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


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 72 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 14 ms]
(4) RelTRS
(5) FlatCCProof [EQUIVALENT, 0 ms]
(6) RelTRS
(7) RootLabelingProof [EQUIVALENT, 0 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 12 ms]
(10) RelTRS
(11) RelTRSRRRProof [EQUIVALENT, 15 ms]
(12) RelTRS
(13) RelTRSRRRProof [EQUIVALENT, 73 ms]
(14) RelTRS
(15) RelTRSRRRProof [EQUIVALENT, 5 ms]
(16) RelTRS
(17) RelTRSRRRProof [EQUIVALENT, 53 ms]
(18) RelTRS
(19) RelTRSRRRProof [EQUIVALENT, 47 ms]
(20) RelTRS
(21) RelTRSRRRProof [EQUIVALENT, 3 ms]
(22) RelTRS
(23) RelTRSRRRProof [EQUIVALENT, 277 ms]
(24) RelTRS
(25) RIsEmptyProof [EQUIVALENT, 0 ms]
(26) YES


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

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

   a(x1) -> b(c(x1))
   a(b(x1)) -> b(a(x1))
   a(c(x1)) -> c(a(x1))

The relative TRS consists of the following S rules:

   d(c(x1)) -> d(a(x1))
   d(a(x1)) -> d(c(b(x1)))


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

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

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

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

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

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

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

   a(x1) -> b(c(x1))
Rules from S:
none




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

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

   a(b(x1)) -> b(a(x1))
   a(c(x1)) -> c(a(x1))

The relative TRS consists of the following S rules:

   d(c(x1)) -> d(a(x1))
   d(a(x1)) -> d(c(b(x1)))


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

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

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

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

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

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

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

   a(c(x1)) -> c(a(x1))
Rules from S:
none




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

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

   a(b(x1)) -> b(a(x1))

The relative TRS consists of the following S rules:

   d(c(x1)) -> d(a(x1))
   d(a(x1)) -> d(c(b(x1)))


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

(5) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]

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

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

   a(a(b(x1))) -> a(b(a(x1)))
   b(a(b(x1))) -> b(b(a(x1)))
   d(a(b(x1))) -> d(b(a(x1)))
   c(a(b(x1))) -> c(b(a(x1)))

The relative TRS consists of the following S rules:

   d(c(x1)) -> d(a(x1))
   d(a(x1)) -> d(c(b(x1)))


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

(7) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled


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

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

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{b_1}(b_{a_1}(a_{d_1}(x1)))
   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
   b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{b_1}(b_{a_1}(a_{d_1}(x1)))
   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
   d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{b_1}(b_{a_1}(a_{a_1}(x1)))
   d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{b_1}(b_{a_1}(a_{b_1}(x1)))
   d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{b_1}(b_{a_1}(a_{d_1}(x1)))
   d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{b_1}(b_{a_1}(a_{c_1}(x1)))
   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1)))
   c_{a_1}(a_{b_1}(b_{d_1}(x1))) -> c_{b_1}(b_{a_1}(a_{d_1}(x1)))
   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))

The relative TRS consists of the following S rules:

   d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(x1))
   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
   d_{c_1}(c_{d_1}(x1)) -> d_{a_1}(a_{d_1}(x1))
   d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{c_1}(x1))
   d_{a_1}(a_{a_1}(x1)) -> d_{c_1}(c_{b_1}(b_{a_1}(x1)))
   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
   d_{a_1}(a_{d_1}(x1)) -> d_{c_1}(c_{b_1}(b_{d_1}(x1)))
   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))


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

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

   POL(a_{a_1}(x_1)) = x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(a_{c_1}(x_1)) = x_1
   POL(a_{d_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = x_1
   POL(b_{b_1}(x_1)) = x_1
   POL(b_{c_1}(x_1)) = x_1
   POL(b_{d_1}(x_1)) = x_1
   POL(c_{a_1}(x_1)) = 1 + x_1
   POL(c_{b_1}(x_1)) = x_1
   POL(c_{c_1}(x_1)) = 1 + x_1
   POL(c_{d_1}(x_1)) = 1 + x_1
   POL(d_{a_1}(x_1)) = x_1
   POL(d_{b_1}(x_1)) = x_1
   POL(d_{c_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1)))
   c_{a_1}(a_{b_1}(b_{d_1}(x1))) -> c_{b_1}(b_{a_1}(a_{d_1}(x1)))
   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))
Rules from S:

   d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(x1))
   d_{c_1}(c_{d_1}(x1)) -> d_{a_1}(a_{d_1}(x1))
   d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{c_1}(x1))




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

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

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{b_1}(b_{a_1}(a_{d_1}(x1)))
   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
   b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{b_1}(b_{a_1}(a_{d_1}(x1)))
   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
   d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{b_1}(b_{a_1}(a_{a_1}(x1)))
   d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{b_1}(b_{a_1}(a_{b_1}(x1)))
   d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{b_1}(b_{a_1}(a_{d_1}(x1)))
   d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{b_1}(b_{a_1}(a_{c_1}(x1)))

The relative TRS consists of the following S rules:

   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
   d_{a_1}(a_{a_1}(x1)) -> d_{c_1}(c_{b_1}(b_{a_1}(x1)))
   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
   d_{a_1}(a_{d_1}(x1)) -> d_{c_1}(c_{b_1}(b_{d_1}(x1)))
   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))


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

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

   POL(a_{a_1}(x_1)) = x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(a_{c_1}(x_1)) = x_1
   POL(a_{d_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = x_1
   POL(b_{b_1}(x_1)) = x_1
   POL(b_{c_1}(x_1)) = x_1
   POL(b_{d_1}(x_1)) = x_1
   POL(c_{b_1}(x_1)) = 1 + x_1
   POL(d_{a_1}(x_1)) = 1 + x_1
   POL(d_{b_1}(x_1)) = x_1
   POL(d_{c_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{b_1}(b_{a_1}(a_{a_1}(x1)))
   d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{b_1}(b_{a_1}(a_{b_1}(x1)))
   d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{b_1}(b_{a_1}(a_{d_1}(x1)))
   d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{b_1}(b_{a_1}(a_{c_1}(x1)))
Rules from S:
none




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

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

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{b_1}(b_{a_1}(a_{d_1}(x1)))
   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
   b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{b_1}(b_{a_1}(a_{d_1}(x1)))
   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))

The relative TRS consists of the following S rules:

   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
   d_{a_1}(a_{a_1}(x1)) -> d_{c_1}(c_{b_1}(b_{a_1}(x1)))
   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
   d_{a_1}(a_{d_1}(x1)) -> d_{c_1}(c_{b_1}(b_{d_1}(x1)))
   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))


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

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

   <<<
 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 1], [2, 2]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 1], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(b_{d_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
>>>

   <<<
 POL(a_{d_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(d_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(d_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

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

   a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{b_1}(b_{a_1}(a_{d_1}(x1)))
   b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{b_1}(b_{a_1}(a_{d_1}(x1)))
Rules from S:
none




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

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

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))

The relative TRS consists of the following S rules:

   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
   d_{a_1}(a_{a_1}(x1)) -> d_{c_1}(c_{b_1}(b_{a_1}(x1)))
   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
   d_{a_1}(a_{d_1}(x1)) -> d_{c_1}(c_{b_1}(b_{d_1}(x1)))
   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))


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

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

   POL(a_{a_1}(x_1)) = x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(a_{c_1}(x_1)) = x_1
   POL(a_{d_1}(x_1)) = 1 + x_1
   POL(b_{a_1}(x_1)) = x_1
   POL(b_{b_1}(x_1)) = x_1
   POL(b_{c_1}(x_1)) = x_1
   POL(b_{d_1}(x_1)) = x_1
   POL(c_{b_1}(x_1)) = x_1
   POL(d_{a_1}(x_1)) = x_1
   POL(d_{c_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   d_{a_1}(a_{d_1}(x1)) -> d_{c_1}(c_{b_1}(b_{d_1}(x1)))




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

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

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))

The relative TRS consists of the following S rules:

   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
   d_{a_1}(a_{a_1}(x1)) -> d_{c_1}(c_{b_1}(b_{a_1}(x1)))
   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))


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

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

   <<<
 POL(a_{a_1}(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(d_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(d_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 1], [0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   d_{a_1}(a_{a_1}(x1)) -> d_{c_1}(c_{b_1}(b_{a_1}(x1)))




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

(18)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))

The relative TRS consists of the following S rules:

   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))


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

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

   <<<
 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{c_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{c_1}(x_1)) =  	[[2], [2]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
>>>

   <<<
 POL(d_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(d_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))




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

(20)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))

The relative TRS consists of the following S rules:

   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))


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

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

   POL(a_{a_1}(x_1)) = 1 + x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(a_{c_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = 1 + x_1
   POL(b_{b_1}(x_1)) = x_1
   POL(b_{c_1}(x_1)) = 1 + x_1
   POL(c_{b_1}(x_1)) = x_1
   POL(d_{a_1}(x_1)) = x_1
   POL(d_{c_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
Rules from S:
none




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

(22)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))

The relative TRS consists of the following S rules:

   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))


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

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

   <<<
 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(d_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(d_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

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

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
Rules from S:
none




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

(24)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))


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

(26)
YES