5.12/2.15	YES
5.12/2.16	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.12/2.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.12/2.16	
5.12/2.16	
5.12/2.16	Termination of the given RelTRS could be proven:
5.12/2.16	
5.12/2.16	(0) RelTRS
5.12/2.16	(1) RelTRSRRRProof [EQUIVALENT, 42 ms]
5.12/2.16	(2) RelTRS
5.12/2.16	(3) RelTRSRRRProof [EQUIVALENT, 21 ms]
5.12/2.16	(4) RelTRS
5.12/2.16	(5) RelTRSRRRProof [EQUIVALENT, 2 ms]
5.12/2.16	(6) RelTRS
5.12/2.16	(7) RIsEmptyProof [EQUIVALENT, 0 ms]
5.12/2.16	(8) YES
5.12/2.16	
5.12/2.16	
5.12/2.16	----------------------------------------
5.12/2.16	
5.12/2.16	(0)
5.12/2.16	Obligation:
5.12/2.16	Relative term rewrite system:
5.12/2.16	The relative TRS consists of the following R rules:
5.12/2.16	
5.12/2.16	   s(a(x)) -> s(b(x))
5.12/2.16	   b(b(x)) -> a(x)
5.12/2.16	
5.12/2.16	The relative TRS consists of the following S rules:
5.12/2.16	
5.12/2.16	   f(s(x), y) -> f(x, s(y))
5.12/2.16	   s(a(x)) -> a(s(x))
5.12/2.16	   s(b(x)) -> b(s(x))
5.12/2.16	   a(s(x)) -> s(a(x))
5.12/2.16	   b(s(x)) -> s(b(x))
5.12/2.16	
5.12/2.16	
5.12/2.16	----------------------------------------
5.12/2.16	
5.12/2.16	(1) RelTRSRRRProof (EQUIVALENT)
5.12/2.16	We used the following monotonic ordering for rule removal:
5.12/2.16	Polynomial interpretation [POLO]:
5.12/2.16	
5.12/2.16	   POL(a(x_1)) = 1 + x_1
5.12/2.16	   POL(b(x_1)) = 1 + x_1
5.12/2.16	   POL(f(x_1, x_2)) = x_1 + x_2
5.12/2.16	   POL(s(x_1)) = x_1
5.12/2.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.12/2.16	Rules from R:
5.12/2.16	
5.12/2.16	   b(b(x)) -> a(x)
5.12/2.16	Rules from S:
5.12/2.16	none
5.12/2.16	
5.12/2.16	
5.12/2.16	
5.12/2.16	
5.12/2.16	----------------------------------------
5.12/2.16	
5.12/2.16	(2)
5.12/2.16	Obligation:
5.12/2.16	Relative term rewrite system:
5.12/2.16	The relative TRS consists of the following R rules:
5.12/2.16	
5.12/2.16	   s(a(x)) -> s(b(x))
5.12/2.16	
5.12/2.16	The relative TRS consists of the following S rules:
5.12/2.16	
5.12/2.16	   f(s(x), y) -> f(x, s(y))
5.12/2.16	   s(a(x)) -> a(s(x))
5.12/2.16	   s(b(x)) -> b(s(x))
5.12/2.16	   a(s(x)) -> s(a(x))
5.12/2.16	   b(s(x)) -> s(b(x))
5.12/2.16	
5.12/2.16	
5.12/2.16	----------------------------------------
5.12/2.16	
5.12/2.16	(3) RelTRSRRRProof (EQUIVALENT)
5.12/2.16	We used the following monotonic ordering for rule removal:
5.12/2.16	s/1(YES)
5.12/2.16	a/1)YES(
5.12/2.16	b/1)YES(
5.12/2.16	f/2(YES,YES)
5.12/2.16	
5.12/2.16	Quasi precedence:
5.12/2.16	f_2 > s_1
5.12/2.16	
5.12/2.16	
5.12/2.16	Status:
5.12/2.16	s_1: multiset status
5.12/2.16	f_2: [1,2]
5.12/2.16	
5.12/2.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.12/2.16	Rules from R:
5.12/2.16	none
5.12/2.16	Rules from S:
5.12/2.16	
5.12/2.16	   f(s(x), y) -> f(x, s(y))
5.12/2.16	
5.12/2.16	
5.12/2.16	
5.12/2.16	
5.12/2.16	----------------------------------------
5.12/2.16	
5.12/2.16	(4)
5.12/2.16	Obligation:
5.12/2.16	Relative term rewrite system:
5.12/2.16	The relative TRS consists of the following R rules:
5.12/2.16	
5.12/2.16	   s(a(x)) -> s(b(x))
5.12/2.16	
5.12/2.16	The relative TRS consists of the following S rules:
5.12/2.16	
5.12/2.16	   s(a(x)) -> a(s(x))
5.12/2.16	   s(b(x)) -> b(s(x))
5.12/2.16	   a(s(x)) -> s(a(x))
5.12/2.16	   b(s(x)) -> s(b(x))
5.12/2.16	
5.12/2.16	
5.12/2.16	----------------------------------------
5.12/2.16	
5.12/2.16	(5) RelTRSRRRProof (EQUIVALENT)
5.12/2.16	We used the following monotonic ordering for rule removal:
5.12/2.16	Polynomial interpretation [POLO]:
5.12/2.16	
5.12/2.16	   POL(a(x_1)) = 1 + x_1
5.12/2.16	   POL(b(x_1)) = x_1
5.12/2.16	   POL(s(x_1)) = x_1
5.12/2.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.12/2.16	Rules from R:
5.12/2.16	
5.12/2.16	   s(a(x)) -> s(b(x))
5.12/2.16	Rules from S:
5.12/2.16	none
5.12/2.16	
5.12/2.16	
5.12/2.16	
5.12/2.16	
5.12/2.16	----------------------------------------
5.12/2.16	
5.12/2.16	(6)
5.12/2.16	Obligation:
5.12/2.16	Relative term rewrite system:
5.12/2.16	R is empty.
5.12/2.16	The relative TRS consists of the following S rules:
5.12/2.16	
5.12/2.16	   s(a(x)) -> a(s(x))
5.12/2.16	   s(b(x)) -> b(s(x))
5.12/2.16	   a(s(x)) -> s(a(x))
5.12/2.16	   b(s(x)) -> s(b(x))
5.12/2.16	
5.12/2.16	
5.12/2.16	----------------------------------------
5.12/2.16	
5.12/2.16	(7) RIsEmptyProof (EQUIVALENT)
5.12/2.16	The TRS R is empty. Hence, termination is trivially proven.
5.12/2.16	----------------------------------------
5.12/2.16	
5.12/2.16	(8)
5.12/2.16	YES
5.34/2.21	EOF