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


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 73 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 10 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 7 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 9 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 34 ms]
(10) RelTRS
(11) RelTRSRRRProof [EQUIVALENT, 21 ms]
(12) RelTRS
(13) RelTRSRRRProof [EQUIVALENT, 15 ms]
(14) RelTRS
(15) RelTRSRRRProof [EQUIVALENT, 23 ms]
(16) RelTRS
(17) RIsEmptyProof [EQUIVALENT, 0 ms]
(18) YES


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

(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l))))
   pred(cons(s(x), nil)) -> cons(x, nil)

The relative TRS consists of the following S rules:

   cons(x, cons(y, l)) -> cons(y, cons(x, l))


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

(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(app(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
   POL(pred(x_1)) = 1 + x_1
   POL(s(x_1)) = 1 + x_1
   POL(sum(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   plus(0, y) -> y
   pred(cons(s(x), nil)) -> cons(x, nil)
Rules from S:
none




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

(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), y) -> s(plus(x, y))
   sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l))))

The relative TRS consists of the following S rules:

   cons(x, cons(y, l)) -> cons(y, cons(x, l))


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

(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(app(x_1, x_2)) = 1 + x_1 + x_2
   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
   POL(pred(x_1)) = 1 + x_1
   POL(s(x_1)) = 1 + x_1
   POL(sum(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   app(nil, k) -> k
   app(l, nil) -> l
Rules from S:
none




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

(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), y) -> s(plus(x, y))
   sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l))))

The relative TRS consists of the following S rules:

   cons(x, cons(y, l)) -> cons(y, cons(x, l))


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

(5) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(app(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
   POL(pred(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(sum(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   sum(plus(cons(0, x), cons(y, l))) -> pred(sum(cons(s(x), cons(y, l))))
Rules from S:
none




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

(6)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), y) -> s(plus(x, y))

The relative TRS consists of the following S rules:

   cons(x, cons(y, l)) -> cons(y, cons(x, l))


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

(7) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(app(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
   POL(nil) = 0
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = x_1
   POL(sum(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
Rules from S:
none




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

(8)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), y) -> s(plus(x, y))

The relative TRS consists of the following S rules:

   cons(x, cons(y, l)) -> cons(y, cons(x, l))


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

(9) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

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

   <<<
 POL(nil) =  	[[0], [0]]
>>>

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

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

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   plus(s(x), y) -> s(plus(x, y))
Rules from S:
none




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

(10)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))

The relative TRS consists of the following S rules:

   cons(x, cons(y, l)) -> cons(y, cons(x, l))


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

(11) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

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

   <<<
 POL(nil) =  	[[0], [0]]
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   app(cons(x, l), k) -> cons(x, app(l, k))
Rules from S:
none




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

(12)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   sum(cons(x, nil)) -> cons(x, nil)
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))

The relative TRS consists of the following S rules:

   cons(x, cons(y, l)) -> cons(y, cons(x, l))


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

(13) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

   <<<
 POL(nil) =  	[[0], [0]]
>>>

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

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   sum(cons(x, nil)) -> cons(x, nil)
Rules from S:
none




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

(14)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))

The relative TRS consists of the following S rules:

   cons(x, cons(y, l)) -> cons(y, cons(x, l))


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

(15) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

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

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
Rules from S:
none




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

(16)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   cons(x, cons(y, l)) -> cons(y, cons(x, l))


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

(17) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(18)
YES