5.36/2.55	MAYBE
5.47/2.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.47/2.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.47/2.55	
5.47/2.55	
5.47/2.55	Quasi decreasingness of the given CTRS could not be shown:
5.47/2.55	
5.47/2.55	(0) CTRS
5.47/2.55	(1) CTRSToQTRSProof [SOUND, 0 ms]
5.47/2.55	(2) QTRS
5.47/2.55	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
5.47/2.55	(4) QDP
5.47/2.55	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
5.47/2.55	(6) QDP
5.47/2.55	(7) UsableRulesProof [EQUIVALENT, 4 ms]
5.47/2.55	(8) QDP
5.47/2.55	(9) MNOCProof [EQUIVALENT, 0 ms]
5.47/2.55	(10) QDP
5.47/2.55	(11) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	(12) QDP
5.47/2.55	(13) UsableRulesProof [EQUIVALENT, 0 ms]
5.47/2.55	(14) QDP
5.47/2.55	(15) QReductionProof [EQUIVALENT, 0 ms]
5.47/2.55	(16) QDP
5.47/2.55	(17) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	(18) QDP
5.47/2.55	(19) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	(20) QDP
5.47/2.55	(21) DependencyGraphProof [EQUIVALENT, 0 ms]
5.47/2.55	(22) QDP
5.47/2.55	(23) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	(24) QDP
5.47/2.55	(25) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	(26) QDP
5.47/2.55	(27) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	(28) QDP
5.47/2.55	(29) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	(30) QDP
5.47/2.55	(31) DependencyGraphProof [EQUIVALENT, 0 ms]
5.47/2.55	(32) AND
5.47/2.55	    (33) QDP
5.47/2.55	        (34) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	        (35) QDP
5.47/2.55	        (36) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	        (37) QDP
5.47/2.55	        (38) NonTerminationLoopProof [COMPLETE, 0 ms]
5.47/2.55	        (39) NO
5.47/2.55	    (40) QDP
5.47/2.55	        (41) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	        (42) QDP
5.47/2.55	        (43) TransformationProof [EQUIVALENT, 0 ms]
5.47/2.55	        (44) QDP
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(0)
5.47/2.55	Obligation:
5.47/2.55	Conditional term rewrite system:
5.47/2.55	The TRS R consists of the following rules:
5.47/2.55	
5.47/2.55	   a -> b
5.47/2.55	
5.47/2.55	The conditional TRS C consists of the following conditional rules:
5.47/2.55	
5.47/2.55	   f(x', x'') -> h(x, f(x, b)) <= x' -> x, x'' -> x
5.47/2.55	   f(g(y'), y'') -> h(y, f(g(y), a)) <= y' -> y, y'' -> y
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(1) CTRSToQTRSProof (SOUND)
5.47/2.55	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(2)
5.47/2.55	Obligation:
5.47/2.55	Q restricted rewrite system:
5.47/2.55	The TRS R consists of the following rules:
5.47/2.55	
5.47/2.55	   f(x', x'') -> U1(x', x'')
5.47/2.55	   U1(x, x'') -> U2(x'', x)
5.47/2.55	   U2(x, x) -> h(x, f(x, b))
5.47/2.55	   f(g(y'), y'') -> U3(y', y'')
5.47/2.55	   U3(y, y'') -> U4(y'', y)
5.47/2.55	   U4(y, y) -> h(y, f(g(y), a))
5.47/2.55	   a -> b
5.47/2.55	
5.47/2.55	Q is empty.
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(3) DependencyPairsProof (EQUIVALENT)
5.47/2.55	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(4)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   F(x', x'') -> U1^1(x', x'')
5.47/2.55	   U1^1(x, x'') -> U2^1(x'', x)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), a)
5.47/2.55	   U4^1(y, y) -> A
5.47/2.55	
5.47/2.55	The TRS R consists of the following rules:
5.47/2.55	
5.47/2.55	   f(x', x'') -> U1(x', x'')
5.47/2.55	   U1(x, x'') -> U2(x'', x)
5.47/2.55	   U2(x, x) -> h(x, f(x, b))
5.47/2.55	   f(g(y'), y'') -> U3(y', y'')
5.47/2.55	   U3(y, y'') -> U4(y'', y)
5.47/2.55	   U4(y, y) -> h(y, f(g(y), a))
5.47/2.55	   a -> b
5.47/2.55	
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(5) DependencyGraphProof (EQUIVALENT)
5.47/2.55	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(6)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U1^1(x, x'') -> U2^1(x'', x)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(x', x'') -> U1^1(x', x'')
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), a)
5.47/2.55	
5.47/2.55	The TRS R consists of the following rules:
5.47/2.55	
5.47/2.55	   f(x', x'') -> U1(x', x'')
5.47/2.55	   U1(x, x'') -> U2(x'', x)
5.47/2.55	   U2(x, x) -> h(x, f(x, b))
5.47/2.55	   f(g(y'), y'') -> U3(y', y'')
5.47/2.55	   U3(y, y'') -> U4(y'', y)
5.47/2.55	   U4(y, y) -> h(y, f(g(y), a))
5.47/2.55	   a -> b
5.47/2.55	
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(7) UsableRulesProof (EQUIVALENT)
5.47/2.55	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.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(8)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U1^1(x, x'') -> U2^1(x'', x)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(x', x'') -> U1^1(x', x'')
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), a)
5.47/2.55	
5.47/2.55	The TRS R consists of the following rules:
5.47/2.55	
5.47/2.55	   a -> b
5.47/2.55	
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(9) MNOCProof (EQUIVALENT)
5.47/2.55	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(10)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U1^1(x, x'') -> U2^1(x'', x)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(x', x'') -> U1^1(x', x'')
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), a)
5.47/2.55	
5.47/2.55	The TRS R consists of the following rules:
5.47/2.55	
5.47/2.55	   a -> b
5.47/2.55	
5.47/2.55	The set Q consists of the following terms:
5.47/2.55	
5.47/2.55	   a
5.47/2.55	
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(11) TransformationProof (EQUIVALENT)
5.47/2.55	By rewriting [LPAR04] the rule U4^1(y, y) -> F(g(y), a) at position [1] we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (U4^1(y, y) -> F(g(y), b),U4^1(y, y) -> F(g(y), b))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(12)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U1^1(x, x'') -> U2^1(x'', x)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(x', x'') -> U1^1(x', x'')
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), b)
5.47/2.55	
5.47/2.55	The TRS R consists of the following rules:
5.47/2.55	
5.47/2.55	   a -> b
5.47/2.55	
5.47/2.55	The set Q consists of the following terms:
5.47/2.55	
5.47/2.55	   a
5.47/2.55	
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(13) UsableRulesProof (EQUIVALENT)
5.47/2.55	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.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(14)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U1^1(x, x'') -> U2^1(x'', x)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(x', x'') -> U1^1(x', x'')
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	The set Q consists of the following terms:
5.47/2.55	
5.47/2.55	   a
5.47/2.55	
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(15) QReductionProof (EQUIVALENT)
5.47/2.55	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
5.47/2.55	
5.47/2.55	   a
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(16)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U1^1(x, x'') -> U2^1(x'', x)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(x', x'') -> U1^1(x', x'')
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(17) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule F(x', x'') -> U1^1(x', x'') we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (F(z0, b) -> U1^1(z0, b),F(z0, b) -> U1^1(z0, b))
5.47/2.55	   (F(g(z0), b) -> U1^1(g(z0), b),F(g(z0), b) -> U1^1(g(z0), b))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(18)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U1^1(x, x'') -> U2^1(x'', x)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), b)
5.47/2.55	   F(z0, b) -> U1^1(z0, b)
5.47/2.55	   F(g(z0), b) -> U1^1(g(z0), b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(19) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule U1^1(x, x'') -> U2^1(x'', x) we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (U1^1(z0, b) -> U2^1(b, z0),U1^1(z0, b) -> U2^1(b, z0))
5.47/2.55	   (U1^1(g(z0), b) -> U2^1(b, g(z0)),U1^1(g(z0), b) -> U2^1(b, g(z0)))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(20)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), b)
5.47/2.55	   F(z0, b) -> U1^1(z0, b)
5.47/2.55	   F(g(z0), b) -> U1^1(g(z0), b)
5.47/2.55	   U1^1(z0, b) -> U2^1(b, z0)
5.47/2.55	   U1^1(g(z0), b) -> U2^1(b, g(z0))
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(21) DependencyGraphProof (EQUIVALENT)
5.47/2.55	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(22)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   F(g(y'), y'') -> U3^1(y', y'')
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), b)
5.47/2.55	   F(z0, b) -> U1^1(z0, b)
5.47/2.55	   U1^1(z0, b) -> U2^1(b, z0)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(g(z0), b) -> U1^1(g(z0), b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(23) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule F(g(y'), y'') -> U3^1(y', y'') we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (F(g(z0), b) -> U3^1(z0, b),F(g(z0), b) -> U3^1(z0, b))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(24)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U3^1(y, y'') -> U4^1(y'', y)
5.47/2.55	   U4^1(y, y) -> F(g(y), b)
5.47/2.55	   F(z0, b) -> U1^1(z0, b)
5.47/2.55	   U1^1(z0, b) -> U2^1(b, z0)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(g(z0), b) -> U1^1(g(z0), b)
5.47/2.55	   F(g(z0), b) -> U3^1(z0, b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(25) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule U3^1(y, y'') -> U4^1(y'', y) we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (U3^1(z0, b) -> U4^1(b, z0),U3^1(z0, b) -> U4^1(b, z0))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(26)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U4^1(y, y) -> F(g(y), b)
5.47/2.55	   F(z0, b) -> U1^1(z0, b)
5.47/2.55	   U1^1(z0, b) -> U2^1(b, z0)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(g(z0), b) -> U1^1(g(z0), b)
5.47/2.55	   F(g(z0), b) -> U3^1(z0, b)
5.47/2.55	   U3^1(z0, b) -> U4^1(b, z0)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(27) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule U4^1(y, y) -> F(g(y), b) we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (U4^1(b, b) -> F(g(b), b),U4^1(b, b) -> F(g(b), b))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(28)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   F(z0, b) -> U1^1(z0, b)
5.47/2.55	   U1^1(z0, b) -> U2^1(b, z0)
5.47/2.55	   U2^1(x, x) -> F(x, b)
5.47/2.55	   F(g(z0), b) -> U1^1(g(z0), b)
5.47/2.55	   F(g(z0), b) -> U3^1(z0, b)
5.47/2.55	   U3^1(z0, b) -> U4^1(b, z0)
5.47/2.55	   U4^1(b, b) -> F(g(b), b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(29) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule U2^1(x, x) -> F(x, b) we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (U2^1(b, b) -> F(b, b),U2^1(b, b) -> F(b, b))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(30)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   F(z0, b) -> U1^1(z0, b)
5.47/2.55	   U1^1(z0, b) -> U2^1(b, z0)
5.47/2.55	   F(g(z0), b) -> U1^1(g(z0), b)
5.47/2.55	   F(g(z0), b) -> U3^1(z0, b)
5.47/2.55	   U3^1(z0, b) -> U4^1(b, z0)
5.47/2.55	   U4^1(b, b) -> F(g(b), b)
5.47/2.55	   U2^1(b, b) -> F(b, b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(31) DependencyGraphProof (EQUIVALENT)
5.47/2.55	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(32)
5.47/2.55	Complex Obligation (AND)
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(33)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U1^1(z0, b) -> U2^1(b, z0)
5.47/2.55	   U2^1(b, b) -> F(b, b)
5.47/2.55	   F(z0, b) -> U1^1(z0, b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(34) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule F(z0, b) -> U1^1(z0, b) we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (F(b, b) -> U1^1(b, b),F(b, b) -> U1^1(b, b))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(35)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U1^1(z0, b) -> U2^1(b, z0)
5.47/2.55	   U2^1(b, b) -> F(b, b)
5.47/2.55	   F(b, b) -> U1^1(b, b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(36) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule U1^1(z0, b) -> U2^1(b, z0) we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (U1^1(b, b) -> U2^1(b, b),U1^1(b, b) -> U2^1(b, b))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(37)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U2^1(b, b) -> F(b, b)
5.47/2.55	   F(b, b) -> U1^1(b, b)
5.47/2.55	   U1^1(b, b) -> U2^1(b, b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(38) NonTerminationLoopProof (COMPLETE)
5.47/2.55	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
5.47/2.55	Found a loop by narrowing to the left:
5.47/2.55	
5.47/2.55	s = F(b, b) evaluates to  t =F(b, b)
5.47/2.55	
5.47/2.55	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
5.47/2.55	* Matcher: [ ]
5.47/2.55	* Semiunifier: [ ]
5.47/2.55	
5.47/2.55	--------------------------------------------------------------------------------
5.47/2.55	Rewriting sequence
5.47/2.55	
5.47/2.55	F(b, b) -> U1^1(b, b)
5.47/2.55	with rule F(b, b) -> U1^1(b, b) at position [] and matcher [ ]
5.47/2.55	
5.47/2.55	U1^1(b, b) -> U2^1(b, b)
5.47/2.55	with rule U1^1(b, b) -> U2^1(b, b) at position [] and matcher [ ]
5.47/2.55	
5.47/2.55	U2^1(b, b) -> F(b, b)
5.47/2.55	with rule U2^1(b, b) -> F(b, b)
5.47/2.55	
5.47/2.55	Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
5.47/2.55	
5.47/2.55	
5.47/2.55	All these steps are and every following step will be a correct step w.r.t to Q.
5.47/2.55	
5.47/2.55	
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(39)
5.47/2.55	NO
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(40)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   F(g(z0), b) -> U3^1(z0, b)
5.47/2.55	   U3^1(z0, b) -> U4^1(b, z0)
5.47/2.55	   U4^1(b, b) -> F(g(b), b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(41) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule F(g(z0), b) -> U3^1(z0, b) we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (F(g(b), b) -> U3^1(b, b),F(g(b), b) -> U3^1(b, b))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(42)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U3^1(z0, b) -> U4^1(b, z0)
5.47/2.55	   U4^1(b, b) -> F(g(b), b)
5.47/2.55	   F(g(b), b) -> U3^1(b, b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(43) TransformationProof (EQUIVALENT)
5.47/2.55	By instantiating [LPAR04] the rule U3^1(z0, b) -> U4^1(b, z0) we obtained the following new rules [LPAR04]:
5.47/2.55	
5.47/2.55	   (U3^1(b, b) -> U4^1(b, b),U3^1(b, b) -> U4^1(b, b))
5.47/2.55	
5.47/2.55	
5.47/2.55	----------------------------------------
5.47/2.55	
5.47/2.55	(44)
5.47/2.55	Obligation:
5.47/2.55	Q DP problem:
5.47/2.55	The TRS P consists of the following rules:
5.47/2.55	
5.47/2.55	   U4^1(b, b) -> F(g(b), b)
5.47/2.55	   F(g(b), b) -> U3^1(b, b)
5.47/2.55	   U3^1(b, b) -> U4^1(b, b)
5.47/2.55	
5.47/2.55	R is empty.
5.47/2.55	Q is empty.
5.47/2.55	We have to consider all minimal (P,Q,R)-chains.
5.47/2.60	EOF