49.27/13.78	YES
49.27/13.79	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
49.27/13.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
49.27/13.79	
49.27/13.79	
49.27/13.79	Termination of the given RelTRS could be proven:
49.27/13.79	
49.27/13.79	(0) RelTRS
49.27/13.79	(1) RelTRSRRRProof [EQUIVALENT, 107 ms]
49.27/13.79	(2) RelTRS
49.27/13.79	(3) RelTRSRRRProof [EQUIVALENT, 7 ms]
49.27/13.79	(4) RelTRS
49.27/13.79	(5) FlatCCProof [EQUIVALENT, 0 ms]
49.27/13.79	(6) RelTRS
49.27/13.79	(7) RootLabelingProof [EQUIVALENT, 0 ms]
49.27/13.79	(8) RelTRS
49.27/13.79	(9) RelTRSRRRProof [EQUIVALENT, 0 ms]
49.27/13.79	(10) RelTRS
49.27/13.79	(11) RelTRSRRRProof [EQUIVALENT, 176 ms]
49.27/13.79	(12) RelTRS
49.27/13.79	(13) RelTRSRRRProof [EQUIVALENT, 193 ms]
49.27/13.79	(14) RelTRS
49.27/13.79	(15) RelTRSRRRProof [EQUIVALENT, 6 ms]
49.27/13.79	(16) RelTRS
49.27/13.79	(17) RelTRSRRRProof [EQUIVALENT, 5 ms]
49.27/13.79	(18) RelTRS
49.27/13.79	(19) RelTRSRRRProof [EQUIVALENT, 65 ms]
49.27/13.79	(20) RelTRS
49.27/13.79	(21) RelTRSRRRProof [EQUIVALENT, 7 ms]
49.27/13.79	(22) RelTRS
49.27/13.79	(23) RelTRSRRRProof [EQUIVALENT, 316 ms]
49.27/13.79	(24) RelTRS
49.27/13.79	(25) RIsEmptyProof [EQUIVALENT, 0 ms]
49.27/13.79	(26) YES
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(0)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a(x1) -> b(c(x1))
49.27/13.79	   a(b(x1)) -> b(a(x1))
49.27/13.79	   a(c(x1)) -> c(a(x1))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d(c(x1)) -> d(a(x1))
49.27/13.79	   d(a(x1)) -> d(c(b(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(1) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(c(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	
49.27/13.79	   a(x1) -> b(c(x1))
49.27/13.79	Rules from S:
49.27/13.79	none
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(2)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a(b(x1)) -> b(a(x1))
49.27/13.79	   a(c(x1)) -> c(a(x1))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d(c(x1)) -> d(a(x1))
49.27/13.79	   d(a(x1)) -> d(c(b(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(3) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(c(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 1]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	
49.27/13.79	   a(c(x1)) -> c(a(x1))
49.27/13.79	Rules from S:
49.27/13.79	none
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(4)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a(b(x1)) -> b(a(x1))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d(c(x1)) -> d(a(x1))
49.27/13.79	   d(a(x1)) -> d(c(b(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(5) FlatCCProof (EQUIVALENT)
49.27/13.79	We used flat context closure [ROOTLAB]
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(6)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a(a(b(x1))) -> a(b(a(x1)))
49.27/13.79	   b(a(b(x1))) -> b(b(a(x1)))
49.27/13.79	   d(a(b(x1))) -> d(b(a(x1)))
49.27/13.79	   c(a(b(x1))) -> c(b(a(x1)))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d(c(x1)) -> d(a(x1))
49.27/13.79	   d(a(x1)) -> d(c(b(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(7) RootLabelingProof (EQUIVALENT)
49.27/13.79	We used plain root labeling [ROOTLAB] with the following heuristic:
49.27/13.79	LabelAll: All function symbols get labeled
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(8)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   c_{a_1}(a_{b_1}(b_{d_1}(x1))) -> c_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(x1))
49.27/13.79	   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
49.27/13.79	   d_{c_1}(c_{d_1}(x1)) -> d_{a_1}(a_{d_1}(x1))
49.27/13.79	   d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{c_1}(x1))
49.27/13.79	   d_{a_1}(a_{a_1}(x1)) -> d_{c_1}(c_{b_1}(b_{a_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{d_1}(x1)) -> d_{c_1}(c_{b_1}(b_{d_1}(x1)))
49.27/13.79	   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(9) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Polynomial interpretation [POLO]:
49.27/13.79	
49.27/13.79	   POL(a_{a_1}(x_1)) = x_1
49.27/13.79	   POL(a_{b_1}(x_1)) = x_1
49.27/13.79	   POL(a_{c_1}(x_1)) = x_1
49.27/13.79	   POL(a_{d_1}(x_1)) = x_1
49.27/13.79	   POL(b_{a_1}(x_1)) = x_1
49.27/13.79	   POL(b_{b_1}(x_1)) = x_1
49.27/13.79	   POL(b_{c_1}(x_1)) = x_1
49.27/13.79	   POL(b_{d_1}(x_1)) = x_1
49.27/13.79	   POL(c_{a_1}(x_1)) = 1 + x_1
49.27/13.79	   POL(c_{b_1}(x_1)) = x_1
49.27/13.79	   POL(c_{c_1}(x_1)) = 1 + x_1
49.27/13.79	   POL(c_{d_1}(x_1)) = 1 + x_1
49.27/13.79	   POL(d_{a_1}(x_1)) = x_1
49.27/13.79	   POL(d_{b_1}(x_1)) = x_1
49.27/13.79	   POL(d_{c_1}(x_1)) = x_1
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	
49.27/13.79	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   c_{a_1}(a_{b_1}(b_{d_1}(x1))) -> c_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	Rules from S:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(x1))
49.27/13.79	   d_{c_1}(c_{d_1}(x1)) -> d_{a_1}(a_{d_1}(x1))
49.27/13.79	   d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{c_1}(x1))
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(10)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
49.27/13.79	   d_{a_1}(a_{a_1}(x1)) -> d_{c_1}(c_{b_1}(b_{a_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{d_1}(x1)) -> d_{c_1}(c_{b_1}(b_{d_1}(x1)))
49.27/13.79	   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(11) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{a_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{a_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{d_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{d_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	none
49.27/13.79	Rules from S:
49.27/13.79	
49.27/13.79	   d_{a_1}(a_{a_1}(x1)) -> d_{c_1}(c_{b_1}(b_{a_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(12)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
49.27/13.79	   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{d_1}(x1)) -> d_{c_1}(c_{b_1}(b_{d_1}(x1)))
49.27/13.79	   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(13) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [2, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{d_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{d_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{a_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{c_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	none
49.27/13.79	Rules from S:
49.27/13.79	
49.27/13.79	   d_{a_1}(a_{d_1}(x1)) -> d_{c_1}(c_{b_1}(b_{d_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(14)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
49.27/13.79	   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(15) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Polynomial interpretation [POLO]:
49.27/13.79	
49.27/13.79	   POL(a_{a_1}(x_1)) = x_1
49.27/13.79	   POL(a_{b_1}(x_1)) = x_1
49.27/13.79	   POL(a_{c_1}(x_1)) = x_1
49.27/13.79	   POL(a_{d_1}(x_1)) = x_1
49.27/13.79	   POL(b_{a_1}(x_1)) = x_1
49.27/13.79	   POL(b_{b_1}(x_1)) = x_1
49.27/13.79	   POL(b_{c_1}(x_1)) = x_1
49.27/13.79	   POL(b_{d_1}(x_1)) = 1 + x_1
49.27/13.79	   POL(c_{b_1}(x_1)) = x_1
49.27/13.79	   POL(d_{a_1}(x_1)) = x_1
49.27/13.79	   POL(d_{b_1}(x_1)) = x_1
49.27/13.79	   POL(d_{c_1}(x_1)) = x_1
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{b_1}(b_{a_1}(a_{d_1}(x1)))
49.27/13.79	Rules from S:
49.27/13.79	none
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(16)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
49.27/13.79	   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(17) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Polynomial interpretation [POLO]:
49.27/13.79	
49.27/13.79	   POL(a_{a_1}(x_1)) = x_1
49.27/13.79	   POL(a_{b_1}(x_1)) = x_1
49.27/13.79	   POL(a_{c_1}(x_1)) = x_1
49.27/13.79	   POL(b_{a_1}(x_1)) = x_1
49.27/13.79	   POL(b_{b_1}(x_1)) = x_1
49.27/13.79	   POL(b_{c_1}(x_1)) = x_1
49.27/13.79	   POL(c_{b_1}(x_1)) = x_1
49.27/13.79	   POL(d_{a_1}(x_1)) = 1 + x_1
49.27/13.79	   POL(d_{b_1}(x_1)) = x_1
49.27/13.79	   POL(d_{c_1}(x_1)) = 1 + x_1
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	
49.27/13.79	   d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	Rules from S:
49.27/13.79	none
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(18)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
49.27/13.79	   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
49.27/13.79	   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(19) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{a_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{c_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{c_1}(x_1)) =  	[[2], [2]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	none
49.27/13.79	Rules from S:
49.27/13.79	
49.27/13.79	   d_{a_1}(a_{c_1}(x1)) -> d_{c_1}(c_{b_1}(b_{c_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(20)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
49.27/13.79	   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(21) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Polynomial interpretation [POLO]:
49.27/13.79	
49.27/13.79	   POL(a_{a_1}(x_1)) = 1 + x_1
49.27/13.79	   POL(a_{b_1}(x_1)) = x_1
49.27/13.79	   POL(a_{c_1}(x_1)) = x_1
49.27/13.79	   POL(b_{a_1}(x_1)) = 1 + x_1
49.27/13.79	   POL(b_{b_1}(x_1)) = x_1
49.27/13.79	   POL(b_{c_1}(x_1)) = 1 + x_1
49.27/13.79	   POL(c_{b_1}(x_1)) = x_1
49.27/13.79	   POL(d_{a_1}(x_1)) = x_1
49.27/13.79	   POL(d_{c_1}(x_1)) = x_1
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1)))
49.27/13.79	Rules from S:
49.27/13.79	none
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(22)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	The relative TRS consists of the following R rules:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
49.27/13.79	   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(23) RelTRSRRRProof (EQUIVALENT)
49.27/13.79	We used the following monotonic ordering for rule removal:
49.27/13.79	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{a_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(b_{b_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	   <<<
49.27/13.79	 POL(d_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
49.27/13.79	>>>
49.27/13.79	
49.27/13.79	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
49.27/13.79	Rules from R:
49.27/13.79	
49.27/13.79	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1)))
49.27/13.79	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1)))
49.27/13.79	Rules from S:
49.27/13.79	none
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(24)
49.27/13.79	Obligation:
49.27/13.79	Relative term rewrite system:
49.27/13.79	R is empty.
49.27/13.79	The relative TRS consists of the following S rules:
49.27/13.79	
49.27/13.79	   d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{b_1}(x1))
49.27/13.79	   d_{a_1}(a_{b_1}(x1)) -> d_{c_1}(c_{b_1}(b_{b_1}(x1)))
49.27/13.79	
49.27/13.79	
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(25) RIsEmptyProof (EQUIVALENT)
49.27/13.79	The TRS R is empty. Hence, termination is trivially proven.
49.27/13.79	----------------------------------------
49.27/13.79	
49.27/13.79	(26)
49.27/13.79	YES
49.27/13.83	EOF