3.60/1.46	YES
3.60/1.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.60/1.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.60/1.47	
3.60/1.47	
3.60/1.47	Outermost Termination of the given OTRS could be proven:
3.60/1.47	
3.60/1.47	(0) OTRS
3.60/1.47	(1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
3.60/1.47	(2) QTRS
3.60/1.47	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.60/1.47	(4) QDP
3.60/1.47	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.60/1.47	(6) QDP
3.60/1.47	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.60/1.47	(8) QDP
3.60/1.47	(9) QReductionProof [EQUIVALENT, 0 ms]
3.60/1.47	(10) QDP
3.60/1.47	(11) TransformationProof [EQUIVALENT, 0 ms]
3.60/1.47	(12) QDP
3.60/1.47	(13) UsableRulesProof [EQUIVALENT, 0 ms]
3.60/1.47	(14) QDP
3.60/1.47	(15) QReductionProof [EQUIVALENT, 0 ms]
3.60/1.47	(16) QDP
3.60/1.47	(17) TransformationProof [EQUIVALENT, 0 ms]
3.60/1.47	(18) QDP
3.60/1.47	(19) UsableRulesProof [EQUIVALENT, 0 ms]
3.60/1.47	(20) QDP
3.60/1.47	(21) QReductionProof [EQUIVALENT, 0 ms]
3.60/1.47	(22) QDP
3.60/1.47	(23) TransformationProof [EQUIVALENT, 0 ms]
3.60/1.47	(24) QDP
3.60/1.47	(25) TransformationProof [EQUIVALENT, 0 ms]
3.60/1.47	(26) QDP
3.60/1.47	(27) UsableRulesProof [EQUIVALENT, 0 ms]
3.60/1.47	(28) QDP
3.60/1.47	(29) QReductionProof [EQUIVALENT, 0 ms]
3.60/1.47	(30) QDP
3.60/1.47	(31) QDPOrderProof [EQUIVALENT, 0 ms]
3.60/1.47	(32) QDP
3.60/1.47	(33) DependencyGraphProof [EQUIVALENT, 0 ms]
3.60/1.47	(34) TRUE
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(0)
3.60/1.47	Obligation:
3.60/1.47	Term rewrite system R:
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   g(x, x) -> g(g(x, x), x)
3.60/1.47	   g(x, y) -> y
3.60/1.47	
3.60/1.47	
3.60/1.47	
3.60/1.47	Outermost Strategy.
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(1) Thiemann-SpecialC-Transformation (EQUIVALENT)
3.60/1.47	We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(2)
3.60/1.47	Obligation:
3.60/1.47	Q restricted rewrite system:
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   top(go_up(x)) -> top(reduce(x))
3.60/1.47	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
3.60/1.47	   redex_g(x, x) -> result_g(g(g(x, x), x))
3.60/1.47	   redex_g(x, y) -> result_g(y)
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
3.60/1.47	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
3.60/1.47	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
3.60/1.47	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   top(go_up(x0))
3.60/1.47	   reduce(g(x0, x1))
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	   in_g_1(go_up(x0), x1)
3.60/1.47	   in_g_2(x0, go_up(x1))
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(3) DependencyPairsProof (EQUIVALENT)
3.60/1.47	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(4)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(x)) -> TOP(reduce(x))
3.60/1.47	   TOP(go_up(x)) -> REDUCE(x)
3.60/1.47	   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
3.60/1.47	   REDUCE(g(x_1, x_2)) -> REDEX_G(x_1, x_2)
3.60/1.47	   CHECK_G(redex_g(x_1, x_2)) -> IN_G_1(reduce(x_1), x_2)
3.60/1.47	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
3.60/1.47	   CHECK_G(redex_g(x_1, x_2)) -> IN_G_2(x_1, reduce(x_2))
3.60/1.47	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   top(go_up(x)) -> top(reduce(x))
3.60/1.47	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
3.60/1.47	   redex_g(x, x) -> result_g(g(g(x, x), x))
3.60/1.47	   redex_g(x, y) -> result_g(y)
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
3.60/1.47	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
3.60/1.47	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
3.60/1.47	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   top(go_up(x0))
3.60/1.47	   reduce(g(x0, x1))
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	   in_g_1(go_up(x0), x1)
3.60/1.47	   in_g_2(x0, go_up(x1))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(5) DependencyGraphProof (EQUIVALENT)
3.60/1.47	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(6)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(x)) -> TOP(reduce(x))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   top(go_up(x)) -> top(reduce(x))
3.60/1.47	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
3.60/1.47	   redex_g(x, x) -> result_g(g(g(x, x), x))
3.60/1.47	   redex_g(x, y) -> result_g(y)
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
3.60/1.47	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
3.60/1.47	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
3.60/1.47	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   top(go_up(x0))
3.60/1.47	   reduce(g(x0, x1))
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	   in_g_1(go_up(x0), x1)
3.60/1.47	   in_g_2(x0, go_up(x1))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(7) UsableRulesProof (EQUIVALENT)
3.60/1.47	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(8)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(x)) -> TOP(reduce(x))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
3.60/1.47	   redex_g(x, x) -> result_g(g(g(x, x), x))
3.60/1.47	   redex_g(x, y) -> result_g(y)
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   top(go_up(x0))
3.60/1.47	   reduce(g(x0, x1))
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	   in_g_1(go_up(x0), x1)
3.60/1.47	   in_g_2(x0, go_up(x1))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(9) QReductionProof (EQUIVALENT)
3.60/1.47	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
3.60/1.47	
3.60/1.47	   top(go_up(x0))
3.60/1.47	   in_g_1(go_up(x0), x1)
3.60/1.47	   in_g_2(x0, go_up(x1))
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(10)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(x)) -> TOP(reduce(x))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
3.60/1.47	   redex_g(x, x) -> result_g(g(g(x, x), x))
3.60/1.47	   redex_g(x, y) -> result_g(y)
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   reduce(g(x0, x1))
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(11) TransformationProof (EQUIVALENT)
3.60/1.47	By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]:
3.60/1.47	
3.60/1.47	   (TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1))),TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1))))
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(12)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
3.60/1.47	   redex_g(x, x) -> result_g(g(g(x, x), x))
3.60/1.47	   redex_g(x, y) -> result_g(y)
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   reduce(g(x0, x1))
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(13) UsableRulesProof (EQUIVALENT)
3.60/1.47	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(14)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   redex_g(x, x) -> result_g(g(g(x, x), x))
3.60/1.47	   redex_g(x, y) -> result_g(y)
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   reduce(g(x0, x1))
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(15) QReductionProof (EQUIVALENT)
3.60/1.47	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
3.60/1.47	
3.60/1.47	   reduce(g(x0, x1))
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(16)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   redex_g(x, x) -> result_g(g(g(x, x), x))
3.60/1.47	   redex_g(x, y) -> result_g(y)
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(17) TransformationProof (EQUIVALENT)
3.60/1.47	By narrowing [LPAR04] the rule TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1))) at position [0,0] we obtained the following new rules [LPAR04]:
3.60/1.47	
3.60/1.47	   (TOP(go_up(g(x0, x0))) -> TOP(check_g(result_g(g(g(x0, x0), x0)))),TOP(go_up(g(x0, x0))) -> TOP(check_g(result_g(g(g(x0, x0), x0)))))
3.60/1.47	   (TOP(go_up(g(x0, x1))) -> TOP(check_g(result_g(x1))),TOP(go_up(g(x0, x1))) -> TOP(check_g(result_g(x1))))
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(18)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x0))) -> TOP(check_g(result_g(g(g(x0, x0), x0))))
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(check_g(result_g(x1)))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   redex_g(x, x) -> result_g(g(g(x, x), x))
3.60/1.47	   redex_g(x, y) -> result_g(y)
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(19) UsableRulesProof (EQUIVALENT)
3.60/1.47	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(20)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x0))) -> TOP(check_g(result_g(g(g(x0, x0), x0))))
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(check_g(result_g(x1)))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(21) QReductionProof (EQUIVALENT)
3.60/1.47	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
3.60/1.47	
3.60/1.47	   redex_g(x0, x1)
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(22)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x0))) -> TOP(check_g(result_g(g(g(x0, x0), x0))))
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(check_g(result_g(x1)))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(23) TransformationProof (EQUIVALENT)
3.60/1.47	By rewriting [LPAR04] the rule TOP(go_up(g(x0, x0))) -> TOP(check_g(result_g(g(g(x0, x0), x0)))) at position [0] we obtained the following new rules [LPAR04]:
3.60/1.47	
3.60/1.47	   (TOP(go_up(g(x0, x0))) -> TOP(go_up(g(g(x0, x0), x0))),TOP(go_up(g(x0, x0))) -> TOP(go_up(g(g(x0, x0), x0))))
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(24)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(check_g(result_g(x1)))
3.60/1.47	   TOP(go_up(g(x0, x0))) -> TOP(go_up(g(g(x0, x0), x0)))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(25) TransformationProof (EQUIVALENT)
3.60/1.47	By rewriting [LPAR04] the rule TOP(go_up(g(x0, x1))) -> TOP(check_g(result_g(x1))) at position [0] we obtained the following new rules [LPAR04]:
3.60/1.47	
3.60/1.47	   (TOP(go_up(g(x0, x1))) -> TOP(go_up(x1)),TOP(go_up(g(x0, x1))) -> TOP(go_up(x1)))
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(26)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x0))) -> TOP(go_up(g(g(x0, x0), x0)))
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(go_up(x1))
3.60/1.47	
3.60/1.47	The TRS R consists of the following rules:
3.60/1.47	
3.60/1.47	   check_g(result_g(x)) -> go_up(x)
3.60/1.47	
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(27) UsableRulesProof (EQUIVALENT)
3.60/1.47	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(28)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x0))) -> TOP(go_up(g(g(x0, x0), x0)))
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(go_up(x1))
3.60/1.47	
3.60/1.47	R is empty.
3.60/1.47	The set Q consists of the following terms:
3.60/1.47	
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(29) QReductionProof (EQUIVALENT)
3.60/1.47	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
3.60/1.47	
3.60/1.47	   check_g(result_g(x0))
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(30)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x0))) -> TOP(go_up(g(g(x0, x0), x0)))
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(go_up(x1))
3.60/1.47	
3.60/1.47	R is empty.
3.60/1.47	Q is empty.
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(31) QDPOrderProof (EQUIVALENT)
3.60/1.47	We use the reduction pair processor [LPAR04,JAR06].
3.60/1.47	
3.60/1.47	
3.60/1.47	The following pairs can be oriented strictly and are deleted.
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x1))) -> TOP(go_up(x1))
3.60/1.47	The remaining pairs can at least be oriented weakly.
3.60/1.47	Used ordering:  Polynomial interpretation [POLO]:
3.60/1.47	
3.60/1.47	   POL(TOP(x_1)) = x_1
3.60/1.47	   POL(g(x_1, x_2)) = 1 + x_2
3.60/1.47	   POL(go_up(x_1)) = x_1
3.60/1.47	
3.60/1.47	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
3.60/1.47	none
3.60/1.47	
3.60/1.47	
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(32)
3.60/1.47	Obligation:
3.60/1.47	Q DP problem:
3.60/1.47	The TRS P consists of the following rules:
3.60/1.47	
3.60/1.47	   TOP(go_up(g(x0, x0))) -> TOP(go_up(g(g(x0, x0), x0)))
3.60/1.47	
3.60/1.47	R is empty.
3.60/1.47	Q is empty.
3.60/1.47	We have to consider all minimal (P,Q,R)-chains.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(33) DependencyGraphProof (EQUIVALENT)
3.60/1.47	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
3.60/1.47	----------------------------------------
3.60/1.47	
3.60/1.47	(34)
3.60/1.47	TRUE
3.60/1.49	EOF