3.36/1.59	YES
3.36/1.60	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.36/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.36/1.60	
3.36/1.60	
3.36/1.60	Outermost Termination of the given OTRS could be proven:
3.36/1.60	
3.36/1.60	(0) OTRS
3.36/1.60	(1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
3.36/1.60	(2) QTRS
3.36/1.60	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.36/1.60	(4) QDP
3.36/1.60	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.36/1.60	(6) AND
3.36/1.60	    (7) QDP
3.36/1.60	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
3.36/1.60	        (9) QDP
3.36/1.60	        (10) QReductionProof [EQUIVALENT, 0 ms]
3.36/1.60	        (11) QDP
3.36/1.60	        (12) MRRProof [EQUIVALENT, 12 ms]
3.36/1.60	        (13) QDP
3.36/1.60	        (14) PisEmptyProof [EQUIVALENT, 0 ms]
3.36/1.60	        (15) YES
3.36/1.60	    (16) QDP
3.36/1.60	        (17) UsableRulesProof [EQUIVALENT, 0 ms]
3.36/1.60	        (18) QDP
3.36/1.60	        (19) QReductionProof [EQUIVALENT, 0 ms]
3.36/1.60	        (20) QDP
3.36/1.60	        (21) QDPOrderProof [EQUIVALENT, 30 ms]
3.36/1.60	        (22) QDP
3.36/1.60	        (23) PisEmptyProof [EQUIVALENT, 0 ms]
3.36/1.60	        (24) YES
3.36/1.60	
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(0)
3.36/1.60	Obligation:
3.36/1.60	Term rewrite system R:
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   f(x, x) -> g(f(x, x))
3.36/1.60	   g(x) -> a
3.36/1.60	
3.36/1.60	
3.36/1.60	
3.36/1.60	Outermost Strategy.
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(1) Thiemann-SpecialC-Transformation (EQUIVALENT)
3.36/1.60	We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(2)
3.36/1.60	Obligation:
3.36/1.60	Q restricted rewrite system:
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   top(go_up(x)) -> top(reduce(x))
3.36/1.60	   reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2))
3.36/1.60	   reduce(g(x_1)) -> check_g(redex_g(x_1))
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	   redex_g(x) -> result_g(a)
3.36/1.60	   check_f(result_f(x)) -> go_up(x)
3.36/1.60	   check_g(result_g(x)) -> go_up(x)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2))
3.36/1.60	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
3.36/1.60	   in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2))
3.36/1.60	   in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2))
3.36/1.60	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   top(go_up(x0))
3.36/1.60	   reduce(f(x0, x1))
3.36/1.60	   reduce(g(x0))
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	   redex_g(x0)
3.36/1.60	   check_f(result_f(x0))
3.36/1.60	   check_g(result_g(x0))
3.36/1.60	   check_f(redex_f(x0, x1))
3.36/1.60	   in_f_1(go_up(x0), x1)
3.36/1.60	   in_f_2(x0, go_up(x1))
3.36/1.60	   in_g_1(go_up(x0))
3.36/1.60	
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(3) DependencyPairsProof (EQUIVALENT)
3.36/1.60	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(4)
3.36/1.60	Obligation:
3.36/1.60	Q DP problem:
3.36/1.60	The TRS P consists of the following rules:
3.36/1.60	
3.36/1.60	   TOP(go_up(x)) -> TOP(reduce(x))
3.36/1.60	   TOP(go_up(x)) -> REDUCE(x)
3.36/1.60	   REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2))
3.36/1.60	   REDUCE(f(x_1, x_2)) -> REDEX_F(x_1, x_2)
3.36/1.60	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
3.36/1.60	   REDUCE(g(x_1)) -> REDEX_G(x_1)
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> IN_F_1(reduce(x_1), x_2)
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1)
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> IN_F_2(x_1, reduce(x_2))
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2)
3.36/1.60	   CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1))
3.36/1.60	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   top(go_up(x)) -> top(reduce(x))
3.36/1.60	   reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2))
3.36/1.60	   reduce(g(x_1)) -> check_g(redex_g(x_1))
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	   redex_g(x) -> result_g(a)
3.36/1.60	   check_f(result_f(x)) -> go_up(x)
3.36/1.60	   check_g(result_g(x)) -> go_up(x)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2))
3.36/1.60	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
3.36/1.60	   in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2))
3.36/1.60	   in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2))
3.36/1.60	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   top(go_up(x0))
3.36/1.60	   reduce(f(x0, x1))
3.36/1.60	   reduce(g(x0))
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	   redex_g(x0)
3.36/1.60	   check_f(result_f(x0))
3.36/1.60	   check_g(result_g(x0))
3.36/1.60	   check_f(redex_f(x0, x1))
3.36/1.60	   in_f_1(go_up(x0), x1)
3.36/1.60	   in_f_2(x0, go_up(x1))
3.36/1.60	   in_g_1(go_up(x0))
3.36/1.60	
3.36/1.60	We have to consider all minimal (P,Q,R)-chains.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(5) DependencyGraphProof (EQUIVALENT)
3.36/1.60	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(6)
3.36/1.60	Complex Obligation (AND)
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(7)
3.36/1.60	Obligation:
3.36/1.60	Q DP problem:
3.36/1.60	The TRS P consists of the following rules:
3.36/1.60	
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1)
3.36/1.60	   REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2))
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2)
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   top(go_up(x)) -> top(reduce(x))
3.36/1.60	   reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2))
3.36/1.60	   reduce(g(x_1)) -> check_g(redex_g(x_1))
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	   redex_g(x) -> result_g(a)
3.36/1.60	   check_f(result_f(x)) -> go_up(x)
3.36/1.60	   check_g(result_g(x)) -> go_up(x)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2))
3.36/1.60	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
3.36/1.60	   in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2))
3.36/1.60	   in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2))
3.36/1.60	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   top(go_up(x0))
3.36/1.60	   reduce(f(x0, x1))
3.36/1.60	   reduce(g(x0))
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	   redex_g(x0)
3.36/1.60	   check_f(result_f(x0))
3.36/1.60	   check_g(result_g(x0))
3.36/1.60	   check_f(redex_f(x0, x1))
3.36/1.60	   in_f_1(go_up(x0), x1)
3.36/1.60	   in_f_2(x0, go_up(x1))
3.36/1.60	   in_g_1(go_up(x0))
3.36/1.60	
3.36/1.60	We have to consider all minimal (P,Q,R)-chains.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(8) UsableRulesProof (EQUIVALENT)
3.36/1.60	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.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(9)
3.36/1.60	Obligation:
3.36/1.60	Q DP problem:
3.36/1.60	The TRS P consists of the following rules:
3.36/1.60	
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1)
3.36/1.60	   REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2))
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2)
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   top(go_up(x0))
3.36/1.60	   reduce(f(x0, x1))
3.36/1.60	   reduce(g(x0))
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	   redex_g(x0)
3.36/1.60	   check_f(result_f(x0))
3.36/1.60	   check_g(result_g(x0))
3.36/1.60	   check_f(redex_f(x0, x1))
3.36/1.60	   in_f_1(go_up(x0), x1)
3.36/1.60	   in_f_2(x0, go_up(x1))
3.36/1.60	   in_g_1(go_up(x0))
3.36/1.60	
3.36/1.60	We have to consider all minimal (P,Q,R)-chains.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(10) QReductionProof (EQUIVALENT)
3.36/1.60	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
3.36/1.60	
3.36/1.60	   top(go_up(x0))
3.36/1.60	   reduce(f(x0, x1))
3.36/1.60	   reduce(g(x0))
3.36/1.60	   redex_g(x0)
3.36/1.60	   check_f(result_f(x0))
3.36/1.60	   check_g(result_g(x0))
3.36/1.60	   check_f(redex_f(x0, x1))
3.36/1.60	   in_f_1(go_up(x0), x1)
3.36/1.60	   in_f_2(x0, go_up(x1))
3.36/1.60	   in_g_1(go_up(x0))
3.36/1.60	
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(11)
3.36/1.60	Obligation:
3.36/1.60	Q DP problem:
3.36/1.60	The TRS P consists of the following rules:
3.36/1.60	
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1)
3.36/1.60	   REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2))
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2)
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	
3.36/1.60	We have to consider all minimal (P,Q,R)-chains.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(12) MRRProof (EQUIVALENT)
3.36/1.60	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.
3.36/1.60	
3.36/1.60	Strictly oriented dependency pairs:
3.36/1.60	
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_1)
3.36/1.60	   REDUCE(f(x_1, x_2)) -> CHECK_F(redex_f(x_1, x_2))
3.36/1.60	   CHECK_F(redex_f(x_1, x_2)) -> REDUCE(x_2)
3.36/1.60	
3.36/1.60	
3.36/1.60	Used ordering: Polynomial interpretation [POLO]:
3.36/1.60	
3.36/1.60	   POL(CHECK_F(x_1)) = 1 + x_1
3.36/1.60	   POL(REDUCE(x_1)) = 2*x_1
3.36/1.60	   POL(f(x_1, x_2)) = 2 + x_1 + 2*x_2
3.36/1.60	   POL(g(x_1)) = x_1
3.36/1.60	   POL(redex_f(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
3.36/1.60	   POL(result_f(x_1)) = x_1
3.36/1.60	
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(13)
3.36/1.60	Obligation:
3.36/1.60	Q DP problem:
3.36/1.60	P is empty.
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	
3.36/1.60	We have to consider all minimal (P,Q,R)-chains.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(14) PisEmptyProof (EQUIVALENT)
3.36/1.60	The TRS P is empty. Hence, there is no (P,Q,R) chain.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(15)
3.36/1.60	YES
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(16)
3.36/1.60	Obligation:
3.36/1.60	Q DP problem:
3.36/1.60	The TRS P consists of the following rules:
3.36/1.60	
3.36/1.60	   TOP(go_up(x)) -> TOP(reduce(x))
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   top(go_up(x)) -> top(reduce(x))
3.36/1.60	   reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2))
3.36/1.60	   reduce(g(x_1)) -> check_g(redex_g(x_1))
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	   redex_g(x) -> result_g(a)
3.36/1.60	   check_f(result_f(x)) -> go_up(x)
3.36/1.60	   check_g(result_g(x)) -> go_up(x)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2))
3.36/1.60	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
3.36/1.60	   in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2))
3.36/1.60	   in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2))
3.36/1.60	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   top(go_up(x0))
3.36/1.60	   reduce(f(x0, x1))
3.36/1.60	   reduce(g(x0))
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	   redex_g(x0)
3.36/1.60	   check_f(result_f(x0))
3.36/1.60	   check_g(result_g(x0))
3.36/1.60	   check_f(redex_f(x0, x1))
3.36/1.60	   in_f_1(go_up(x0), x1)
3.36/1.60	   in_f_2(x0, go_up(x1))
3.36/1.60	   in_g_1(go_up(x0))
3.36/1.60	
3.36/1.60	We have to consider all minimal (P,Q,R)-chains.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(17) UsableRulesProof (EQUIVALENT)
3.36/1.60	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.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(18)
3.36/1.60	Obligation:
3.36/1.60	Q DP problem:
3.36/1.60	The TRS P consists of the following rules:
3.36/1.60	
3.36/1.60	   TOP(go_up(x)) -> TOP(reduce(x))
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2))
3.36/1.60	   reduce(g(x_1)) -> check_g(redex_g(x_1))
3.36/1.60	   redex_g(x) -> result_g(a)
3.36/1.60	   check_g(result_g(x)) -> go_up(x)
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	   check_f(result_f(x)) -> go_up(x)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2))
3.36/1.60	   in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2))
3.36/1.60	   in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   top(go_up(x0))
3.36/1.60	   reduce(f(x0, x1))
3.36/1.60	   reduce(g(x0))
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	   redex_g(x0)
3.36/1.60	   check_f(result_f(x0))
3.36/1.60	   check_g(result_g(x0))
3.36/1.60	   check_f(redex_f(x0, x1))
3.36/1.60	   in_f_1(go_up(x0), x1)
3.36/1.60	   in_f_2(x0, go_up(x1))
3.36/1.60	   in_g_1(go_up(x0))
3.36/1.60	
3.36/1.60	We have to consider all minimal (P,Q,R)-chains.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(19) QReductionProof (EQUIVALENT)
3.36/1.60	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
3.36/1.60	
3.36/1.60	   top(go_up(x0))
3.36/1.60	   in_g_1(go_up(x0))
3.36/1.60	
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(20)
3.36/1.60	Obligation:
3.36/1.60	Q DP problem:
3.36/1.60	The TRS P consists of the following rules:
3.36/1.60	
3.36/1.60	   TOP(go_up(x)) -> TOP(reduce(x))
3.36/1.60	
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2))
3.36/1.60	   reduce(g(x_1)) -> check_g(redex_g(x_1))
3.36/1.60	   redex_g(x) -> result_g(a)
3.36/1.60	   check_g(result_g(x)) -> go_up(x)
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	   check_f(result_f(x)) -> go_up(x)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2))
3.36/1.60	   in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2))
3.36/1.60	   in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   reduce(f(x0, x1))
3.36/1.60	   reduce(g(x0))
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	   redex_g(x0)
3.36/1.60	   check_f(result_f(x0))
3.36/1.60	   check_g(result_g(x0))
3.36/1.60	   check_f(redex_f(x0, x1))
3.36/1.60	   in_f_1(go_up(x0), x1)
3.36/1.60	   in_f_2(x0, go_up(x1))
3.36/1.60	
3.36/1.60	We have to consider all minimal (P,Q,R)-chains.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(21) QDPOrderProof (EQUIVALENT)
3.36/1.60	We use the reduction pair processor [LPAR04,JAR06].
3.36/1.60	
3.36/1.60	
3.36/1.60	The following pairs can be oriented strictly and are deleted.
3.36/1.60	
3.36/1.60	   TOP(go_up(x)) -> TOP(reduce(x))
3.36/1.60	The remaining pairs can at least be oriented weakly.
3.36/1.60	Used ordering:  Matrix interpretation [MATRO]:
3.36/1.60	
3.36/1.60	Non-tuple symbols: 
3.36/1.60	<<<
3.36/1.60	 M( result_f_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 1], [0, 0]] 	* 	x_1
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( a ) =  	[[0], [0]]
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( reduce_1(x_1) ) =  	[[1], [0]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( in_f_2_2(x_1, x_2) ) =  	[[0], [1]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1 	+ 	[[1, 0], [1, 1]] 	* 	x_2
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( redex_f_2(x_1, x_2) ) =  	[[1], [0]] 	+ 	[[1, 1], [0, 0]] 	* 	x_1 	+ 	[[1, 1], [0, 0]] 	* 	x_2
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( check_f_1(x_1) ) =  	[[1], [1]] 	+ 	[[0, 0], [1, 0]] 	* 	x_1
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( g_1(x_1) ) =  	[[1], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( go_up_1(x_1) ) =  	[[1], [1]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( result_g_1(x_1) ) =  	[[0], [1]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( in_f_1_2(x_1, x_2) ) =  	[[0], [1]] 	+ 	[[1, 0], [1, 1]] 	* 	x_1 	+ 	[[0, 0], [1, 1]] 	* 	x_2
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( f_2(x_1, x_2) ) =  	[[1], [1]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1 	+ 	[[1, 0], [0, 1]] 	* 	x_2
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( check_g_1(x_1) ) =  	[[1], [0]] 	+ 	[[0, 0], [0, 1]] 	* 	x_1
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	<<<
3.36/1.60	 M( redex_g_1(x_1) ) =  	[[0], [1]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	Tuple symbols: 
3.36/1.60	<<<
3.36/1.60	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[0, 1]] 	* 	x_1
3.36/1.60	>>>
3.36/1.60	
3.36/1.60	
3.36/1.60	
3.36/1.60	Matrix type: 
3.36/1.60	
3.36/1.60	We used a basic matrix type which is not further parametrizeable.
3.36/1.60	
3.36/1.60	
3.36/1.60	
3.36/1.60	
3.36/1.60	
3.36/1.60	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
3.36/1.60	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
3.36/1.60	
3.36/1.60	   reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2))
3.36/1.60	   reduce(g(x_1)) -> check_g(redex_g(x_1))
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2)
3.36/1.60	   in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2))
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	   check_f(result_f(x)) -> go_up(x)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2))
3.36/1.60	   in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2))
3.36/1.60	   redex_g(x) -> result_g(a)
3.36/1.60	   check_g(result_g(x)) -> go_up(x)
3.36/1.60	
3.36/1.60	
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(22)
3.36/1.60	Obligation:
3.36/1.60	Q DP problem:
3.36/1.60	P is empty.
3.36/1.60	The TRS R consists of the following rules:
3.36/1.60	
3.36/1.60	   reduce(f(x_1, x_2)) -> check_f(redex_f(x_1, x_2))
3.36/1.60	   reduce(g(x_1)) -> check_g(redex_g(x_1))
3.36/1.60	   redex_g(x) -> result_g(a)
3.36/1.60	   check_g(result_g(x)) -> go_up(x)
3.36/1.60	   redex_f(x, x) -> result_f(g(f(x, x)))
3.36/1.60	   check_f(result_f(x)) -> go_up(x)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_1(reduce(x_1), x_2)
3.36/1.60	   check_f(redex_f(x_1, x_2)) -> in_f_2(x_1, reduce(x_2))
3.36/1.60	   in_f_2(x_1, go_up(x_2)) -> go_up(f(x_1, x_2))
3.36/1.60	   in_f_1(go_up(x_1), x_2) -> go_up(f(x_1, x_2))
3.36/1.60	
3.36/1.60	The set Q consists of the following terms:
3.36/1.60	
3.36/1.60	   reduce(f(x0, x1))
3.36/1.60	   reduce(g(x0))
3.36/1.60	   redex_f(x0, x0)
3.36/1.60	   redex_g(x0)
3.36/1.60	   check_f(result_f(x0))
3.36/1.60	   check_g(result_g(x0))
3.36/1.60	   check_f(redex_f(x0, x1))
3.36/1.60	   in_f_1(go_up(x0), x1)
3.36/1.60	   in_f_2(x0, go_up(x1))
3.36/1.60	
3.36/1.60	We have to consider all minimal (P,Q,R)-chains.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(23) PisEmptyProof (EQUIVALENT)
3.36/1.60	The TRS P is empty. Hence, there is no (P,Q,R) chain.
3.36/1.60	----------------------------------------
3.36/1.60	
3.36/1.60	(24)
3.36/1.60	YES
3.63/1.63	EOF