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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 39 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 14 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 6 ms]
(6) RelTRS
(7) RIsEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

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

   l(m(x)) -> m(l(x))
   m(r(x)) -> r(m(x))
   f(m(x), y) -> f(x, m(y))

The relative TRS consists of the following S rules:

   b -> l(b)
   f(x, y) -> f(x, r(y))


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

(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
l/1)YES(
m/1(YES)
r/1)YES(
f/2(YES,YES)
b/0)

Quasi precedence:
f_2 > m_1


Status:
m_1: [1]
f_2: [1,2]
b: multiset status

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

   f(m(x), y) -> f(x, m(y))
Rules from S:
none




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

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

   l(m(x)) -> m(l(x))
   m(r(x)) -> r(m(x))

The relative TRS consists of the following S rules:

   b -> l(b)
   f(x, y) -> f(x, r(y))


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

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

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

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

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

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

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

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

   l(m(x)) -> m(l(x))
Rules from S:
none




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

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

   m(r(x)) -> r(m(x))

The relative TRS consists of the following S rules:

   b -> l(b)
   f(x, y) -> f(x, r(y))


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

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

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

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

   <<<
 POL(b) =  	[[1], [1]]
>>>

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

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

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

   m(r(x)) -> r(m(x))
Rules from S:
none




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

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

   b -> l(b)
   f(x, y) -> f(x, r(y))


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

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

(8)
YES