/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 119 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 13 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   U21(tt, M, N) -> U22(tt, activate(M), activate(N))
   U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)
   activate(X) -> X

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
U11/3(YES,YES,YES)
tt/0)
U12/3(YES,YES,YES)
activate/1)YES(
s/1(YES)
plus/2(YES,YES)
U21/3(YES,YES,YES)
U22/3(YES,YES,YES)
x/2(YES,YES)
0/0)

Quasi precedence:
[U21_3, U22_3, x_2] > [U11_3, U12_3, plus_2] > s_1 > [tt, 0]


Status:
U11_3: multiset status
tt: multiset status
U12_3: multiset status
s_1: multiset status
plus_2: multiset status
U21_3: multiset status
U22_3: multiset status
x_2: multiset status
0: multiset status

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

   U12(tt, M, N) -> s(plus(activate(N), activate(M)))
   U22(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N))
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)




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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U21(tt, M, N) -> U22(tt, activate(M), activate(N))
   activate(X) -> X

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(U11(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
   POL(U12(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3
   POL(U21(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U22(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
   POL(activate(x_1)) = x_1
   POL(tt) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U11(tt, M, N) -> U12(tt, activate(M), activate(N))
   U21(tt, M, N) -> U22(tt, activate(M), activate(N))




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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   activate(X) -> X

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(activate(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   activate(X) -> X




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(6)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(7) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(8)
YES