6.08/2.50	YES
6.08/2.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.08/2.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.08/2.51	
6.08/2.51	
6.08/2.51	Termination of the given ETRS could be proven:
6.08/2.51	
6.08/2.51	(0) ETRS
6.08/2.51	(1) EquationalDependencyPairsProof [EQUIVALENT, 0 ms]
6.08/2.51	(2) EDP
6.08/2.51	(3) EDependencyGraphProof [EQUIVALENT, 0 ms]
6.08/2.51	(4) AND
6.08/2.51	    (5) EDP
6.08/2.51	        (6) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
6.08/2.51	        (7) EDP
6.08/2.51	        (8) EUsableRulesReductionPairsProof [EQUIVALENT, 8 ms]
6.08/2.51	        (9) EDP
6.08/2.51	        (10) EDPProblemToQDPProblemProof [EQUIVALENT, 0 ms]
6.08/2.51	        (11) QDP
6.08/2.51	        (12) MNOCProof [EQUIVALENT, 0 ms]
6.08/2.51	        (13) QDP
6.08/2.51	        (14) MRRProof [EQUIVALENT, 0 ms]
6.08/2.51	        (15) QDP
6.08/2.51	        (16) DependencyGraphProof [EQUIVALENT, 0 ms]
6.08/2.51	        (17) TRUE
6.08/2.51	    (18) EDP
6.08/2.51	        (19) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
6.08/2.51	        (20) EDP
6.08/2.51	        (21) EDPPoloProof [EQUIVALENT, 10 ms]
6.08/2.51	        (22) EDP
6.08/2.51	        (23) PisEmptyProof [EQUIVALENT, 0 ms]
6.08/2.51	        (24) YES
6.08/2.51	    (25) EDP
6.08/2.51	        (26) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
6.08/2.51	        (27) EDP
6.08/2.51	        (28) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
6.08/2.51	        (29) EDP
6.08/2.51	        (30) ERuleRemovalProof [EQUIVALENT, 0 ms]
6.08/2.51	        (31) EDP
6.08/2.51	        (32) EDPPoloProof [EQUIVALENT, 4 ms]
6.08/2.51	        (33) EDP
6.08/2.51	        (34) PisEmptyProof [EQUIVALENT, 0 ms]
6.08/2.51	        (35) YES
6.08/2.51	    (36) EDP
6.08/2.51	        (37) ESharpUsableEquationsProof [EQUIVALENT, 5 ms]
6.08/2.51	        (38) EDP
6.08/2.51	        (39) EDPPoloProof [EQUIVALENT, 0 ms]
6.08/2.51	        (40) EDP
6.08/2.51	        (41) EDPPoloProof [EQUIVALENT, 4 ms]
6.08/2.51	        (42) EDP
6.08/2.51	        (43) PisEmptyProof [EQUIVALENT, 0 ms]
6.08/2.51	        (44) YES
6.08/2.51	
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(0)
6.08/2.51	Obligation:
6.08/2.51	Equational rewrite system:
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(1) EquationalDependencyPairsProof (EQUIVALENT)
6.08/2.51	Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   PLUS(x, s(y)) -> PLUS(x, y)
6.08/2.51	   TIMES(x, s(y)) -> PLUS(x, times(x, y))
6.08/2.51	   TIMES(x, s(y)) -> TIMES(x, y)
6.08/2.51	   MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))
6.08/2.51	   MINUS(s(x), s(y)) -> P(s(x))
6.08/2.51	   MINUS(s(x), s(y)) -> P(s(y))
6.08/2.51	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
6.08/2.51	   DIV(s(x), s(y)) -> MINUS(x, y)
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
6.08/2.51	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
6.08/2.51	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
6.08/2.51	   TIMES(times(x, s(y)), ext) -> PLUS(x, times(x, y))
6.08/2.51	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	The set E# consists of the following equations:
6.08/2.51	
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	   TIMES(x, y) == TIMES(y, x)
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(2)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   PLUS(x, s(y)) -> PLUS(x, y)
6.08/2.51	   TIMES(x, s(y)) -> PLUS(x, times(x, y))
6.08/2.51	   TIMES(x, s(y)) -> TIMES(x, y)
6.08/2.51	   MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))
6.08/2.51	   MINUS(s(x), s(y)) -> P(s(x))
6.08/2.51	   MINUS(s(x), s(y)) -> P(s(y))
6.08/2.51	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
6.08/2.51	   DIV(s(x), s(y)) -> MINUS(x, y)
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
6.08/2.51	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
6.08/2.51	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
6.08/2.51	   TIMES(times(x, s(y)), ext) -> PLUS(x, times(x, y))
6.08/2.51	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	The set E# consists of the following equations:
6.08/2.51	
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	   TIMES(x, y) == TIMES(y, x)
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(3) EDependencyGraphProof (EQUIVALENT)
6.08/2.51	The approximation of the Equational Dependency Graph [DA_STEIN] contains 4 SCCs with 5 less nodes.
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(4)
6.08/2.51	Complex Obligation (AND)
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(5)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	The set E# consists of the following equations:
6.08/2.51	
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	   TIMES(x, y) == TIMES(y, x)
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(6) ESharpUsableEquationsProof (EQUIVALENT)
6.08/2.51	We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	   TIMES(x, y) == TIMES(y, x)
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(7)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	E# is empty.
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(8) EUsableRulesReductionPairsProof (EQUIVALENT)
6.08/2.51	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.
6.08/2.51	
6.08/2.51	No dependency pairs are removed.
6.08/2.51	
6.08/2.51	The following rules are removed from R:
6.08/2.51	
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	The following equations are removed from E:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	Used ordering: POLO with Polynomial interpretation [POLO]:
6.08/2.51	
6.08/2.51	   POL(MINUS(x_1, x_2)) = x_1 + x_2
6.08/2.51	   POL(p(x_1)) = x_1
6.08/2.51	   POL(s(x_1)) = x_1
6.08/2.51	
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(9)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	
6.08/2.51	E is empty.
6.08/2.51	E# is empty.
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(10) EDPProblemToQDPProblemProof (EQUIVALENT)
6.08/2.51	The EDP problem does not contain equations anymore, so we can transform it with the EDP to QDP problem processor [DA_STEIN] into a QDP problem.
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(11)
6.08/2.51	Obligation:
6.08/2.51	Q DP problem:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	
6.08/2.51	Q is empty.
6.08/2.51	We have to consider all minimal (P,Q,R)-chains.
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(12) MNOCProof (EQUIVALENT)
6.08/2.51	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(13)
6.08/2.51	Obligation:
6.08/2.51	Q DP problem:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	
6.08/2.51	The set Q consists of the following terms:
6.08/2.51	
6.08/2.51	   p(s(x0))
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,Q,R)-chains.
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(14) MRRProof (EQUIVALENT)
6.08/2.51	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
6.08/2.51	
6.08/2.51	
6.08/2.51	Strictly oriented rules of the TRS R:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	
6.08/2.51	Used ordering: Polynomial interpretation [POLO]:
6.08/2.51	
6.08/2.51	   POL(MINUS(x_1, x_2)) = 2*x_1 + 2*x_2
6.08/2.51	   POL(p(x_1)) = x_1
6.08/2.51	   POL(s(x_1)) = 2 + 2*x_1
6.08/2.51	
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(15)
6.08/2.51	Obligation:
6.08/2.51	Q DP problem:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   MINUS(s(x), s(y)) -> MINUS(p(s(x)), p(s(y)))
6.08/2.51	
6.08/2.51	R is empty.
6.08/2.51	The set Q consists of the following terms:
6.08/2.51	
6.08/2.51	   p(s(x0))
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,Q,R)-chains.
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(16) DependencyGraphProof (EQUIVALENT)
6.08/2.51	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(17)
6.08/2.51	TRUE
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(18)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	The set E# consists of the following equations:
6.08/2.51	
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	   TIMES(x, y) == TIMES(y, x)
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(19) ESharpUsableEquationsProof (EQUIVALENT)
6.08/2.51	We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	   TIMES(x, y) == TIMES(y, x)
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(20)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	E# is empty.
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(21) EDPPoloProof (EQUIVALENT)
6.08/2.51	We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented.
6.08/2.51	
6.08/2.51	
6.08/2.51	   DIV(s(x), s(y)) -> DIV(minus(x, y), s(y))
6.08/2.51	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
6.08/2.51	
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	There is no equation of E#.
6.08/2.51	
6.08/2.51	
6.08/2.51	With the implicit AFS there is no usable equation of E.
6.08/2.51	
6.08/2.51	
6.08/2.51	Used ordering: POLO with Polynomial interpretation [POLO]:
6.08/2.51	
6.08/2.51	   POL(0) = 0
6.08/2.51	   POL(DIV(x_1, x_2)) = x_1
6.08/2.51	   POL(minus(x_1, x_2)) = 2*x_1
6.08/2.51	   POL(p(x_1)) = x_1
6.08/2.51	   POL(s(x_1)) = 1 + 2*x_1
6.08/2.51	
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(22)
6.08/2.51	Obligation:
6.08/2.51	P is empty.
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	E# is empty.
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(23) PisEmptyProof (EQUIVALENT)
6.08/2.51	The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(24)
6.08/2.51	YES
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(25)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
6.08/2.51	   PLUS(x, s(y)) -> PLUS(x, y)
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	The set E# consists of the following equations:
6.08/2.51	
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	   TIMES(x, y) == TIMES(y, x)
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(26) ESharpUsableEquationsProof (EQUIVALENT)
6.08/2.51	We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
6.08/2.51	   TIMES(x, y) == TIMES(y, x)
6.08/2.51	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(27)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
6.08/2.51	   PLUS(x, s(y)) -> PLUS(x, y)
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	
6.08/2.51	The set E# consists of the following equations:
6.08/2.51	
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(28) EUsableRulesReductionPairsProof (EQUIVALENT)
6.08/2.51	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.
6.08/2.51	
6.08/2.51	No dependency pairs are removed.
6.08/2.51	
6.08/2.51	The following rules are removed from R:
6.08/2.51	
6.08/2.51	   p(s(x)) -> x
6.08/2.51	   plus(x, 0) -> x
6.08/2.51	   times(x, 0) -> 0
6.08/2.51	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.51	   minus(x, 0) -> x
6.08/2.51	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.51	   div(0, s(y)) -> 0
6.08/2.51	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.51	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.51	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.51	The following equations are removed from E:
6.08/2.51	
6.08/2.51	   times(x, y) == times(y, x)
6.08/2.51	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.51	Used ordering: POLO with Polynomial interpretation [POLO]:
6.08/2.51	
6.08/2.51	   POL(0) = 0
6.08/2.51	   POL(PLUS(x_1, x_2)) = x_1 + x_2
6.08/2.51	   POL(plus(x_1, x_2)) = x_1 + x_2
6.08/2.51	   POL(s(x_1)) = x_1
6.08/2.51	
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(29)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
6.08/2.51	   PLUS(x, s(y)) -> PLUS(x, y)
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	
6.08/2.51	The set E# consists of the following equations:
6.08/2.51	
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(30) ERuleRemovalProof (EQUIVALENT)
6.08/2.51	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.
6.08/2.51	
6.08/2.51	Strictly oriented dependency pairs:
6.08/2.51	
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)
6.08/2.51	   PLUS(x, s(y)) -> PLUS(x, y)
6.08/2.51	
6.08/2.51	
6.08/2.51	Used ordering: POLO with Polynomial interpretation [POLO]:
6.08/2.51	
6.08/2.51	   POL(PLUS(x_1, x_2)) = x_1 + x_2
6.08/2.51	   POL(plus(x_1, x_2)) = 2 + x_1 + x_2
6.08/2.51	   POL(s(x_1)) = 2 + x_1
6.08/2.51	
6.08/2.51	
6.08/2.51	----------------------------------------
6.08/2.51	
6.08/2.51	(31)
6.08/2.51	Obligation:
6.08/2.51	The TRS P consists of the following rules:
6.08/2.51	
6.08/2.51	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
6.08/2.51	
6.08/2.51	The TRS R consists of the following rules:
6.08/2.51	
6.08/2.51	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.51	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.51	
6.08/2.51	The set E consists of the following equations:
6.08/2.51	
6.08/2.51	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.51	   plus(x, y) == plus(y, x)
6.08/2.51	
6.08/2.51	The set E# consists of the following equations:
6.08/2.51	
6.08/2.51	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.51	   PLUS(x, y) == PLUS(y, x)
6.08/2.51	
6.08/2.51	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(32) EDPPoloProof (EQUIVALENT)
6.08/2.52	We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented.
6.08/2.52	
6.08/2.52	
6.08/2.52	   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
6.08/2.52	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
6.08/2.52	
6.08/2.52	
6.08/2.52	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.52	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.52	We had to orient the following equations of E# equivalently.
6.08/2.52	
6.08/2.52	
6.08/2.52	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.52	   PLUS(x, y) == PLUS(y, x)
6.08/2.52	With the implicit AFS we had to orient the following usable equations of E equivalently.
6.08/2.52	
6.08/2.52	
6.08/2.52	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.52	   plus(x, y) == plus(y, x)
6.08/2.52	Used ordering: POLO with Polynomial interpretation [POLO]:
6.08/2.52	
6.08/2.52	   POL(PLUS(x_1, x_2)) = x_1 + x_2
6.08/2.52	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
6.08/2.52	   POL(s(x_1)) = 0
6.08/2.52	
6.08/2.52	
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(33)
6.08/2.52	Obligation:
6.08/2.52	P is empty.
6.08/2.52	The TRS R consists of the following rules:
6.08/2.52	
6.08/2.52	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.52	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.52	
6.08/2.52	The set E consists of the following equations:
6.08/2.52	
6.08/2.52	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.52	   plus(x, y) == plus(y, x)
6.08/2.52	
6.08/2.52	The set E# consists of the following equations:
6.08/2.52	
6.08/2.52	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.52	   PLUS(x, y) == PLUS(y, x)
6.08/2.52	
6.08/2.52	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(34) PisEmptyProof (EQUIVALENT)
6.08/2.52	The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(35)
6.08/2.52	YES
6.08/2.52	
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(36)
6.08/2.52	Obligation:
6.08/2.52	The TRS P consists of the following rules:
6.08/2.52	
6.08/2.52	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
6.08/2.52	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
6.08/2.52	   TIMES(x, s(y)) -> TIMES(x, y)
6.08/2.52	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
6.08/2.52	
6.08/2.52	The TRS R consists of the following rules:
6.08/2.52	
6.08/2.52	   p(s(x)) -> x
6.08/2.52	   plus(x, 0) -> x
6.08/2.52	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.52	   times(x, 0) -> 0
6.08/2.52	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.52	   minus(x, 0) -> x
6.08/2.52	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.52	   div(0, s(y)) -> 0
6.08/2.52	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.52	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.52	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.52	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.52	
6.08/2.52	The set E consists of the following equations:
6.08/2.52	
6.08/2.52	   plus(x, y) == plus(y, x)
6.08/2.52	   times(x, y) == times(y, x)
6.08/2.52	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.52	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.52	
6.08/2.52	The set E# consists of the following equations:
6.08/2.52	
6.08/2.52	   PLUS(x, y) == PLUS(y, x)
6.08/2.52	   TIMES(x, y) == TIMES(y, x)
6.08/2.52	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.52	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.52	
6.08/2.52	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(37) ESharpUsableEquationsProof (EQUIVALENT)
6.08/2.52	We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
6.08/2.52	   PLUS(x, y) == PLUS(y, x)
6.08/2.52	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
6.08/2.52	
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(38)
6.08/2.52	Obligation:
6.08/2.52	The TRS P consists of the following rules:
6.08/2.52	
6.08/2.52	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
6.08/2.52	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
6.08/2.52	   TIMES(x, s(y)) -> TIMES(x, y)
6.08/2.52	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
6.08/2.52	
6.08/2.52	The TRS R consists of the following rules:
6.08/2.52	
6.08/2.52	   p(s(x)) -> x
6.08/2.52	   plus(x, 0) -> x
6.08/2.52	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.52	   times(x, 0) -> 0
6.08/2.52	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.52	   minus(x, 0) -> x
6.08/2.52	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.52	   div(0, s(y)) -> 0
6.08/2.52	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.52	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.52	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.52	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.52	
6.08/2.52	The set E consists of the following equations:
6.08/2.52	
6.08/2.52	   plus(x, y) == plus(y, x)
6.08/2.52	   times(x, y) == times(y, x)
6.08/2.52	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.52	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.52	
6.08/2.52	The set E# consists of the following equations:
6.08/2.52	
6.08/2.52	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.52	   TIMES(x, y) == TIMES(y, x)
6.08/2.52	
6.08/2.52	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(39) EDPPoloProof (EQUIVALENT)
6.08/2.52	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.
6.08/2.52	
6.08/2.52	
6.08/2.52	   TIMES(times(x, s(y)), ext) -> TIMES(plus(x, times(x, y)), ext)
6.08/2.52	   TIMES(times(x, s(y)), ext) -> TIMES(x, y)
6.08/2.52	   TIMES(x, s(y)) -> TIMES(x, y)
6.08/2.52	The remaining Dependency Pairs were at least non-strictly oriented.
6.08/2.52	
6.08/2.52	
6.08/2.52	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
6.08/2.52	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
6.08/2.52	
6.08/2.52	
6.08/2.52	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.52	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.52	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.52	   times(x, 0) -> 0
6.08/2.52	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.52	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.52	   plus(x, 0) -> x
6.08/2.52	We had to orient the following equations of E# equivalently.
6.08/2.52	
6.08/2.52	
6.08/2.52	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.52	   TIMES(x, y) == TIMES(y, x)
6.08/2.52	With the implicit AFS we had to orient the following usable equations of E equivalently.
6.08/2.52	
6.08/2.52	
6.08/2.52	   times(x, y) == times(y, x)
6.08/2.52	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.52	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.52	   plus(x, y) == plus(y, x)
6.08/2.52	Used ordering: POLO with Polynomial interpretation [POLO]:
6.08/2.52	
6.08/2.52	   POL(0) = 0
6.08/2.52	   POL(TIMES(x_1, x_2)) = x_1 + x_1*x_2 + x_2
6.08/2.52	   POL(plus(x_1, x_2)) = x_1 + x_2
6.08/2.52	   POL(s(x_1)) = 1 + x_1
6.08/2.52	   POL(times(x_1, x_2)) = x_1 + x_1*x_2 + x_2
6.08/2.52	
6.08/2.52	
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(40)
6.08/2.52	Obligation:
6.08/2.52	The TRS P consists of the following rules:
6.08/2.52	
6.08/2.52	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
6.08/2.52	
6.08/2.52	The TRS R consists of the following rules:
6.08/2.52	
6.08/2.52	   p(s(x)) -> x
6.08/2.52	   plus(x, 0) -> x
6.08/2.52	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.52	   times(x, 0) -> 0
6.08/2.52	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.52	   minus(x, 0) -> x
6.08/2.52	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.52	   div(0, s(y)) -> 0
6.08/2.52	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.52	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.52	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.52	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.52	
6.08/2.52	The set E consists of the following equations:
6.08/2.52	
6.08/2.52	   plus(x, y) == plus(y, x)
6.08/2.52	   times(x, y) == times(y, x)
6.08/2.52	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.52	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.52	
6.08/2.52	The set E# consists of the following equations:
6.08/2.52	
6.08/2.52	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.52	   TIMES(x, y) == TIMES(y, x)
6.08/2.52	
6.08/2.52	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(41) EDPPoloProof (EQUIVALENT)
6.08/2.52	We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented.
6.08/2.52	
6.08/2.52	
6.08/2.52	   TIMES(times(x, 0), ext) -> TIMES(0, ext)
6.08/2.52	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
6.08/2.52	
6.08/2.52	
6.08/2.52	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.52	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.52	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.52	   times(x, 0) -> 0
6.08/2.52	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.52	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.52	   plus(x, 0) -> x
6.08/2.52	We had to orient the following equations of E# equivalently.
6.08/2.52	
6.08/2.52	
6.08/2.52	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.52	   TIMES(x, y) == TIMES(y, x)
6.08/2.52	With the implicit AFS we had to orient the following usable equations of E equivalently.
6.08/2.52	
6.08/2.52	
6.08/2.52	   times(x, y) == times(y, x)
6.08/2.52	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.52	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.52	   plus(x, y) == plus(y, x)
6.08/2.52	Used ordering: POLO with Polynomial interpretation [POLO]:
6.08/2.52	
6.08/2.52	   POL(0) = 2
6.08/2.52	   POL(TIMES(x_1, x_2)) = 2*x_1 + x_1*x_2 + 2*x_2
6.08/2.52	   POL(plus(x_1, x_2)) = x_1 + x_2
6.08/2.52	   POL(s(x_1)) = 2 + x_1
6.08/2.52	   POL(times(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2
6.08/2.52	
6.08/2.52	
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(42)
6.08/2.52	Obligation:
6.08/2.52	P is empty.
6.08/2.52	The TRS R consists of the following rules:
6.08/2.52	
6.08/2.52	   p(s(x)) -> x
6.08/2.52	   plus(x, 0) -> x
6.08/2.52	   plus(x, s(y)) -> s(plus(x, y))
6.08/2.52	   times(x, 0) -> 0
6.08/2.52	   times(x, s(y)) -> plus(x, times(x, y))
6.08/2.52	   minus(x, 0) -> x
6.08/2.52	   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
6.08/2.52	   div(0, s(y)) -> 0
6.08/2.52	   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
6.08/2.52	   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
6.08/2.52	   times(times(x, 0), ext) -> times(0, ext)
6.08/2.52	   times(times(x, s(y)), ext) -> times(plus(x, times(x, y)), ext)
6.08/2.52	
6.08/2.52	The set E consists of the following equations:
6.08/2.52	
6.08/2.52	   plus(x, y) == plus(y, x)
6.08/2.52	   times(x, y) == times(y, x)
6.08/2.52	   plus(plus(x, y), z) == plus(x, plus(y, z))
6.08/2.52	   times(times(x, y), z) == times(x, times(y, z))
6.08/2.52	
6.08/2.52	The set E# consists of the following equations:
6.08/2.52	
6.08/2.52	   TIMES(times(x, y), z) == TIMES(x, times(y, z))
6.08/2.52	   TIMES(x, y) == TIMES(y, x)
6.08/2.52	
6.08/2.52	We have to consider all minimal (P,E#,R,E)-chains
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(43) PisEmptyProof (EQUIVALENT)
6.08/2.52	The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
6.08/2.52	----------------------------------------
6.08/2.52	
6.08/2.52	(44)
6.08/2.52	YES
6.08/2.54	EOF