YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 52 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   and(tt, X) -> X
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))

The replacement map contains the following entries:

and: {1}
tt: empty set
plus: {1, 2}
0: empty set
s: {1}

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

(1) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   and(tt, X) -> X
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))

The replacement map contains the following entries:

and: {1}
tt: empty set
plus: {1, 2}
0: empty set
s: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   plus(N, 0) -> N




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

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   and(tt, X) -> X
   plus(N, s(M)) -> s(plus(N, M))

The replacement map contains the following entries:

and: {1}
tt: empty set
plus: {1, 2}
s: {1}

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

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   and(tt, X) -> X
   plus(N, s(M)) -> s(plus(N, M))

The replacement map contains the following entries:

and: {1}
tt: empty set
plus: {1, 2}
s: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(and(x_1, x_2)) = 1 + x_1 + x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   and(tt, X) -> X




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

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   plus(N, s(M)) -> s(plus(N, M))

The replacement map contains the following entries:

plus: {1, 2}
s: {1}

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

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   plus(N, s(M)) -> s(plus(N, M))

The replacement map contains the following entries:

plus: {1, 2}
s: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(s(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:

   plus(N, s(M)) -> s(plus(N, M))




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

(6)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(7) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(8)
YES