6.15/2.29	NO
6.28/2.30	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
6.28/2.30	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.28/2.30	
6.28/2.30	
6.28/2.30	Outermost Termination of the given OTRS could be disproven:
6.28/2.30	
6.28/2.30	(0) OTRS
6.28/2.30	(1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
6.28/2.30	(2) QTRS
6.28/2.30	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.28/2.30	(4) QDP
6.28/2.30	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.28/2.30	(6) AND
6.28/2.30	    (7) QDP
6.28/2.30	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.28/2.30	        (9) QDP
6.28/2.30	        (10) QReductionProof [EQUIVALENT, 0 ms]
6.28/2.30	        (11) QDP
6.28/2.30	        (12) UsableRulesReductionPairsProof [EQUIVALENT, 9 ms]
6.28/2.30	        (13) QDP
6.28/2.30	        (14) DependencyGraphProof [EQUIVALENT, 0 ms]
6.28/2.30	        (15) TRUE
6.28/2.30	    (16) QDP
6.28/2.30	        (17) UsableRulesProof [EQUIVALENT, 0 ms]
6.28/2.30	        (18) QDP
6.28/2.30	        (19) QReductionProof [EQUIVALENT, 0 ms]
6.28/2.30	        (20) QDP
6.28/2.30	        (21) TransformationProof [EQUIVALENT, 0 ms]
6.28/2.30	        (22) QDP
6.28/2.30	        (23) UsableRulesProof [EQUIVALENT, 0 ms]
6.28/2.30	        (24) QDP
6.28/2.30	        (25) NonTerminationLoopProof [COMPLETE, 0 ms]
6.28/2.30	        (26) NO
6.28/2.30	
6.28/2.30	
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(0)
6.28/2.30	Obligation:
6.28/2.30	Term rewrite system R:
6.28/2.30	The TRS R consists of the following rules:
6.28/2.30	
6.28/2.30	   b -> f(f(b))
6.28/2.30	   f(b) -> b
6.28/2.30	   f(f(f(x))) -> b
6.28/2.30	
6.28/2.30	
6.28/2.30	
6.28/2.30	Outermost Strategy.
6.28/2.30	
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(1) Thiemann-SpecialC-Transformation (EQUIVALENT)
6.28/2.30	We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(2)
6.28/2.30	Obligation:
6.28/2.30	Q restricted rewrite system:
6.28/2.30	The TRS R consists of the following rules:
6.28/2.30	
6.28/2.30	   top(go_up(x)) -> top(reduce(x))
6.28/2.30	   reduce(f(x_1)) -> check_f(redex_f(x_1))
6.28/2.30	   reduce(b) -> go_up(f(f(b)))
6.28/2.30	   redex_f(b) -> result_f(b)
6.28/2.30	   redex_f(f(f(x))) -> result_f(b)
6.28/2.30	   check_f(result_f(x)) -> go_up(x)
6.28/2.30	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
6.28/2.30	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
6.28/2.30	
6.28/2.30	The set Q consists of the following terms:
6.28/2.30	
6.28/2.30	   top(go_up(x0))
6.28/2.30	   reduce(f(x0))
6.28/2.30	   reduce(b)
6.28/2.30	   redex_f(b)
6.28/2.30	   redex_f(f(f(x0)))
6.28/2.30	   check_f(result_f(x0))
6.28/2.30	   check_f(redex_f(x0))
6.28/2.30	   in_f_1(go_up(x0))
6.28/2.30	
6.28/2.30	
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(3) DependencyPairsProof (EQUIVALENT)
6.28/2.30	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(4)
6.28/2.30	Obligation:
6.28/2.30	Q DP problem:
6.28/2.30	The TRS P consists of the following rules:
6.28/2.30	
6.28/2.30	   TOP(go_up(x)) -> TOP(reduce(x))
6.28/2.30	   TOP(go_up(x)) -> REDUCE(x)
6.28/2.30	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
6.28/2.30	   REDUCE(f(x_1)) -> REDEX_F(x_1)
6.28/2.30	   CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1))
6.28/2.30	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
6.28/2.30	
6.28/2.30	The TRS R consists of the following rules:
6.28/2.30	
6.28/2.30	   top(go_up(x)) -> top(reduce(x))
6.28/2.30	   reduce(f(x_1)) -> check_f(redex_f(x_1))
6.28/2.30	   reduce(b) -> go_up(f(f(b)))
6.28/2.30	   redex_f(b) -> result_f(b)
6.28/2.30	   redex_f(f(f(x))) -> result_f(b)
6.28/2.30	   check_f(result_f(x)) -> go_up(x)
6.28/2.30	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
6.28/2.30	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
6.28/2.30	
6.28/2.30	The set Q consists of the following terms:
6.28/2.30	
6.28/2.30	   top(go_up(x0))
6.28/2.30	   reduce(f(x0))
6.28/2.30	   reduce(b)
6.28/2.30	   redex_f(b)
6.28/2.30	   redex_f(f(f(x0)))
6.28/2.30	   check_f(result_f(x0))
6.28/2.30	   check_f(redex_f(x0))
6.28/2.30	   in_f_1(go_up(x0))
6.28/2.30	
6.28/2.30	We have to consider all minimal (P,Q,R)-chains.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(5) DependencyGraphProof (EQUIVALENT)
6.28/2.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(6)
6.28/2.30	Complex Obligation (AND)
6.28/2.30	
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(7)
6.28/2.30	Obligation:
6.28/2.30	Q DP problem:
6.28/2.30	The TRS P consists of the following rules:
6.28/2.30	
6.28/2.30	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
6.28/2.30	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
6.28/2.30	
6.28/2.30	The TRS R consists of the following rules:
6.28/2.30	
6.28/2.30	   top(go_up(x)) -> top(reduce(x))
6.28/2.30	   reduce(f(x_1)) -> check_f(redex_f(x_1))
6.28/2.30	   reduce(b) -> go_up(f(f(b)))
6.28/2.30	   redex_f(b) -> result_f(b)
6.28/2.30	   redex_f(f(f(x))) -> result_f(b)
6.28/2.30	   check_f(result_f(x)) -> go_up(x)
6.28/2.30	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
6.28/2.30	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
6.28/2.30	
6.28/2.30	The set Q consists of the following terms:
6.28/2.30	
6.28/2.30	   top(go_up(x0))
6.28/2.30	   reduce(f(x0))
6.28/2.30	   reduce(b)
6.28/2.30	   redex_f(b)
6.28/2.30	   redex_f(f(f(x0)))
6.28/2.30	   check_f(result_f(x0))
6.28/2.30	   check_f(redex_f(x0))
6.28/2.30	   in_f_1(go_up(x0))
6.28/2.30	
6.28/2.30	We have to consider all minimal (P,Q,R)-chains.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(8) UsableRulesProof (EQUIVALENT)
6.28/2.30	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.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(9)
6.28/2.30	Obligation:
6.28/2.30	Q DP problem:
6.28/2.30	The TRS P consists of the following rules:
6.28/2.30	
6.28/2.30	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
6.28/2.30	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
6.28/2.30	
6.28/2.30	The TRS R consists of the following rules:
6.28/2.30	
6.28/2.30	   redex_f(b) -> result_f(b)
6.28/2.30	   redex_f(f(f(x))) -> result_f(b)
6.28/2.30	
6.28/2.30	The set Q consists of the following terms:
6.28/2.30	
6.28/2.30	   top(go_up(x0))
6.28/2.30	   reduce(f(x0))
6.28/2.30	   reduce(b)
6.28/2.30	   redex_f(b)
6.28/2.30	   redex_f(f(f(x0)))
6.28/2.30	   check_f(result_f(x0))
6.28/2.30	   check_f(redex_f(x0))
6.28/2.30	   in_f_1(go_up(x0))
6.28/2.30	
6.28/2.30	We have to consider all minimal (P,Q,R)-chains.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(10) QReductionProof (EQUIVALENT)
6.28/2.30	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.28/2.30	
6.28/2.30	   top(go_up(x0))
6.28/2.30	   reduce(f(x0))
6.28/2.30	   reduce(b)
6.28/2.30	   check_f(result_f(x0))
6.28/2.30	   check_f(redex_f(x0))
6.28/2.30	   in_f_1(go_up(x0))
6.28/2.30	
6.28/2.30	
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(11)
6.28/2.30	Obligation:
6.28/2.30	Q DP problem:
6.28/2.30	The TRS P consists of the following rules:
6.28/2.30	
6.28/2.30	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
6.28/2.30	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
6.28/2.30	
6.28/2.30	The TRS R consists of the following rules:
6.28/2.30	
6.28/2.30	   redex_f(b) -> result_f(b)
6.28/2.30	   redex_f(f(f(x))) -> result_f(b)
6.28/2.30	
6.28/2.30	The set Q consists of the following terms:
6.28/2.30	
6.28/2.30	   redex_f(b)
6.28/2.30	   redex_f(f(f(x0)))
6.28/2.30	
6.28/2.30	We have to consider all minimal (P,Q,R)-chains.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(12) UsableRulesReductionPairsProof (EQUIVALENT)
6.28/2.30	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules 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.
6.28/2.30	
6.28/2.30	The following dependency pairs can be deleted:
6.28/2.30	
6.28/2.30	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
6.28/2.30	The following rules are removed from R:
6.28/2.30	
6.28/2.30	   redex_f(f(f(x))) -> result_f(b)
6.28/2.30	Used ordering: POLO with Polynomial interpretation [POLO]:
6.28/2.30	
6.28/2.30	   POL(CHECK_F(x_1)) = x_1
6.28/2.30	   POL(REDUCE(x_1)) = 2*x_1
6.28/2.30	   POL(b) = 0
6.28/2.30	   POL(f(x_1)) = 2*x_1
6.28/2.30	   POL(redex_f(x_1)) = 2*x_1
6.28/2.30	   POL(result_f(x_1)) = 2*x_1
6.28/2.30	
6.28/2.30	
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(13)
6.28/2.30	Obligation:
6.28/2.30	Q DP problem:
6.28/2.30	The TRS P consists of the following rules:
6.28/2.30	
6.28/2.30	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
6.28/2.30	
6.28/2.30	The TRS R consists of the following rules:
6.28/2.30	
6.28/2.30	   redex_f(b) -> result_f(b)
6.28/2.30	
6.28/2.30	The set Q consists of the following terms:
6.28/2.30	
6.28/2.30	   redex_f(b)
6.28/2.30	   redex_f(f(f(x0)))
6.28/2.30	
6.28/2.30	We have to consider all minimal (P,Q,R)-chains.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(14) DependencyGraphProof (EQUIVALENT)
6.28/2.30	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(15)
6.28/2.30	TRUE
6.28/2.30	
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(16)
6.28/2.30	Obligation:
6.28/2.30	Q DP problem:
6.28/2.30	The TRS P consists of the following rules:
6.28/2.30	
6.28/2.30	   TOP(go_up(x)) -> TOP(reduce(x))
6.28/2.30	
6.28/2.30	The TRS R consists of the following rules:
6.28/2.30	
6.28/2.30	   top(go_up(x)) -> top(reduce(x))
6.28/2.30	   reduce(f(x_1)) -> check_f(redex_f(x_1))
6.28/2.30	   reduce(b) -> go_up(f(f(b)))
6.28/2.30	   redex_f(b) -> result_f(b)
6.28/2.30	   redex_f(f(f(x))) -> result_f(b)
6.28/2.30	   check_f(result_f(x)) -> go_up(x)
6.28/2.30	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
6.28/2.30	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
6.28/2.30	
6.28/2.30	The set Q consists of the following terms:
6.28/2.30	
6.28/2.30	   top(go_up(x0))
6.28/2.30	   reduce(f(x0))
6.28/2.30	   reduce(b)
6.28/2.30	   redex_f(b)
6.28/2.30	   redex_f(f(f(x0)))
6.28/2.30	   check_f(result_f(x0))
6.28/2.30	   check_f(redex_f(x0))
6.28/2.30	   in_f_1(go_up(x0))
6.28/2.30	
6.28/2.30	We have to consider all minimal (P,Q,R)-chains.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(17) UsableRulesProof (EQUIVALENT)
6.28/2.30	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.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(18)
6.28/2.30	Obligation:
6.28/2.30	Q DP problem:
6.28/2.30	The TRS P consists of the following rules:
6.28/2.30	
6.28/2.30	   TOP(go_up(x)) -> TOP(reduce(x))
6.28/2.30	
6.28/2.30	The TRS R consists of the following rules:
6.28/2.30	
6.28/2.30	   reduce(f(x_1)) -> check_f(redex_f(x_1))
6.28/2.30	   reduce(b) -> go_up(f(f(b)))
6.28/2.30	   redex_f(b) -> result_f(b)
6.28/2.30	   redex_f(f(f(x))) -> result_f(b)
6.28/2.30	   check_f(result_f(x)) -> go_up(x)
6.28/2.30	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
6.28/2.30	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
6.28/2.30	
6.28/2.30	The set Q consists of the following terms:
6.28/2.30	
6.28/2.30	   top(go_up(x0))
6.28/2.30	   reduce(f(x0))
6.28/2.30	   reduce(b)
6.28/2.30	   redex_f(b)
6.28/2.30	   redex_f(f(f(x0)))
6.28/2.30	   check_f(result_f(x0))
6.28/2.30	   check_f(redex_f(x0))
6.28/2.30	   in_f_1(go_up(x0))
6.28/2.30	
6.28/2.30	We have to consider all minimal (P,Q,R)-chains.
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(19) QReductionProof (EQUIVALENT)
6.28/2.30	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.28/2.30	
6.28/2.30	   top(go_up(x0))
6.28/2.30	
6.28/2.30	
6.28/2.30	----------------------------------------
6.28/2.30	
6.28/2.30	(20)
6.28/2.30	Obligation:
6.28/2.30	Q DP problem:
6.28/2.30	The TRS P consists of the following rules:
6.28/2.31	
6.28/2.31	   TOP(go_up(x)) -> TOP(reduce(x))
6.28/2.31	
6.28/2.31	The TRS R consists of the following rules:
6.28/2.31	
6.28/2.31	   reduce(f(x_1)) -> check_f(redex_f(x_1))
6.28/2.31	   reduce(b) -> go_up(f(f(b)))
6.28/2.31	   redex_f(b) -> result_f(b)
6.28/2.31	   redex_f(f(f(x))) -> result_f(b)
6.28/2.31	   check_f(result_f(x)) -> go_up(x)
6.28/2.31	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
6.28/2.31	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
6.28/2.31	
6.28/2.31	The set Q consists of the following terms:
6.28/2.31	
6.28/2.31	   reduce(f(x0))
6.28/2.31	   reduce(b)
6.28/2.31	   redex_f(b)
6.28/2.31	   redex_f(f(f(x0)))
6.28/2.31	   check_f(result_f(x0))
6.28/2.31	   check_f(redex_f(x0))
6.28/2.31	   in_f_1(go_up(x0))
6.28/2.31	
6.28/2.31	We have to consider all minimal (P,Q,R)-chains.
6.28/2.31	----------------------------------------
6.28/2.31	
6.28/2.31	(21) TransformationProof (EQUIVALENT)
6.28/2.31	By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]:
6.28/2.31	
6.28/2.31	   (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))))
6.28/2.31	   (TOP(go_up(b)) -> TOP(go_up(f(f(b)))),TOP(go_up(b)) -> TOP(go_up(f(f(b)))))
6.28/2.31	
6.28/2.31	
6.28/2.31	----------------------------------------
6.28/2.31	
6.28/2.31	(22)
6.28/2.31	Obligation:
6.28/2.31	Q DP problem:
6.28/2.31	The TRS P consists of the following rules:
6.28/2.31	
6.28/2.31	   TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))
6.28/2.31	   TOP(go_up(b)) -> TOP(go_up(f(f(b))))
6.28/2.31	
6.28/2.31	The TRS R consists of the following rules:
6.28/2.31	
6.28/2.31	   reduce(f(x_1)) -> check_f(redex_f(x_1))
6.28/2.31	   reduce(b) -> go_up(f(f(b)))
6.28/2.31	   redex_f(b) -> result_f(b)
6.28/2.31	   redex_f(f(f(x))) -> result_f(b)
6.28/2.31	   check_f(result_f(x)) -> go_up(x)
6.28/2.31	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
6.28/2.31	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
6.28/2.31	
6.28/2.31	The set Q consists of the following terms:
6.28/2.31	
6.28/2.31	   reduce(f(x0))
6.28/2.31	   reduce(b)
6.28/2.31	   redex_f(b)
6.28/2.31	   redex_f(f(f(x0)))
6.28/2.31	   check_f(result_f(x0))
6.28/2.31	   check_f(redex_f(x0))
6.28/2.31	   in_f_1(go_up(x0))
6.28/2.31	
6.28/2.31	We have to consider all minimal (P,Q,R)-chains.
6.28/2.31	----------------------------------------
6.28/2.31	
6.28/2.31	(23) UsableRulesProof (EQUIVALENT)
6.28/2.31	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.
6.28/2.31	----------------------------------------
6.28/2.31	
6.28/2.31	(24)
6.28/2.31	Obligation:
6.28/2.31	Q DP problem:
6.28/2.31	The TRS P consists of the following rules:
6.28/2.31	
6.28/2.31	   TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))
6.28/2.31	   TOP(go_up(b)) -> TOP(go_up(f(f(b))))
6.28/2.31	
6.28/2.31	The TRS R consists of the following rules:
6.28/2.31	
6.28/2.31	   redex_f(b) -> result_f(b)
6.28/2.31	   redex_f(f(f(x))) -> result_f(b)
6.28/2.31	   check_f(result_f(x)) -> go_up(x)
6.28/2.31	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
6.28/2.31	   reduce(f(x_1)) -> check_f(redex_f(x_1))
6.28/2.31	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
6.28/2.31	
6.28/2.31	The set Q consists of the following terms:
6.28/2.31	
6.28/2.31	   reduce(f(x0))
6.28/2.31	   reduce(b)
6.28/2.31	   redex_f(b)
6.28/2.31	   redex_f(f(f(x0)))
6.28/2.31	   check_f(result_f(x0))
6.28/2.31	   check_f(redex_f(x0))
6.28/2.31	   in_f_1(go_up(x0))
6.28/2.31	
6.28/2.31	We have to consider all minimal (P,Q,R)-chains.
6.28/2.31	----------------------------------------
6.28/2.31	
6.28/2.31	(25) NonTerminationLoopProof (COMPLETE)
6.28/2.31	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
6.28/2.31	Found a loop by narrowing to the right:
6.28/2.31	
6.28/2.31	s = TOP(go_up(f(f(b)))) evaluates to  t =TOP(go_up(f(f(b))))
6.28/2.31	
6.28/2.31	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
6.28/2.31	* Matcher: [ ]
6.28/2.31	* Semiunifier: [ ]
6.28/2.31	
6.28/2.31	--------------------------------------------------------------------------------
6.28/2.31	Rewriting sequence
6.28/2.31	
6.28/2.31	TOP(go_up(f(f(b)))) -> TOP(check_f(redex_f(f(b))))
6.28/2.31	 with rule TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) and matcher [x0 / f(b)].
6.28/2.31	
6.28/2.31	TOP(check_f(redex_f(f(b)))) -> TOP(in_f_1(reduce(f(b))))
6.28/2.31	with rule check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) at position [0] and matcher [x_1 / f(b)]
6.28/2.31	
6.28/2.31	TOP(in_f_1(reduce(f(b)))) -> TOP(in_f_1(check_f(redex_f(b))))
6.28/2.31	with rule reduce(f(x_1')) -> check_f(redex_f(x_1')) at position [0,0] and matcher [x_1' / b]
6.28/2.31	
6.28/2.31	TOP(in_f_1(check_f(redex_f(b)))) -> TOP(in_f_1(check_f(result_f(b))))
6.28/2.31	with rule redex_f(b) -> result_f(b) at position [0,0,0] and matcher [ ]
6.28/2.31	
6.28/2.31	TOP(in_f_1(check_f(result_f(b)))) -> TOP(in_f_1(go_up(b)))
6.28/2.31	with rule check_f(result_f(x)) -> go_up(x) at position [0,0] and matcher [x / b]
6.28/2.31	
6.28/2.31	TOP(in_f_1(go_up(b))) -> TOP(go_up(f(b)))
6.28/2.31	with rule in_f_1(go_up(x_1)) -> go_up(f(x_1)) at position [0] and matcher [x_1 / b]
6.28/2.31	
6.28/2.31	TOP(go_up(f(b))) -> TOP(check_f(redex_f(b)))
6.28/2.31	with rule TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) at position [] and matcher [x0 / b]
6.28/2.31	
6.28/2.31	TOP(check_f(redex_f(b))) -> TOP(check_f(result_f(b)))
6.28/2.31	with rule redex_f(b) -> result_f(b) at position [0,0] and matcher [ ]
6.28/2.31	
6.28/2.31	TOP(check_f(result_f(b))) -> TOP(go_up(b))
6.28/2.31	with rule check_f(result_f(x)) -> go_up(x) at position [0] and matcher [x / b]
6.28/2.31	
6.28/2.31	TOP(go_up(b)) -> TOP(go_up(f(f(b))))
6.28/2.31	with rule TOP(go_up(b)) -> TOP(go_up(f(f(b)))) at position [] and matcher [ ]
6.28/2.31	
6.28/2.31	Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
6.28/2.31	
6.28/2.31	
6.28/2.31	All these steps are and every following step will be a correct step w.r.t to Q.
6.28/2.31	
6.28/2.31	
6.28/2.31	
6.28/2.31	
6.28/2.31	----------------------------------------
6.28/2.31	
6.28/2.31	(26)
6.28/2.31	NO
6.36/2.38	EOF