5.89/2.42	YES
5.89/2.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.89/2.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.89/2.43	
5.89/2.43	
5.89/2.43	Termination of the given ETRS could be proven:
5.89/2.43	
5.89/2.43	(0) ETRS
5.89/2.43	(1) EquationalDependencyPairsProof [EQUIVALENT, 0 ms]
5.89/2.43	(2) EDP
5.89/2.43	(3) EDependencyGraphProof [EQUIVALENT, 0 ms]
5.89/2.43	(4) AND
5.89/2.43	    (5) EDP
5.89/2.43	        (6) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
5.89/2.43	        (7) EDP
5.89/2.43	        (8) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
5.89/2.43	        (9) EDP
5.89/2.43	        (10) PisEmptyProof [EQUIVALENT, 0 ms]
5.89/2.43	        (11) YES
5.89/2.43	    (12) EDP
5.89/2.43	        (13) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
5.89/2.43	        (14) EDP
5.89/2.43	        (15) EDPPoloProof [EQUIVALENT, 0 ms]
5.89/2.43	        (16) EDP
5.89/2.43	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
5.89/2.43	        (18) YES
5.89/2.43	    (19) EDP
5.89/2.43	        (20) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
5.89/2.43	        (21) EDP
5.89/2.43	        (22) EUsableRulesReductionPairsProof [EQUIVALENT, 21 ms]
5.89/2.43	        (23) EDP
5.89/2.43	        (24) ERuleRemovalProof [EQUIVALENT, 0 ms]
5.89/2.43	        (25) EDP
5.89/2.43	        (26) EDPPoloProof [EQUIVALENT, 0 ms]
5.89/2.43	        (27) EDP
5.89/2.43	        (28) PisEmptyProof [EQUIVALENT, 0 ms]
5.89/2.43	        (29) YES
5.89/2.43	    (30) EDP
5.89/2.43	        (31) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
5.89/2.43	        (32) EDP
5.89/2.43	        (33) EDPPoloProof [EQUIVALENT, 0 ms]
5.89/2.43	        (34) EDP
5.89/2.43	        (35) EDPPoloProof [EQUIVALENT, 0 ms]
5.89/2.43	        (36) EDP
5.89/2.43	        (37) PisEmptyProof [EQUIVALENT, 0 ms]
5.89/2.43	        (38) YES
5.89/2.43	
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(0)
5.89/2.43	Obligation:
5.89/2.43	Equational rewrite system:
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(1) EquationalDependencyPairsProof (EQUIVALENT)
5.89/2.43	Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem:
5.89/2.43	The TRS P consists of the following rules:
5.89/2.43	
5.89/2.43	   PLUS(x, s(y)) -> PLUS(x, y)
5.89/2.43	   TIMES(x, s(y)) -> PLUS(x, times(x, y))
5.89/2.43	   TIMES(x, s(y)) -> TIMES(x, y)
5.89/2.43	   MINUS(s(x), s(y)) -> MINUS(x, y)
5.89/2.43	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
5.89/2.43	   DIV(s(x), s(y)) -> MINUS(x, y)
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
5.89/2.43	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
5.89/2.43	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
5.89/2.43	   TIMES(times(x, s(y)), ext) -> PLUS(x, times(x, y))
5.89/2.43	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
5.89/2.43	
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	The set E# consists of the following equations:
5.89/2.43	
5.89/2.43	   PLUS(x, y) == PLUS(y, x)
5.89/2.43	   TIMES(x, y) == TIMES(y, x)
5.89/2.43	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.43	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.43	
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(2)
5.89/2.43	Obligation:
5.89/2.43	The TRS P consists of the following rules:
5.89/2.43	
5.89/2.43	   PLUS(x, s(y)) -> PLUS(x, y)
5.89/2.43	   TIMES(x, s(y)) -> PLUS(x, times(x, y))
5.89/2.43	   TIMES(x, s(y)) -> TIMES(x, y)
5.89/2.43	   MINUS(s(x), s(y)) -> MINUS(x, y)
5.89/2.43	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
5.89/2.43	   DIV(s(x), s(y)) -> MINUS(x, y)
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
5.89/2.43	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
5.89/2.43	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
5.89/2.43	   TIMES(times(x, s(y)), ext) -> PLUS(x, times(x, y))
5.89/2.43	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
5.89/2.43	
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	The set E# consists of the following equations:
5.89/2.43	
5.89/2.43	   PLUS(x, y) == PLUS(y, x)
5.89/2.43	   TIMES(x, y) == TIMES(y, x)
5.89/2.43	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.43	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.43	
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(3) EDependencyGraphProof (EQUIVALENT)
5.89/2.43	The approximation of the Equational Dependency Graph [DA_STEIN] contains 4 SCCs with 3 less nodes.
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(4)
5.89/2.43	Complex Obligation (AND)
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(5)
5.89/2.43	Obligation:
5.89/2.43	The TRS P consists of the following rules:
5.89/2.43	
5.89/2.43	   MINUS(s(x), s(y)) -> MINUS(x, y)
5.89/2.43	
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	The set E# consists of the following equations:
5.89/2.43	
5.89/2.43	   PLUS(x, y) == PLUS(y, x)
5.89/2.43	   TIMES(x, y) == TIMES(y, x)
5.89/2.43	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.43	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.43	
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(6) ESharpUsableEquationsProof (EQUIVALENT)
5.89/2.43	We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
5.89/2.43	   PLUS(x, y) == PLUS(y, x)
5.89/2.43	   TIMES(x, y) == TIMES(y, x)
5.89/2.43	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.43	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(7)
5.89/2.43	Obligation:
5.89/2.43	The TRS P consists of the following rules:
5.89/2.43	
5.89/2.43	   MINUS(s(x), s(y)) -> MINUS(x, y)
5.89/2.43	
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	E# is empty.
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(8) EUsableRulesReductionPairsProof (EQUIVALENT)
5.89/2.43	By using the usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules can be oriented non-strictly, the  usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
5.89/2.43	
5.89/2.43	The following dependency pairs can be deleted:
5.89/2.43	
5.89/2.43	   MINUS(s(x), s(y)) -> MINUS(x, y)
5.89/2.43	The following rules are removed from R:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	The following equations are removed from E:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	Used ordering: POLO with Polynomial interpretation [POLO]:
5.89/2.43	
5.89/2.43	   POL(MINUS(x_1, x_2)) = 3*x_1 + 3*x_2
5.89/2.43	   POL(s(x_1)) = x_1
5.89/2.43	
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(9)
5.89/2.43	Obligation:
5.89/2.43	P is empty.
5.89/2.43	R is empty.
5.89/2.43	E is empty.
5.89/2.43	E# is empty.
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(10) PisEmptyProof (EQUIVALENT)
5.89/2.43	The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(11)
5.89/2.43	YES
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(12)
5.89/2.43	Obligation:
5.89/2.43	The TRS P consists of the following rules:
5.89/2.43	
5.89/2.43	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
5.89/2.43	
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	The set E# consists of the following equations:
5.89/2.43	
5.89/2.43	   PLUS(x, y) == PLUS(y, x)
5.89/2.43	   TIMES(x, y) == TIMES(y, x)
5.89/2.43	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.43	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.43	
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(13) ESharpUsableEquationsProof (EQUIVALENT)
5.89/2.43	We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
5.89/2.43	   PLUS(x, y) == PLUS(y, x)
5.89/2.43	   TIMES(x, y) == TIMES(y, x)
5.89/2.43	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.43	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(14)
5.89/2.43	Obligation:
5.89/2.43	The TRS P consists of the following rules:
5.89/2.43	
5.89/2.43	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
5.89/2.43	
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	E# is empty.
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(15) EDPPoloProof (EQUIVALENT)
5.89/2.43	We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented.
5.89/2.43	
5.89/2.43	
5.89/2.43	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
5.89/2.43	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
5.89/2.43	
5.89/2.43	
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	There is no equation of E#.
5.89/2.43	
5.89/2.43	
5.89/2.43	With the implicit AFS there is no usable equation of E.
5.89/2.43	
5.89/2.43	
5.89/2.43	Used ordering: POLO with Polynomial interpretation [POLO]:
5.89/2.43	
5.89/2.43	   POL(0) = 0
5.89/2.43	   POL(DIV(x_1, x_2)) = 2*x_1
5.89/2.43	   POL(minus(x_1, x_2)) = 2*x_1
5.89/2.43	   POL(s(x_1)) = 2 + 2*x_1
5.89/2.43	
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(16)
5.89/2.43	Obligation:
5.89/2.43	P is empty.
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	E# is empty.
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(17) PisEmptyProof (EQUIVALENT)
5.89/2.43	The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(18)
5.89/2.43	YES
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(19)
5.89/2.43	Obligation:
5.89/2.43	The TRS P consists of the following rules:
5.89/2.43	
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
5.89/2.43	   PLUS(x, s(y)) -> PLUS(x, y)
5.89/2.43	
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	The set E# consists of the following equations:
5.89/2.43	
5.89/2.43	   PLUS(x, y) == PLUS(y, x)
5.89/2.43	   TIMES(x, y) == TIMES(y, x)
5.89/2.43	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.43	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.43	
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(20) ESharpUsableEquationsProof (EQUIVALENT)
5.89/2.43	We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
5.89/2.43	   TIMES(x, y) == TIMES(y, x)
5.89/2.43	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(21)
5.89/2.43	Obligation:
5.89/2.43	The TRS P consists of the following rules:
5.89/2.43	
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
5.89/2.43	   PLUS(x, s(y)) -> PLUS(x, y)
5.89/2.43	
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	
5.89/2.43	The set E consists of the following equations:
5.89/2.43	
5.89/2.43	   plus(x, y) == plus(y, x)
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	
5.89/2.43	The set E# consists of the following equations:
5.89/2.43	
5.89/2.43	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.43	   PLUS(x, y) == PLUS(y, x)
5.89/2.43	
5.89/2.43	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(22) EUsableRulesReductionPairsProof (EQUIVALENT)
5.89/2.43	By using the usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules can be oriented non-strictly, the  usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
5.89/2.43	
5.89/2.43	No dependency pairs are removed.
5.89/2.43	
5.89/2.43	The following rules are removed from R:
5.89/2.43	
5.89/2.43	   plus(x, 0) -> x
5.89/2.43	   times(x, 0) -> 0
5.89/2.43	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.43	   minus(x, 0) -> x
5.89/2.43	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.43	   div(0, s(y)) -> 0
5.89/2.43	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.43	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.43	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.43	The following equations are removed from E:
5.89/2.43	
5.89/2.43	   times(x, y) == times(y, x)
5.89/2.43	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.43	Used ordering: POLO with Polynomial interpretation [POLO]:
5.89/2.43	
5.89/2.43	   POL(0) = 0
5.89/2.43	   POL(PLUS(x_1, x_2)) = x_1 + x_2
5.89/2.43	   POL(plus(x_1, x_2)) = x_1 + x_2
5.89/2.43	   POL(s(x_1)) = x_1
5.89/2.43	
5.89/2.43	
5.89/2.43	----------------------------------------
5.89/2.43	
5.89/2.43	(23)
5.89/2.43	Obligation:
5.89/2.43	The TRS P consists of the following rules:
5.89/2.43	
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
5.89/2.43	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
5.89/2.43	   PLUS(x, s(y)) -> PLUS(x, y)
5.89/2.43	
5.89/2.43	The TRS R consists of the following rules:
5.89/2.43	
5.89/2.43	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.43	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	
5.89/2.44	The set E consists of the following equations:
5.89/2.44	
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	
5.89/2.44	The set E# consists of the following equations:
5.89/2.44	
5.89/2.44	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.44	   PLUS(x, y) == PLUS(y, x)
5.89/2.44	
5.89/2.44	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(24) ERuleRemovalProof (EQUIVALENT)
5.89/2.44	By using the rule removal processor [DA_STEIN] with the following polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this EDP problem can be strictly oriented.
5.89/2.44	
5.89/2.44	Strictly oriented dependency pairs:
5.89/2.44	
5.89/2.44	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
5.89/2.44	   PLUS(x, s(y)) -> PLUS(x, y)
5.89/2.44	
5.89/2.44	
5.89/2.44	Used ordering: POLO with Polynomial interpretation [POLO]:
5.89/2.44	
5.89/2.44	   POL(PLUS(x_1, x_2)) = x_1 + x_2
5.89/2.44	   POL(plus(x_1, x_2)) = 2 + x_1 + x_2
5.89/2.44	   POL(s(x_1)) = 2 + x_1
5.89/2.44	
5.89/2.44	
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(25)
5.89/2.44	Obligation:
5.89/2.44	The TRS P consists of the following rules:
5.89/2.44	
5.89/2.44	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
5.89/2.44	
5.89/2.44	The TRS R consists of the following rules:
5.89/2.44	
5.89/2.44	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.44	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	
5.89/2.44	The set E consists of the following equations:
5.89/2.44	
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	
5.89/2.44	The set E# consists of the following equations:
5.89/2.44	
5.89/2.44	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.44	   PLUS(x, y) == PLUS(y, x)
5.89/2.44	
5.89/2.44	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(26) EDPPoloProof (EQUIVALENT)
5.89/2.44	We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented.
5.89/2.44	
5.89/2.44	
5.89/2.44	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
5.89/2.44	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
5.89/2.44	
5.89/2.44	
5.89/2.44	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.44	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	We had to orient the following equations of E# equivalently.
5.89/2.44	
5.89/2.44	
5.89/2.44	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.44	   PLUS(x, y) == PLUS(y, x)
5.89/2.44	With the implicit AFS we had to orient the following usable equations of E equivalently.
5.89/2.44	
5.89/2.44	
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	Used ordering: POLO with Polynomial interpretation [POLO]:
5.89/2.44	
5.89/2.44	   POL(PLUS(x_1, x_2)) = x_1 + x_2
5.89/2.44	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
5.89/2.44	   POL(s(x_1)) = 0
5.89/2.44	
5.89/2.44	
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(27)
5.89/2.44	Obligation:
5.89/2.44	P is empty.
5.89/2.44	The TRS R consists of the following rules:
5.89/2.44	
5.89/2.44	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.44	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	
5.89/2.44	The set E consists of the following equations:
5.89/2.44	
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	
5.89/2.44	The set E# consists of the following equations:
5.89/2.44	
5.89/2.44	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.44	   PLUS(x, y) == PLUS(y, x)
5.89/2.44	
5.89/2.44	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(28) PisEmptyProof (EQUIVALENT)
5.89/2.44	The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(29)
5.89/2.44	YES
5.89/2.44	
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(30)
5.89/2.44	Obligation:
5.89/2.44	The TRS P consists of the following rules:
5.89/2.44	
5.89/2.44	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
5.89/2.44	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
5.89/2.44	   TIMES(x, s(y)) -> TIMES(x, y)
5.89/2.44	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
5.89/2.44	
5.89/2.44	The TRS R consists of the following rules:
5.89/2.44	
5.89/2.44	   plus(x, 0) -> x
5.89/2.44	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.44	   times(x, 0) -> 0
5.89/2.44	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.44	   minus(x, 0) -> x
5.89/2.44	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.44	   div(0, s(y)) -> 0
5.89/2.44	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.44	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.44	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.44	
5.89/2.44	The set E consists of the following equations:
5.89/2.44	
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	   times(x, y) == times(y, x)
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.44	
5.89/2.44	The set E# consists of the following equations:
5.89/2.44	
5.89/2.44	   PLUS(x, y) == PLUS(y, x)
5.89/2.44	   TIMES(x, y) == TIMES(y, x)
5.89/2.44	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.44	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.44	
5.89/2.44	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(31) ESharpUsableEquationsProof (EQUIVALENT)
5.89/2.44	We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
5.89/2.44	   PLUS(x, y) == PLUS(y, x)
5.89/2.44	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
5.89/2.44	
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(32)
5.89/2.44	Obligation:
5.89/2.44	The TRS P consists of the following rules:
5.89/2.44	
5.89/2.44	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
5.89/2.44	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
5.89/2.44	   TIMES(x, s(y)) -> TIMES(x, y)
5.89/2.44	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
5.89/2.44	
5.89/2.44	The TRS R consists of the following rules:
5.89/2.44	
5.89/2.44	   plus(x, 0) -> x
5.89/2.44	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.44	   times(x, 0) -> 0
5.89/2.44	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.44	   minus(x, 0) -> x
5.89/2.44	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.44	   div(0, s(y)) -> 0
5.89/2.44	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.44	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.44	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.44	
5.89/2.44	The set E consists of the following equations:
5.89/2.44	
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	   times(x, y) == times(y, x)
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.44	
5.89/2.44	The set E# consists of the following equations:
5.89/2.44	
5.89/2.44	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.44	   TIMES(x, y) == TIMES(y, x)
5.89/2.44	
5.89/2.44	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(33) EDPPoloProof (EQUIVALENT)
5.89/2.44	We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. The following set of Dependency Pairs of this DP problem can be strictly oriented.
5.89/2.44	
5.89/2.44	
5.89/2.44	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
5.89/2.44	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
5.89/2.44	   TIMES(x, s(y)) -> TIMES(x, y)
5.89/2.44	The remaining Dependency Pairs were at least non-strictly oriented.
5.89/2.44	
5.89/2.44	
5.89/2.44	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
5.89/2.44	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
5.89/2.44	
5.89/2.44	
5.89/2.44	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.44	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	   plus(x, 0) -> x
5.89/2.44	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.44	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.44	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.44	   times(x, 0) -> 0
5.89/2.44	We had to orient the following equations of E# equivalently.
5.89/2.44	
5.89/2.44	
5.89/2.44	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.44	   TIMES(x, y) == TIMES(y, x)
5.89/2.44	With the implicit AFS we had to orient the following usable equations of E equivalently.
5.89/2.44	
5.89/2.44	
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	   times(x, y) == times(y, x)
5.89/2.44	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.44	Used ordering: POLO with Polynomial interpretation [POLO]:
5.89/2.44	
5.89/2.44	   POL(0) = 0
5.89/2.44	   POL(TIMES(x_1, x_2)) = x_1 + x_1*x_2 + x_2
5.89/2.44	   POL(plus(x_1, x_2)) = x_1 + x_2
5.89/2.44	   POL(s(x_1)) = 1 + x_1
5.89/2.44	   POL(times(x_1, x_2)) = x_1 + x_1*x_2 + x_2
5.89/2.44	
5.89/2.44	
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(34)
5.89/2.44	Obligation:
5.89/2.44	The TRS P consists of the following rules:
5.89/2.44	
5.89/2.44	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
5.89/2.44	
5.89/2.44	The TRS R consists of the following rules:
5.89/2.44	
5.89/2.44	   plus(x, 0) -> x
5.89/2.44	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.44	   times(x, 0) -> 0
5.89/2.44	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.44	   minus(x, 0) -> x
5.89/2.44	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.44	   div(0, s(y)) -> 0
5.89/2.44	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.44	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.44	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.44	
5.89/2.44	The set E consists of the following equations:
5.89/2.44	
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	   times(x, y) == times(y, x)
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.44	
5.89/2.44	The set E# consists of the following equations:
5.89/2.44	
5.89/2.44	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.44	   TIMES(x, y) == TIMES(y, x)
5.89/2.44	
5.89/2.44	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(35) EDPPoloProof (EQUIVALENT)
5.89/2.44	We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented.
5.89/2.44	
5.89/2.44	
5.89/2.44	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
5.89/2.44	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
5.89/2.44	
5.89/2.44	
5.89/2.44	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.44	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.44	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.44	   times(x, 0) -> 0
5.89/2.44	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.44	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	   plus(x, 0) -> x
5.89/2.44	We had to orient the following equations of E# equivalently.
5.89/2.44	
5.89/2.44	
5.89/2.44	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.44	   TIMES(x, y) == TIMES(y, x)
5.89/2.44	With the implicit AFS we had to orient the following usable equations of E equivalently.
5.89/2.44	
5.89/2.44	
5.89/2.44	   times(x, y) == times(y, x)
5.89/2.44	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	Used ordering: POLO with Polynomial interpretation [POLO]:
5.89/2.44	
5.89/2.44	   POL(0) = 2
5.89/2.44	   POL(TIMES(x_1, x_2)) = 2*x_1 + x_1*x_2 + 2*x_2
5.89/2.44	   POL(plus(x_1, x_2)) = x_1 + x_2
5.89/2.44	   POL(s(x_1)) = 2 + x_1
5.89/2.44	   POL(times(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2
5.89/2.44	
5.89/2.44	
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(36)
5.89/2.44	Obligation:
5.89/2.44	P is empty.
5.89/2.44	The TRS R consists of the following rules:
5.89/2.44	
5.89/2.44	   plus(x, 0) -> x
5.89/2.44	   plus(x, s(y)) -> s(plus(x, y))
5.89/2.44	   times(x, 0) -> 0
5.89/2.44	   times(x, s(y)) -> plus(x, times(x, y))
5.89/2.44	   minus(x, 0) -> x
5.89/2.44	   minus(s(x), s(y)) -> minus(x, y)
5.89/2.44	   div(0, s(y)) -> 0
5.89/2.44	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
5.89/2.44	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
5.89/2.44	   times(times(x, 0), ext) -> times(0, ext)
5.89/2.44	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
5.89/2.44	
5.89/2.44	The set E consists of the following equations:
5.89/2.44	
5.89/2.44	   plus(x, y) == plus(y, x)
5.89/2.44	   times(x, y) == times(y, x)
5.89/2.44	   plus(plus(x, y), z) == plus(x, plus(y, z))
5.89/2.44	   times(times(x, y), z) == times(x, times(y, z))
5.89/2.44	
5.89/2.44	The set E# consists of the following equations:
5.89/2.44	
5.89/2.44	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
5.89/2.44	   TIMES(x, y) == TIMES(y, x)
5.89/2.44	
5.89/2.44	We have to consider all minimal (P,E#,R,E)-chains
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(37) PisEmptyProof (EQUIVALENT)
5.89/2.44	The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
5.89/2.44	----------------------------------------
5.89/2.44	
5.89/2.44	(38)
5.89/2.44	YES
6.16/2.67	EOF