166.67/111.04	MAYBE
166.67/111.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
166.67/111.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
166.67/111.05	
166.67/111.05	
166.67/111.05	Outermost Termination of the given OTRS could not be shown:
166.67/111.05	
166.67/111.05	(0) OTRS
166.67/111.05	(1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
166.67/111.05	(2) QTRS
166.67/111.05	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
166.67/111.05	    (4) QDP
166.67/111.05	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
166.67/111.05	    (6) AND
166.67/111.05	        (7) QDP
166.67/111.05	            (8) UsableRulesProof [EQUIVALENT, 1 ms]
166.67/111.05	            (9) QDP
166.67/111.05	            (10) QReductionProof [EQUIVALENT, 0 ms]
166.67/111.05	            (11) QDP
166.67/111.05	            (12) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
166.67/111.05	            (13) QDP
166.67/111.05	            (14) DependencyGraphProof [EQUIVALENT, 0 ms]
166.67/111.05	            (15) TRUE
166.67/111.05	        (16) QDP
166.67/111.05	            (17) UsableRulesProof [EQUIVALENT, 0 ms]
166.67/111.05	            (18) QDP
166.67/111.05	                (19) QReductionProof [EQUIVALENT, 0 ms]
166.67/111.05	                (20) QDP
166.67/111.05	                (21) TransformationProof [EQUIVALENT, 0 ms]
166.67/111.05	                (22) QDP
166.67/111.05	                (23) TransformationProof [EQUIVALENT, 0 ms]
166.67/111.05	                (24) QDP
166.67/111.05	                (25) TransformationProof [EQUIVALENT, 0 ms]
166.67/111.05	                (26) QDP
166.67/111.05	                (27) QDPOrderProof [EQUIVALENT, 8 ms]
166.67/111.05	                (28) QDP
166.67/111.05	            (29) UsableRulesProof [EQUIVALENT, 0 ms]
166.67/111.05	            (30) QDP
166.67/111.05	                (31) QReductionProof [EQUIVALENT, 0 ms]
166.67/111.05	                (32) QDP
166.67/111.05	(33) Trivial-Transformation [SOUND, 0 ms]
166.67/111.05	(34) QTRS
166.67/111.05	    (35) DependencyPairsProof [EQUIVALENT, 0 ms]
166.67/111.05	    (36) QDP
166.67/111.05	    (37) DependencyGraphProof [EQUIVALENT, 0 ms]
166.67/111.05	    (38) QDP
166.67/111.05	    (39) UsableRulesProof [EQUIVALENT, 0 ms]
166.67/111.05	    (40) QDP
166.67/111.05	    (41) TransformationProof [EQUIVALENT, 0 ms]
166.67/111.05	    (42) QDP
166.67/111.05	    (43) TransformationProof [EQUIVALENT, 0 ms]
166.67/111.05	    (44) QDP
166.67/111.05	    (45) NonTerminationLoopProof [COMPLETE, 0 ms]
166.67/111.05	    (46) NO
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(0)
166.67/111.05	Obligation:
166.67/111.05	Term rewrite system R:
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   cons(x, cons(y, z)) -> big
166.67/111.05	   inf(x) -> cons(x, inf(s(x)))
166.67/111.05	   g(x, x) -> b
166.67/111.05	
166.67/111.05	
166.67/111.05	
166.67/111.05	Outermost Strategy.
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(1) Thiemann-SpecialC-Transformation (EQUIVALENT)
166.67/111.05	We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(2)
166.67/111.05	Obligation:
166.67/111.05	Q restricted rewrite system:
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   top(go_up(x)) -> top(reduce(x))
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   check_inf(redex_inf(x_1)) -> in_inf_1(reduce(x_1))
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_inf_1(go_up(x_1)) -> go_up(inf(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(3) DependencyPairsProof (EQUIVALENT)
166.67/111.05	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(4)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(x)) -> TOP(reduce(x))
166.67/111.05	   TOP(go_up(x)) -> REDUCE(x)
166.67/111.05	   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
166.67/111.05	   REDUCE(cons(x_1, x_2)) -> REDEX_CONS(x_1, x_2)
166.67/111.05	   REDUCE(inf(x_1)) -> CHECK_INF(redex_inf(x_1))
166.67/111.05	   REDUCE(inf(x_1)) -> REDEX_INF(x_1)
166.67/111.05	   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
166.67/111.05	   REDUCE(g(x_1, x_2)) -> REDEX_G(x_1, x_2)
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> IN_CONS_1(reduce(x_1), x_2)
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> IN_CONS_2(x_1, reduce(x_2))
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	   CHECK_INF(redex_inf(x_1)) -> IN_INF_1(reduce(x_1))
166.67/111.05	   CHECK_INF(redex_inf(x_1)) -> REDUCE(x_1)
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> IN_G_1(reduce(x_1), x_2)
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> IN_G_2(x_1, reduce(x_2))
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	   REDUCE(s(x_1)) -> IN_S_1(reduce(x_1))
166.67/111.05	   REDUCE(s(x_1)) -> REDUCE(x_1)
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   top(go_up(x)) -> top(reduce(x))
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   check_inf(redex_inf(x_1)) -> in_inf_1(reduce(x_1))
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_inf_1(go_up(x_1)) -> go_up(inf(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(5) DependencyGraphProof (EQUIVALENT)
166.67/111.05	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 12 less nodes.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(6)
166.67/111.05	Complex Obligation (AND)
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(7)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   REDUCE(s(x_1)) -> REDUCE(x_1)
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   top(go_up(x)) -> top(reduce(x))
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   check_inf(redex_inf(x_1)) -> in_inf_1(reduce(x_1))
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_inf_1(go_up(x_1)) -> go_up(inf(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(8) UsableRulesProof (EQUIVALENT)
166.67/111.05	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.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(9)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   REDUCE(s(x_1)) -> REDUCE(x_1)
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(10) QReductionProof (EQUIVALENT)
166.67/111.05	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(11)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   REDUCE(s(x_1)) -> REDUCE(x_1)
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(12) UsableRulesReductionPairsProof (EQUIVALENT)
166.67/111.05	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.
166.67/111.05	
166.67/111.05	The following dependency pairs can be deleted:
166.67/111.05	
166.67/111.05	   REDUCE(cons(x_1, x_2)) -> CHECK_CONS(redex_cons(x_1, x_2))
166.67/111.05	   REDUCE(g(x_1, x_2)) -> CHECK_G(redex_g(x_1, x_2))
166.67/111.05	   REDUCE(s(x_1)) -> REDUCE(x_1)
166.67/111.05	The following rules are removed from R:
166.67/111.05	
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	Used ordering: POLO with Polynomial interpretation [POLO]:
166.67/111.05	
166.67/111.05	   POL(CHECK_CONS(x_1)) = x_1
166.67/111.05	   POL(CHECK_G(x_1)) = 2*x_1
166.67/111.05	   POL(REDUCE(x_1)) = 2*x_1
166.67/111.05	   POL(b) = 0
166.67/111.05	   POL(big) = 0
166.67/111.05	   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
166.67/111.05	   POL(g(x_1, x_2)) = 2*x_1 + 2*x_2
166.67/111.05	   POL(redex_cons(x_1, x_2)) = 2*x_1 + 2*x_2
166.67/111.05	   POL(redex_g(x_1, x_2)) = x_1 + x_2
166.67/111.05	   POL(result_cons(x_1)) = x_1
166.67/111.05	   POL(result_g(x_1)) = x_1
166.67/111.05	   POL(s(x_1)) = 2*x_1
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(13)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   CHECK_CONS(redex_cons(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_1)
166.67/111.05	   CHECK_G(redex_g(x_1, x_2)) -> REDUCE(x_2)
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(14) DependencyGraphProof (EQUIVALENT)
166.67/111.05	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(15)
166.67/111.05	TRUE
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(16)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(x)) -> TOP(reduce(x))
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   top(go_up(x)) -> top(reduce(x))
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   check_inf(redex_inf(x_1)) -> in_inf_1(reduce(x_1))
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_inf_1(go_up(x_1)) -> go_up(inf(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(17) UsableRulesProof (EQUIVALENT)
166.67/111.05	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.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(18)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(x)) -> TOP(reduce(x))
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(19) QReductionProof (EQUIVALENT)
166.67/111.05	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(20)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(x)) -> TOP(reduce(x))
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(21) TransformationProof (EQUIVALENT)
166.67/111.05	By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]:
166.67/111.05	
166.67/111.05	   (TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1))),TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1))))
166.67/111.05	   (TOP(go_up(inf(x0))) -> TOP(check_inf(redex_inf(x0))),TOP(go_up(inf(x0))) -> TOP(check_inf(redex_inf(x0))))
166.67/111.05	   (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))))
166.67/111.05	   (TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0))),TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0))))
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(22)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1)))
166.67/111.05	   TOP(go_up(inf(x0))) -> TOP(check_inf(redex_inf(x0)))
166.67/111.05	   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
166.67/111.05	   TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0)))
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(23) TransformationProof (EQUIVALENT)
166.67/111.05	By rewriting [LPAR04] the rule TOP(go_up(inf(x0))) -> TOP(check_inf(redex_inf(x0))) at position [0,0] we obtained the following new rules [LPAR04]:
166.67/111.05	
166.67/111.05	   (TOP(go_up(inf(x0))) -> TOP(check_inf(result_inf(cons(x0, inf(s(x0)))))),TOP(go_up(inf(x0))) -> TOP(check_inf(result_inf(cons(x0, inf(s(x0)))))))
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(24)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1)))
166.67/111.05	   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
166.67/111.05	   TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0)))
166.67/111.05	   TOP(go_up(inf(x0))) -> TOP(check_inf(result_inf(cons(x0, inf(s(x0))))))
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(25) TransformationProof (EQUIVALENT)
166.67/111.05	By rewriting [LPAR04] the rule TOP(go_up(inf(x0))) -> TOP(check_inf(result_inf(cons(x0, inf(s(x0)))))) at position [0] we obtained the following new rules [LPAR04]:
166.67/111.05	
166.67/111.05	   (TOP(go_up(inf(x0))) -> TOP(go_up(cons(x0, inf(s(x0))))),TOP(go_up(inf(x0))) -> TOP(go_up(cons(x0, inf(s(x0))))))
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(26)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1)))
166.67/111.05	   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
166.67/111.05	   TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0)))
166.67/111.05	   TOP(go_up(inf(x0))) -> TOP(go_up(cons(x0, inf(s(x0)))))
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(27) QDPOrderProof (EQUIVALENT)
166.67/111.05	We use the reduction pair processor [LPAR04,JAR06].
166.67/111.05	
166.67/111.05	
166.67/111.05	The following pairs can be oriented strictly and are deleted.
166.67/111.05	
166.67/111.05	   TOP(go_up(inf(x0))) -> TOP(go_up(cons(x0, inf(s(x0)))))
166.67/111.05	The remaining pairs can at least be oriented weakly.
166.67/111.05	Used ordering:  Polynomial interpretation [POLO]:
166.67/111.05	
166.67/111.05	   POL(TOP(x_1)) = x_1
166.67/111.05	   POL(b) = 0
166.67/111.05	   POL(big) = 0
166.67/111.05	   POL(check_cons(x_1)) = x_1
166.67/111.05	   POL(check_g(x_1)) = x_1
166.67/111.05	   POL(check_inf(x_1)) = 1
166.67/111.05	   POL(cons(x_1, x_2)) = 0
166.67/111.05	   POL(g(x_1, x_2)) = 0
166.67/111.05	   POL(go_up(x_1)) = x_1
166.67/111.05	   POL(in_cons_1(x_1, x_2)) = 0
166.67/111.05	   POL(in_cons_2(x_1, x_2)) = 0
166.67/111.05	   POL(in_g_1(x_1, x_2)) = 0
166.67/111.05	   POL(in_g_2(x_1, x_2)) = 0
166.67/111.05	   POL(in_s_1(x_1)) = 0
166.67/111.05	   POL(inf(x_1)) = 1
166.67/111.05	   POL(redex_cons(x_1, x_2)) = 0
166.67/111.05	   POL(redex_g(x_1, x_2)) = 0
166.67/111.05	   POL(redex_inf(x_1)) = x_1
166.67/111.05	   POL(reduce(x_1)) = 0
166.67/111.05	   POL(result_cons(x_1)) = x_1
166.67/111.05	   POL(result_g(x_1)) = x_1
166.67/111.05	   POL(result_inf(x_1)) = 1
166.67/111.05	   POL(s(x_1)) = 0
166.67/111.05	
166.67/111.05	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
166.67/111.05	
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(28)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(cons(x0, x1))) -> TOP(check_cons(redex_cons(x0, x1)))
166.67/111.05	   TOP(go_up(g(x0, x1))) -> TOP(check_g(redex_g(x0, x1)))
166.67/111.05	   TOP(go_up(s(x0))) -> TOP(in_s_1(reduce(x0)))
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(29) UsableRulesProof (EQUIVALENT)
166.67/111.05	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.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(30)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(x)) -> TOP(reduce(x))
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(31) QReductionProof (EQUIVALENT)
166.67/111.05	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
166.67/111.05	
166.67/111.05	   top(go_up(x0))
166.67/111.05	   in_inf_1(go_up(x0))
166.67/111.05	
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(32)
166.67/111.05	Obligation:
166.67/111.05	Q DP problem:
166.67/111.05	The TRS P consists of the following rules:
166.67/111.05	
166.67/111.05	   TOP(go_up(x)) -> TOP(reduce(x))
166.67/111.05	
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   reduce(cons(x_1, x_2)) -> check_cons(redex_cons(x_1, x_2))
166.67/111.05	   reduce(inf(x_1)) -> check_inf(redex_inf(x_1))
166.67/111.05	   reduce(g(x_1, x_2)) -> check_g(redex_g(x_1, x_2))
166.67/111.05	   reduce(s(x_1)) -> in_s_1(reduce(x_1))
166.67/111.05	   in_s_1(go_up(x_1)) -> go_up(s(x_1))
166.67/111.05	   redex_g(x, x) -> result_g(b)
166.67/111.05	   check_g(result_g(x)) -> go_up(x)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_1(reduce(x_1), x_2)
166.67/111.05	   check_g(redex_g(x_1, x_2)) -> in_g_2(x_1, reduce(x_2))
166.67/111.05	   in_g_2(x_1, go_up(x_2)) -> go_up(g(x_1, x_2))
166.67/111.05	   in_g_1(go_up(x_1), x_2) -> go_up(g(x_1, x_2))
166.67/111.05	   redex_inf(x) -> result_inf(cons(x, inf(s(x))))
166.67/111.05	   check_inf(result_inf(x)) -> go_up(x)
166.67/111.05	   redex_cons(x, cons(y, z)) -> result_cons(big)
166.67/111.05	   check_cons(result_cons(x)) -> go_up(x)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_1(reduce(x_1), x_2)
166.67/111.05	   check_cons(redex_cons(x_1, x_2)) -> in_cons_2(x_1, reduce(x_2))
166.67/111.05	   in_cons_2(x_1, go_up(x_2)) -> go_up(cons(x_1, x_2))
166.67/111.05	   in_cons_1(go_up(x_1), x_2) -> go_up(cons(x_1, x_2))
166.67/111.05	
166.67/111.05	The set Q consists of the following terms:
166.67/111.05	
166.67/111.05	   reduce(cons(x0, x1))
166.67/111.05	   reduce(inf(x0))
166.67/111.05	   reduce(g(x0, x1))
166.67/111.05	   redex_cons(x0, cons(x1, x2))
166.67/111.05	   redex_inf(x0)
166.67/111.05	   redex_g(x0, x0)
166.67/111.05	   check_cons(result_cons(x0))
166.67/111.05	   check_inf(result_inf(x0))
166.67/111.05	   check_g(result_g(x0))
166.67/111.05	   check_cons(redex_cons(x0, x1))
166.67/111.05	   check_g(redex_g(x0, x1))
166.67/111.05	   reduce(s(x0))
166.67/111.05	   in_cons_1(go_up(x0), x1)
166.67/111.05	   in_cons_2(x0, go_up(x1))
166.67/111.05	   in_s_1(go_up(x0))
166.67/111.05	   in_g_1(go_up(x0), x1)
166.67/111.05	   in_g_2(x0, go_up(x1))
166.67/111.05	
166.67/111.05	We have to consider all minimal (P,Q,R)-chains.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(33) Trivial-Transformation (SOUND)
166.67/111.05	We applied the Trivial transformation  to transform the outermost TRS to a standard TRS.
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(34)
166.67/111.05	Obligation:
166.67/111.05	Q restricted rewrite system:
166.67/111.05	The TRS R consists of the following rules:
166.67/111.05	
166.67/111.05	   cons(x, cons(y, z)) -> big
166.67/111.05	   inf(x) -> cons(x, inf(s(x)))
166.67/111.05	   g(x, x) -> b
166.67/111.05	
166.67/111.05	Q is empty.
166.67/111.05	
166.67/111.05	----------------------------------------
166.67/111.05	
166.67/111.05	(35) DependencyPairsProof (EQUIVALENT)
166.67/111.05	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
166.67/111.05	----------------------------------------
166.67/111.06	
166.67/111.06	(36)
166.67/111.06	Obligation:
166.67/111.06	Q DP problem:
166.67/111.06	The TRS P consists of the following rules:
166.67/111.06	
166.67/111.06	   INF(x) -> CONS(x, inf(s(x)))
166.67/111.06	   INF(x) -> INF(s(x))
166.67/111.06	
166.67/111.06	The TRS R consists of the following rules:
166.67/111.06	
166.67/111.06	   cons(x, cons(y, z)) -> big
166.67/111.06	   inf(x) -> cons(x, inf(s(x)))
166.67/111.06	   g(x, x) -> b
166.67/111.06	
166.67/111.06	Q is empty.
166.67/111.06	We have to consider all minimal (P,Q,R)-chains.
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(37) DependencyGraphProof (EQUIVALENT)
166.67/111.06	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(38)
166.67/111.06	Obligation:
166.67/111.06	Q DP problem:
166.67/111.06	The TRS P consists of the following rules:
166.67/111.06	
166.67/111.06	   INF(x) -> INF(s(x))
166.67/111.06	
166.67/111.06	The TRS R consists of the following rules:
166.67/111.06	
166.67/111.06	   cons(x, cons(y, z)) -> big
166.67/111.06	   inf(x) -> cons(x, inf(s(x)))
166.67/111.06	   g(x, x) -> b
166.67/111.06	
166.67/111.06	Q is empty.
166.67/111.06	We have to consider all minimal (P,Q,R)-chains.
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(39) UsableRulesProof (EQUIVALENT)
166.67/111.06	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(40)
166.67/111.06	Obligation:
166.67/111.06	Q DP problem:
166.67/111.06	The TRS P consists of the following rules:
166.67/111.06	
166.67/111.06	   INF(x) -> INF(s(x))
166.67/111.06	
166.67/111.06	R is empty.
166.67/111.06	Q is empty.
166.67/111.06	We have to consider all minimal (P,Q,R)-chains.
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(41) TransformationProof (EQUIVALENT)
166.67/111.06	By instantiating [LPAR04] the rule INF(x) -> INF(s(x)) we obtained the following new rules [LPAR04]:
166.67/111.06	
166.67/111.06	   (INF(s(z0)) -> INF(s(s(z0))),INF(s(z0)) -> INF(s(s(z0))))
166.67/111.06	
166.67/111.06	
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(42)
166.67/111.06	Obligation:
166.67/111.06	Q DP problem:
166.67/111.06	The TRS P consists of the following rules:
166.67/111.06	
166.67/111.06	   INF(s(z0)) -> INF(s(s(z0)))
166.67/111.06	
166.67/111.06	R is empty.
166.67/111.06	Q is empty.
166.67/111.06	We have to consider all minimal (P,Q,R)-chains.
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(43) TransformationProof (EQUIVALENT)
166.67/111.06	By instantiating [LPAR04] the rule INF(s(z0)) -> INF(s(s(z0))) we obtained the following new rules [LPAR04]:
166.67/111.06	
166.67/111.06	   (INF(s(s(z0))) -> INF(s(s(s(z0)))),INF(s(s(z0))) -> INF(s(s(s(z0)))))
166.67/111.06	
166.67/111.06	
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(44)
166.67/111.06	Obligation:
166.67/111.06	Q DP problem:
166.67/111.06	The TRS P consists of the following rules:
166.67/111.06	
166.67/111.06	   INF(s(s(z0))) -> INF(s(s(s(z0))))
166.67/111.06	
166.67/111.06	R is empty.
166.67/111.06	Q is empty.
166.67/111.06	We have to consider all minimal (P,Q,R)-chains.
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(45) NonTerminationLoopProof (COMPLETE)
166.67/111.06	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
166.67/111.06	Found a loop by semiunifying a rule from P directly.
166.67/111.06	
166.67/111.06	s = INF(s(s(z0))) evaluates to  t =INF(s(s(s(z0))))
166.67/111.06	
166.67/111.06	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
166.67/111.06	* Matcher: [z0 / s(z0)]
166.67/111.06	* Semiunifier: [ ]
166.67/111.06	
166.67/111.06	--------------------------------------------------------------------------------
166.67/111.06	Rewriting sequence
166.67/111.06	
166.67/111.06	The DP semiunifies directly so there is only one rewrite step from INF(s(s(z0))) to INF(s(s(s(z0)))).
166.67/111.06	
166.67/111.06	
166.67/111.06	
166.67/111.06	
166.67/111.06	----------------------------------------
166.67/111.06	
166.67/111.06	(46)
166.67/111.06	NO
166.76/111.11	EOF