10.83/5.34	YES
10.83/5.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
10.83/5.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.83/5.35	
10.83/5.35	
10.83/5.35	Termination of the given ETRS could be proven:
10.83/5.35	
10.83/5.35	(0) ETRS
10.83/5.35	(1) RRRPoloETRSProof [EQUIVALENT, 154 ms]
10.83/5.35	(2) ETRS
10.83/5.35	(3) RRRPoloETRSProof [EQUIVALENT, 45 ms]
10.83/5.35	(4) ETRS
10.83/5.35	(5) RRRPoloETRSProof [EQUIVALENT, 40 ms]
10.83/5.35	(6) ETRS
10.83/5.35	(7) RRRPoloETRSProof [EQUIVALENT, 1052 ms]
10.83/5.35	(8) ETRS
10.83/5.35	(9) EquationalDependencyPairsProof [EQUIVALENT, 66 ms]
10.83/5.35	(10) EDP
10.83/5.35	(11) EUsableRulesReductionPairsProof [EQUIVALENT, 17 ms]
10.83/5.35	(12) EDP
10.83/5.35	(13) ERuleRemovalProof [EQUIVALENT, 2 ms]
10.83/5.35	(14) EDP
10.83/5.35	(15) EDPPoloProof [EQUIVALENT, 0 ms]
10.83/5.35	(16) EDP
10.83/5.35	(17) PisEmptyProof [EQUIVALENT, 0 ms]
10.83/5.35	(18) YES
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(0)
10.83/5.35	Obligation:
10.83/5.35	Equational rewrite system:
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   0(S) -> S
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(0(x), 0(y)) -> 0(plus(x, y))
10.83/5.35	   plus(0(x), 1(y)) -> 1(plus(x, y))
10.83/5.35	   plus(0(x), j(y)) -> j(plus(x, y))
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   plus(1(x), j(y)) -> 0(plus(x, y))
10.83/5.35	   opp(S) -> S
10.83/5.35	   opp(0(x)) -> 0(opp(x))
10.83/5.35	   opp(1(x)) -> j(opp(x))
10.83/5.35	   opp(j(x)) -> 1(opp(x))
10.83/5.35	   minus(x, y) -> plus(opp(y), x)
10.83/5.35	   times(S, x) -> S
10.83/5.35	   times(0(x), y) -> 0(times(x, y))
10.83/5.35	   times(1(x), y) -> plus(0(times(x, y)), y)
10.83/5.35	   times(j(x), y) -> minus(0(times(x, y)), y)
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(1) RRRPoloETRSProof (EQUIVALENT)
10.83/5.35	The following E TRS is given: Equational rewrite system:
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   0(S) -> S
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(0(x), 0(y)) -> 0(plus(x, y))
10.83/5.35	   plus(0(x), 1(y)) -> 1(plus(x, y))
10.83/5.35	   plus(0(x), j(y)) -> j(plus(x, y))
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   plus(1(x), j(y)) -> 0(plus(x, y))
10.83/5.35	   opp(S) -> S
10.83/5.35	   opp(0(x)) -> 0(opp(x))
10.83/5.35	   opp(1(x)) -> j(opp(x))
10.83/5.35	   opp(j(x)) -> 1(opp(x))
10.83/5.35	   minus(x, y) -> plus(opp(y), x)
10.83/5.35	   times(S, x) -> S
10.83/5.35	   times(0(x), y) -> 0(times(x, y))
10.83/5.35	   times(1(x), y) -> plus(0(times(x, y)), y)
10.83/5.35	   times(j(x), y) -> minus(0(times(x, y)), y)
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
10.83/5.35	
10.83/5.35	   plus(1(x), j(y)) -> 0(plus(x, y))
10.83/5.35	   opp(1(x)) -> j(opp(x))
10.83/5.35	   opp(j(x)) -> 1(opp(x))
10.83/5.35	   minus(x, y) -> plus(opp(y), x)
10.83/5.35	   times(1(x), y) -> plus(0(times(x, y)), y)
10.83/5.35	Used ordering:
10.83/5.35	Polynomial interpretation [POLO]:
10.83/5.35	
10.83/5.35	   POL(0(x_1)) = x_1
10.83/5.35	   POL(1(x_1)) = 3 + x_1
10.83/5.35	   POL(S) = 0
10.83/5.35	   POL(j(x_1)) = 3 + x_1
10.83/5.35	   POL(minus(x_1, x_2)) = 3 + x_1 + 3*x_2
10.83/5.35	   POL(opp(x_1)) = 2*x_1
10.83/5.35	   POL(plus(x_1, x_2)) = x_1 + x_2
10.83/5.35	   POL(times(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2
10.83/5.35	
10.83/5.35	
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(2)
10.83/5.35	Obligation:
10.83/5.35	Equational rewrite system:
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   0(S) -> S
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(0(x), 0(y)) -> 0(plus(x, y))
10.83/5.35	   plus(0(x), 1(y)) -> 1(plus(x, y))
10.83/5.35	   plus(0(x), j(y)) -> j(plus(x, y))
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   opp(S) -> S
10.83/5.35	   opp(0(x)) -> 0(opp(x))
10.83/5.35	   times(S, x) -> S
10.83/5.35	   times(0(x), y) -> 0(times(x, y))
10.83/5.35	   times(j(x), y) -> minus(0(times(x, y)), y)
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(3) RRRPoloETRSProof (EQUIVALENT)
10.83/5.35	The following E TRS is given: Equational rewrite system:
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   0(S) -> S
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(0(x), 0(y)) -> 0(plus(x, y))
10.83/5.35	   plus(0(x), 1(y)) -> 1(plus(x, y))
10.83/5.35	   plus(0(x), j(y)) -> j(plus(x, y))
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   opp(S) -> S
10.83/5.35	   opp(0(x)) -> 0(opp(x))
10.83/5.35	   times(S, x) -> S
10.83/5.35	   times(0(x), y) -> 0(times(x, y))
10.83/5.35	   times(j(x), y) -> minus(0(times(x, y)), y)
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
10.83/5.35	
10.83/5.35	   times(S, x) -> S
10.83/5.35	   times(j(x), y) -> minus(0(times(x, y)), y)
10.83/5.35	Used ordering:
10.83/5.35	Polynomial interpretation [POLO]:
10.83/5.35	
10.83/5.35	   POL(0(x_1)) = x_1
10.83/5.35	   POL(1(x_1)) = 2 + x_1
10.83/5.35	   POL(S) = 0
10.83/5.35	   POL(j(x_1)) = 2 + x_1
10.83/5.35	   POL(minus(x_1, x_2)) = x_1 + x_2
10.83/5.35	   POL(opp(x_1)) = 2*x_1
10.83/5.35	   POL(plus(x_1, x_2)) = x_1 + x_2
10.83/5.35	   POL(times(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2
10.83/5.35	
10.83/5.35	
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(4)
10.83/5.35	Obligation:
10.83/5.35	Equational rewrite system:
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   0(S) -> S
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(0(x), 0(y)) -> 0(plus(x, y))
10.83/5.35	   plus(0(x), 1(y)) -> 1(plus(x, y))
10.83/5.35	   plus(0(x), j(y)) -> j(plus(x, y))
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   opp(S) -> S
10.83/5.35	   opp(0(x)) -> 0(opp(x))
10.83/5.35	   times(0(x), y) -> 0(times(x, y))
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(5) RRRPoloETRSProof (EQUIVALENT)
10.83/5.35	The following E TRS is given: Equational rewrite system:
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   0(S) -> S
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(0(x), 0(y)) -> 0(plus(x, y))
10.83/5.35	   plus(0(x), 1(y)) -> 1(plus(x, y))
10.83/5.35	   plus(0(x), j(y)) -> j(plus(x, y))
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   opp(S) -> S
10.83/5.35	   opp(0(x)) -> 0(opp(x))
10.83/5.35	   times(0(x), y) -> 0(times(x, y))
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
10.83/5.35	
10.83/5.35	   0(S) -> S
10.83/5.35	   plus(0(x), 0(y)) -> 0(plus(x, y))
10.83/5.35	   plus(0(x), 1(y)) -> 1(plus(x, y))
10.83/5.35	   plus(0(x), j(y)) -> j(plus(x, y))
10.83/5.35	   opp(S) -> S
10.83/5.35	Used ordering:
10.83/5.35	Polynomial interpretation [POLO]:
10.83/5.35	
10.83/5.35	   POL(0(x_1)) = 1 + 3*x_1
10.83/5.35	   POL(1(x_1)) = 2*x_1
10.83/5.35	   POL(S) = 0
10.83/5.35	   POL(j(x_1)) = 2*x_1
10.83/5.35	   POL(opp(x_1)) = 1 + 3*x_1
10.83/5.35	   POL(plus(x_1, x_2)) = x_1 + x_2
10.83/5.35	   POL(times(x_1, x_2)) = x_1 + 3*x_1*x_2 + x_2
10.83/5.35	
10.83/5.35	
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(6)
10.83/5.35	Obligation:
10.83/5.35	Equational rewrite system:
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   opp(0(x)) -> 0(opp(x))
10.83/5.35	   times(0(x), y) -> 0(times(x, y))
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(7) RRRPoloETRSProof (EQUIVALENT)
10.83/5.35	The following E TRS is given: Equational rewrite system:
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   opp(0(x)) -> 0(opp(x))
10.83/5.35	   times(0(x), y) -> 0(times(x, y))
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:
10.83/5.35	
10.83/5.35	   opp(0(x)) -> 0(opp(x))
10.83/5.35	   times(0(x), y) -> 0(times(x, y))
10.83/5.35	Used ordering:
10.83/5.35	Polynomial interpretation [POLO]:
10.83/5.35	
10.83/5.35	   POL(0(x_1)) = 2 + x_1
10.83/5.35	   POL(1(x_1)) = 2*x_1
10.83/5.35	   POL(S) = 0
10.83/5.35	   POL(j(x_1)) = 2*x_1
10.83/5.35	   POL(opp(x_1)) = 2 + 2*x_1^2
10.83/5.35	   POL(plus(x_1, x_2)) = x_1 + x_2
10.83/5.35	   POL(times(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2
10.83/5.35	
10.83/5.35	
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(8)
10.83/5.35	Obligation:
10.83/5.35	Equational rewrite system:
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(9) EquationalDependencyPairsProof (EQUIVALENT)
10.83/5.35	Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem:
10.83/5.35	The TRS P consists of the following rules:
10.83/5.35	
10.83/5.35	   PLUS(1(x), 1(y)) -> PLUS(1(S), plus(x, y))
10.83/5.35	   PLUS(1(x), 1(y)) -> PLUS(x, y)
10.83/5.35	   PLUS(j(x), j(y)) -> PLUS(j(S), plus(x, y))
10.83/5.35	   PLUS(j(x), j(y)) -> PLUS(x, y)
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(1(S), plus(x, y))
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(x, y)
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(j(S), plus(x, y))
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(x, y)
10.83/5.35	
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	The set E# consists of the following equations:
10.83/5.35	
10.83/5.35	   PLUS(x, y) == PLUS(y, x)
10.83/5.35	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
10.83/5.35	
10.83/5.35	We have to consider all minimal (P,E#,R,E)-chains
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(10)
10.83/5.35	Obligation:
10.83/5.35	The TRS P consists of the following rules:
10.83/5.35	
10.83/5.35	   PLUS(1(x), 1(y)) -> PLUS(1(S), plus(x, y))
10.83/5.35	   PLUS(1(x), 1(y)) -> PLUS(x, y)
10.83/5.35	   PLUS(j(x), j(y)) -> PLUS(j(S), plus(x, y))
10.83/5.35	   PLUS(j(x), j(y)) -> PLUS(x, y)
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(1(S), plus(x, y))
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(x, y)
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(j(S), plus(x, y))
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(x, y)
10.83/5.35	
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	
10.83/5.35	The set E# consists of the following equations:
10.83/5.35	
10.83/5.35	   PLUS(x, y) == PLUS(y, x)
10.83/5.35	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
10.83/5.35	
10.83/5.35	We have to consider all minimal (P,E#,R,E)-chains
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(11) EUsableRulesReductionPairsProof (EQUIVALENT)
10.83/5.35	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.
10.83/5.35	
10.83/5.35	No dependency pairs are removed.
10.83/5.35	
10.83/5.35	No rules are removed from R.
10.83/5.35	
10.83/5.35	The following equations are removed from E:
10.83/5.35	
10.83/5.35	   times(x, y) == times(y, x)
10.83/5.35	   times(times(x, y), z) == times(x, times(y, z))
10.83/5.35	Used ordering: POLO with Polynomial interpretation [POLO]:
10.83/5.35	
10.83/5.35	   POL(1(x_1)) = x_1
10.83/5.35	   POL(PLUS(x_1, x_2)) = x_1 + x_2
10.83/5.35	   POL(S) = 0
10.83/5.35	   POL(j(x_1)) = x_1
10.83/5.35	   POL(plus(x_1, x_2)) = x_1 + x_2
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(12)
10.83/5.35	Obligation:
10.83/5.35	The TRS P consists of the following rules:
10.83/5.35	
10.83/5.35	   PLUS(1(x), 1(y)) -> PLUS(1(S), plus(x, y))
10.83/5.35	   PLUS(1(x), 1(y)) -> PLUS(x, y)
10.83/5.35	   PLUS(j(x), j(y)) -> PLUS(j(S), plus(x, y))
10.83/5.35	   PLUS(j(x), j(y)) -> PLUS(x, y)
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(1(S), plus(x, y))
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(x, y)
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(j(S), plus(x, y))
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(x, y)
10.83/5.35	
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	
10.83/5.35	The set E# consists of the following equations:
10.83/5.35	
10.83/5.35	   PLUS(x, y) == PLUS(y, x)
10.83/5.35	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
10.83/5.35	
10.83/5.35	We have to consider all minimal (P,E#,R,E)-chains
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(13) ERuleRemovalProof (EQUIVALENT)
10.83/5.35	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.
10.83/5.35	
10.83/5.35	Strictly oriented dependency pairs:
10.83/5.35	
10.83/5.35	   PLUS(1(x), 1(y)) -> PLUS(1(S), plus(x, y))
10.83/5.35	   PLUS(1(x), 1(y)) -> PLUS(x, y)
10.83/5.35	   PLUS(j(x), j(y)) -> PLUS(j(S), plus(x, y))
10.83/5.35	   PLUS(j(x), j(y)) -> PLUS(x, y)
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(1(S), plus(x, y))
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(x, y)
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(j(S), plus(x, y))
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(x, y)
10.83/5.35	
10.83/5.35	
10.83/5.35	Used ordering: POLO with Polynomial interpretation [POLO]:
10.83/5.35	
10.83/5.35	   POL(1(x_1)) = 3 + x_1
10.83/5.35	   POL(PLUS(x_1, x_2)) = x_1 + x_2
10.83/5.35	   POL(S) = 0
10.83/5.35	   POL(j(x_1)) = 3 + x_1
10.83/5.35	   POL(plus(x_1, x_2)) = x_1 + x_2
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(14)
10.83/5.35	Obligation:
10.83/5.35	The TRS P consists of the following rules:
10.83/5.35	
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	
10.83/5.35	The set E# consists of the following equations:
10.83/5.35	
10.83/5.35	   PLUS(x, y) == PLUS(y, x)
10.83/5.35	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
10.83/5.35	
10.83/5.35	We have to consider all minimal (P,E#,R,E)-chains
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(15) EDPPoloProof (EQUIVALENT)
10.83/5.35	We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented.
10.83/5.35	
10.83/5.35	
10.83/5.35	   PLUS(plus(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	With the implicit AFS we had to orient the following set of usable rules of R non-strictly.
10.83/5.35	
10.83/5.35	
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	   plus(S, x) -> x
10.83/5.35	We had to orient the following equations of E# equivalently.
10.83/5.35	
10.83/5.35	
10.83/5.35	   PLUS(x, y) == PLUS(y, x)
10.83/5.35	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
10.83/5.35	With the implicit AFS we had to orient the following usable equations of E equivalently.
10.83/5.35	
10.83/5.35	
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	Used ordering: POLO with Polynomial interpretation [POLO]:
10.83/5.35	
10.83/5.35	   POL(1(x_1)) = 3
10.83/5.35	   POL(PLUS(x_1, x_2)) = 2*x_1 + 2*x_2
10.83/5.35	   POL(S) = 0
10.83/5.35	   POL(j(x_1)) = 2
10.83/5.35	   POL(plus(x_1, x_2)) = 2 + x_1 + x_2
10.83/5.35	
10.83/5.35	
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(16)
10.83/5.35	Obligation:
10.83/5.35	P is empty.
10.83/5.35	The TRS R consists of the following rules:
10.83/5.35	
10.83/5.35	   plus(S, x) -> x
10.83/5.35	   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
10.83/5.35	   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
10.83/5.35	   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
10.83/5.35	   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)
10.83/5.35	
10.83/5.35	The set E consists of the following equations:
10.83/5.35	
10.83/5.35	   plus(plus(x, y), z) == plus(x, plus(y, z))
10.83/5.35	   plus(x, y) == plus(y, x)
10.83/5.35	
10.83/5.35	The set E# consists of the following equations:
10.83/5.35	
10.83/5.35	   PLUS(x, y) == PLUS(y, x)
10.83/5.35	   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
10.83/5.35	
10.83/5.35	We have to consider all minimal (P,E#,R,E)-chains
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(17) PisEmptyProof (EQUIVALENT)
10.83/5.35	The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
10.83/5.35	----------------------------------------
10.83/5.35	
10.83/5.35	(18)
10.83/5.35	YES
11.02/5.39	EOF