/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 122 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> tt
   U21(tt, V2) -> U22(isList(activate(V2)))
   U22(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNeList(activate(V2)))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isList(activate(V2)))
   U52(tt) -> tt
   U61(tt) -> tt
   U71(tt, P) -> U72(isPal(activate(P)))
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(activate(V)))
   isList(n__nil) -> tt
   isList(n____(V1, V2)) -> U21(isList(activate(V1)), activate(V2))
   isNeList(V) -> U31(isQid(activate(V)))
   isNeList(n____(V1, V2)) -> U41(isList(activate(V1)), activate(V2))
   isNeList(n____(V1, V2)) -> U51(isNeList(activate(V1)), activate(V2))
   isNePal(V) -> U61(isQid(activate(V)))
   isNePal(n____(I, __(P, I))) -> U71(isQid(activate(I)), activate(P))
   isPal(V) -> U81(isNePal(activate(V)))
   isPal(n__nil) -> tt
   isQid(n__a) -> tt
   isQid(n__e) -> tt
   isQid(n__i) -> tt
   isQid(n__o) -> tt
   isQid(n__u) -> tt
   nil -> n__nil
   __(X1, X2) -> n____(X1, X2)
   a -> n__a
   e -> n__e
   i -> n__i
   o -> n__o
   u -> n__u
   activate(n__nil) -> nil
   activate(n____(X1, X2)) -> __(X1, X2)
   activate(n__a) -> a
   activate(n__e) -> e
   activate(n__i) -> i
   activate(n__o) -> o
   activate(n__u) -> u
   activate(X) -> X

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:tt > activate_1 > n__e > isNeList_1 > u > o > isList_1 > n__a > isNePal_1 > isQid_1 > n_____2 > isPal_1 > U31_1 > U51_2 > U81_1 > i > e > a > U71_2 > U52_1 > U42_1 > U41_2 > U22_1 > n__u > n__nil > n__o > nil > U11_1 > U72_1 > ___2 > U21_2 > n__i > U61_1

and weight map:

   nil=5
   tt=11
   n__nil=4
   n__a=11
   n__e=11
   n__i=11
   n__o=11
   n__u=11
   a=13
   e=12
   i=12
   o=12
   u=12
   U11_1=1
   U22_1=1
   isList_1=9
   activate_1=2
   U31_1=1
   U42_1=1
   isNeList_1=5
   U52_1=3
   U61_1=1
   U72_1=18
   isPal_1=8
   U81_1=1
   isQid_1=1
   isNePal_1=5
   ___2=9
   U21_2=3
   U41_2=0
   U51_2=3
   U71_2=18
   n_____2=8

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   __(__(X, Y), Z) -> __(X, __(Y, Z))
   __(X, nil) -> X
   __(nil, X) -> X
   U11(tt) -> tt
   U21(tt, V2) -> U22(isList(activate(V2)))
   U22(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNeList(activate(V2)))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isList(activate(V2)))
   U52(tt) -> tt
   U61(tt) -> tt
   U71(tt, P) -> U72(isPal(activate(P)))
   U72(tt) -> tt
   U81(tt) -> tt
   isList(V) -> U11(isNeList(activate(V)))
   isList(n__nil) -> tt
   isList(n____(V1, V2)) -> U21(isList(activate(V1)), activate(V2))
   isNeList(V) -> U31(isQid(activate(V)))
   isNeList(n____(V1, V2)) -> U41(isList(activate(V1)), activate(V2))
   isNeList(n____(V1, V2)) -> U51(isNeList(activate(V1)), activate(V2))
   isNePal(V) -> U61(isQid(activate(V)))
   isNePal(n____(I, __(P, I))) -> U71(isQid(activate(I)), activate(P))
   isPal(V) -> U81(isNePal(activate(V)))
   isPal(n__nil) -> tt
   isQid(n__a) -> tt
   isQid(n__e) -> tt
   isQid(n__i) -> tt
   isQid(n__o) -> tt
   isQid(n__u) -> tt
   nil -> n__nil
   __(X1, X2) -> n____(X1, X2)
   a -> n__a
   e -> n__e
   i -> n__i
   o -> n__o
   u -> n__u
   activate(n__nil) -> nil
   activate(n____(X1, X2)) -> __(X1, X2)
   activate(n__a) -> a
   activate(n__e) -> e
   activate(n__i) -> i
   activate(n__o) -> o
   activate(n__u) -> u
   activate(X) -> X




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES