/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 238 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 31 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 10 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:

   active(U11(tt, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(plus(N, 0)) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(0) -> active(0)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(mark(X1), X2, X3) -> U21(X1, X2, X3)
   U21(X1, mark(X2), X3) -> U21(X1, X2, X3)
   U21(X1, X2, mark(X3)) -> U21(X1, X2, X3)
   U21(active(X1), X2, X3) -> U21(X1, X2, X3)
   U21(X1, active(X2), X3) -> U21(X1, X2, X3)
   U21(X1, X2, active(X3)) -> U21(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
active/1)YES(
U11/2(YES,YES)
tt/0)
mark/1)YES(
U21/3(YES,YES,YES)
s/1(YES)
plus/2(YES,YES)
and/2(YES,YES)
isNat/1(YES)
0/0)

Quasi precedence:
[tt, U21_3, plus_2, 0] > U11_2
[tt, U21_3, plus_2, 0] > [s_1, and_2, isNat_1]


Status:
U11_2: multiset status
tt: multiset status
U21_3: [3,2,1]
s_1: multiset status
plus_2: [1,2]
and_2: multiset status
isNat_1: 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:

   active(U11(tt, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(plus(N, 0)) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))




----------------------------------------

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

   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(0) -> active(0)
   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(mark(X1), X2, X3) -> U21(X1, X2, X3)
   U21(X1, mark(X2), X3) -> U21(X1, X2, X3)
   U21(X1, X2, mark(X3)) -> U21(X1, X2, X3)
   U21(active(X1), X2, X3) -> U21(X1, X2, X3)
   U21(X1, active(X2), X3) -> U21(X1, X2, X3)
   U21(X1, X2, active(X3)) -> U21(X1, X2, X3)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)
   isNat(active(X)) -> isNat(X)

Q is empty.

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

   POL(0) = 2
   POL(U11(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(U21(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
   POL(active(x_1)) = 1 + x_1
   POL(and(x_1, x_2)) = 2 + 2*x_1 + x_2
   POL(isNat(x_1)) = 2 + x_1
   POL(mark(x_1)) = 2*x_1
   POL(plus(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(s(x_1)) = 2 + 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:

   mark(U11(X1, X2)) -> active(U11(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(U21(X1, X2, X3)) -> active(U21(mark(X1), X2, X3))
   mark(s(X)) -> active(s(mark(X)))
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(isNat(X)) -> active(isNat(X))
   mark(0) -> active(0)
   U11(active(X1), X2) -> U11(X1, X2)
   U11(X1, active(X2)) -> U11(X1, X2)
   U21(active(X1), X2, X3) -> U21(X1, X2, X3)
   U21(X1, active(X2), X3) -> U21(X1, X2, X3)
   U21(X1, X2, active(X3)) -> U21(X1, X2, X3)
   s(active(X)) -> s(X)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   isNat(active(X)) -> isNat(X)




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

   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U21(mark(X1), X2, X3) -> U21(X1, X2, X3)
   U21(X1, mark(X2), X3) -> U21(X1, X2, X3)
   U21(X1, X2, mark(X3)) -> U21(X1, X2, X3)
   s(mark(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(X)

Q is empty.

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

   POL(U11(x_1, x_2)) = x_1 + x_2
   POL(U21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = x_1
   POL(mark(x_1)) = 2 + 2*x_1
   POL(plus(x_1, x_2)) = x_1 + 2*x_2
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U11(mark(X1), X2) -> U11(X1, X2)
   U11(X1, mark(X2)) -> U11(X1, X2)
   U21(mark(X1), X2, X3) -> U21(X1, X2, X3)
   U21(X1, mark(X2), X3) -> U21(X1, X2, X3)
   U21(X1, X2, mark(X3)) -> U21(X1, X2, X3)
   s(mark(X)) -> s(X)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   isNat(mark(X)) -> isNat(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