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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRS Reverse [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 49 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 7 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 48 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 44 ms]
(10) RelTRS
(11) RIsEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

   p(s(x1)) -> s(x1)
   n(s(x1)) -> s(x1)
   o(s(x1)) -> s(x1)
   o(n(x1)) -> n(o(x1))
   o(m(x1)) -> n(o(x1))
   n(o(p(x1))) -> o(n(x1))

The relative TRS consists of the following S rules:

   t(x1) -> t(c(n(x1)))
   p(n(x1)) -> m(p(x1))
   p(m(x1)) -> m(p(x1))
   n(p(x1)) -> p(n(x1))
   c(p(x1)) -> p(c(x1))
   c(m(x1)) -> m(c(x1))
   c(n(x1)) -> n(c(x1))
   c(o(x1)) -> o(c(x1))
   c(o(x1)) -> o(x1)


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

(1) RelTRS Reverse (EQUIVALENT)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is 
   p(s(x1)) -> s(x1)
   n(s(x1)) -> s(x1)
   o(s(x1)) -> s(x1)
   o(n(x1)) -> n(o(x1))
   o(m(x1)) -> n(o(x1))
   n(o(p(x1))) -> o(n(x1))

The set of rules S is 
   t(x1) -> t(c(n(x1)))
   p(n(x1)) -> m(p(x1))
   p(m(x1)) -> m(p(x1))
   n(p(x1)) -> p(n(x1))
   c(p(x1)) -> p(c(x1))
   c(m(x1)) -> m(c(x1))
   c(n(x1)) -> n(c(x1))
   c(o(x1)) -> o(c(x1))
   c(o(x1)) -> o(x1)

We have obtained the following relative TRS:
The set of rules R is 
   s(p(x1)) -> s(x1)
   s(n(x1)) -> s(x1)
   s(o(x1)) -> s(x1)
   n(o(x1)) -> o(n(x1))
   m(o(x1)) -> o(n(x1))
   p(o(n(x1))) -> n(o(x1))

The set of rules S is 
   t(x1) -> n(c(t(x1)))
   n(p(x1)) -> p(m(x1))
   m(p(x1)) -> p(m(x1))
   p(n(x1)) -> n(p(x1))
   p(c(x1)) -> c(p(x1))
   m(c(x1)) -> c(m(x1))
   n(c(x1)) -> c(n(x1))
   o(c(x1)) -> c(o(x1))
   o(c(x1)) -> o(x1)


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

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

   s(p(x1)) -> s(x1)
   s(n(x1)) -> s(x1)
   s(o(x1)) -> s(x1)
   n(o(x1)) -> o(n(x1))
   m(o(x1)) -> o(n(x1))
   p(o(n(x1))) -> n(o(x1))

The relative TRS consists of the following S rules:

   t(x1) -> n(c(t(x1)))
   n(p(x1)) -> p(m(x1))
   m(p(x1)) -> p(m(x1))
   p(n(x1)) -> n(p(x1))
   p(c(x1)) -> c(p(x1))
   m(c(x1)) -> c(m(x1))
   n(c(x1)) -> c(n(x1))
   o(c(x1)) -> c(o(x1))
   o(c(x1)) -> o(x1)


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

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

   POL(c(x_1)) = x_1
   POL(m(x_1)) = x_1
   POL(n(x_1)) = x_1
   POL(o(x_1)) = x_1
   POL(p(x_1)) = 1 + x_1
   POL(s(x_1)) = x_1
   POL(t(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   s(p(x1)) -> s(x1)
   p(o(n(x1))) -> n(o(x1))
Rules from S:
none




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

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

   s(n(x1)) -> s(x1)
   s(o(x1)) -> s(x1)
   n(o(x1)) -> o(n(x1))
   m(o(x1)) -> o(n(x1))

The relative TRS consists of the following S rules:

   t(x1) -> n(c(t(x1)))
   n(p(x1)) -> p(m(x1))
   m(p(x1)) -> p(m(x1))
   p(n(x1)) -> n(p(x1))
   p(c(x1)) -> c(p(x1))
   m(c(x1)) -> c(m(x1))
   n(c(x1)) -> c(n(x1))
   o(c(x1)) -> c(o(x1))
   o(c(x1)) -> o(x1)


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

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

   POL(c(x_1)) = x_1
   POL(m(x_1)) = x_1
   POL(n(x_1)) = x_1
   POL(o(x_1)) = 1 + x_1
   POL(p(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(t(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   s(o(x1)) -> s(x1)
Rules from S:
none




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

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

   s(n(x1)) -> s(x1)
   n(o(x1)) -> o(n(x1))
   m(o(x1)) -> o(n(x1))

The relative TRS consists of the following S rules:

   t(x1) -> n(c(t(x1)))
   n(p(x1)) -> p(m(x1))
   m(p(x1)) -> p(m(x1))
   p(n(x1)) -> n(p(x1))
   p(c(x1)) -> c(p(x1))
   m(c(x1)) -> c(m(x1))
   n(c(x1)) -> c(n(x1))
   o(c(x1)) -> c(o(x1))
   o(c(x1)) -> o(x1)


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

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

   <<<
 POL(s(x_1)) =  	[[0], [2]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(n(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
>>>

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

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

   <<<
 POL(t(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

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

   <<<
 POL(p(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
>>>

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

   s(n(x1)) -> s(x1)
Rules from S:
none




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

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

   n(o(x1)) -> o(n(x1))
   m(o(x1)) -> o(n(x1))

The relative TRS consists of the following S rules:

   t(x1) -> n(c(t(x1)))
   n(p(x1)) -> p(m(x1))
   m(p(x1)) -> p(m(x1))
   p(n(x1)) -> n(p(x1))
   p(c(x1)) -> c(p(x1))
   m(c(x1)) -> c(m(x1))
   n(c(x1)) -> c(n(x1))
   o(c(x1)) -> c(o(x1))
   o(c(x1)) -> o(x1)


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

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

   <<<
 POL(n(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(o(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
>>>

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

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

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

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

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

   n(o(x1)) -> o(n(x1))
   m(o(x1)) -> o(n(x1))
Rules from S:
none




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

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

   t(x1) -> n(c(t(x1)))
   n(p(x1)) -> p(m(x1))
   m(p(x1)) -> p(m(x1))
   p(n(x1)) -> n(p(x1))
   p(c(x1)) -> c(p(x1))
   m(c(x1)) -> c(m(x1))
   n(c(x1)) -> c(n(x1))
   o(c(x1)) -> c(o(x1))
   o(c(x1)) -> o(x1)


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

(12)
YES