PeRCenTX,PeRCenTY,nil()) -> nil() || 14: replace(PeRCenTX,PeRCenTY,cons(PeRCenTZ,PeRCenTU)) -> if!fac6220replace(eq(PeRCenTX,PeRCenTZ),PeRCenTX,PeRCenTY,cons(PeRCenTZ,PeRCenTU)) || 15: if!fac6220replace(true(),PeRCenTX,PeRCenTY,cons(PeRCenTZ,PeRCenTU)) -> cons(PeRCenTY,PeRCenTU) || 16: if!fac6220replace(false(),PeRCenTX,PeRCenTY,cons(PeRCenTZ,PeRCenTU)) -> cons(PeRCenTZ,replace(PeRCenTX,PeRCenTY,PeRCenTU)) || 17: sort(nil()) -> nil() || 18: sort(cons(PeRCenTX,PeRCenTY)) -> cons(min(cons(PeRCenTX,PeRCenTY)),sort(replace(min(cons(PeRCenTX,PeRCenTY)),PeRCenTX,PeRCenTY))) || Number of strict rules: 18 || Direct POLO(bPol) ... failed. || Uncurrying min || 1: eq(0(),0()) -> true() || 2: eq(0(),s(PeRCenTX)) -> false()
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