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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) QTRS
(1) QTRSToCSRProof [SOUND, 0 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 41 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

   active(and(tt, X)) -> mark(X)
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(s(plus(N, M)))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(s(X)) -> s(active(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   s(mark(X)) -> mark(s(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   s(ok(X)) -> ok(s(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(and(x0, x1))
   active(plus(x0, x1))
   active(s(x0))
   and(mark(x0), x1)
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   s(mark(x0))
   proper(and(x0, x1))
   proper(tt)
   proper(plus(x0, x1))
   proper(0)
   proper(s(x0))
   and(ok(x0), ok(x1))
   plus(ok(x0), ok(x1))
   s(ok(x0))
   top(mark(x0))
   top(ok(x0))


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(1) QTRSToCSRProof (SOUND)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

   active(and(tt, X)) -> mark(X)
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(s(plus(N, M)))
   active(and(X1, X2)) -> and(active(X1), X2)
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(s(X)) -> s(active(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   s(mark(X)) -> mark(s(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   s(ok(X)) -> ok(s(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

The set Q consists of the following terms:

   active(and(x0, x1))
   active(plus(x0, x1))
   active(s(x0))
   and(mark(x0), x1)
   plus(mark(x0), x1)
   plus(x0, mark(x1))
   s(mark(x0))
   proper(and(x0, x1))
   proper(tt)
   proper(plus(x0, x1))
   proper(0)
   proper(s(x0))
   and(ok(x0), ok(x1))
   plus(ok(x0), ok(x1))
   s(ok(x0))
   top(mark(x0))
   top(ok(x0))

Special symbols used for the transformation (see [GM04]):
top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
The replacement map contains the following entries:

and: {1}
tt: empty set
plus: {1, 2}
0: empty set
s: {1}
The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
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(2)
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}

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(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, 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




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(4)
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}

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(5) 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)) = 2*x_1 + x_2
   POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(s(x_1)) = x_1
   POL(tt) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   and(tt, X) -> X




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

(6)
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}

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(7) 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:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(plus(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [0, 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))




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(8)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(9) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(10)
YES