5.06/2.19	MAYBE
5.06/2.19	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.06/2.19	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.06/2.19	
5.06/2.19	
5.06/2.19	Quasi decreasingness of the given CTRS could not be shown:
5.06/2.19	
5.06/2.19	(0) CTRS
5.06/2.19	(1) CTRSToQTRSProof [SOUND, 0 ms]
5.06/2.19	(2) QTRS
5.06/2.19	(3) Overlay + Local Confluence [EQUIVALENT, 0 ms]
5.06/2.19	(4) QTRS
5.06/2.19	(5) DependencyPairsProof [EQUIVALENT, 0 ms]
5.06/2.19	(6) QDP
5.06/2.19	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
5.06/2.19	(8) AND
5.06/2.19	    (9) QDP
5.06/2.19	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
5.06/2.19	        (11) QDP
5.06/2.19	        (12) QReductionProof [EQUIVALENT, 0 ms]
5.06/2.19	        (13) QDP
5.06/2.19	        (14) NonTerminationLoopProof [COMPLETE, 0 ms]
5.06/2.19	        (15) NO
5.06/2.19	    (16) QDP
5.06/2.19	        (17) UsableRulesProof [EQUIVALENT, 0 ms]
5.06/2.19	        (18) QDP
5.06/2.19	        (19) QReductionProof [EQUIVALENT, 0 ms]
5.06/2.19	        (20) QDP
5.06/2.19	        (21) NonTerminationLoopProof [COMPLETE, 0 ms]
5.06/2.19	        (22) NO
5.06/2.19	
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(0)
5.06/2.19	Obligation:
5.06/2.19	Conditional term rewrite system:
5.06/2.19	The TRS R consists of the following rules:
5.06/2.19	
5.06/2.19	   a -> f(a)
5.06/2.19	
5.06/2.19	The conditional TRS C consists of the following conditional rules:
5.06/2.19	
5.06/2.19	   f(x) -> b <= f(x) -> z, x -> z
5.06/2.19	
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(1) CTRSToQTRSProof (SOUND)
5.06/2.19	The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(2)
5.06/2.19	Obligation:
5.06/2.19	Q restricted rewrite system:
5.06/2.19	The TRS R consists of the following rules:
5.06/2.19	
5.06/2.19	   f(x) -> U1(f(x), x)
5.06/2.19	   U1(z, x) -> U2(x)
5.06/2.19	   U2(z) -> b
5.06/2.19	   a -> f(a)
5.06/2.19	
5.06/2.19	Q is empty.
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(3) Overlay + Local Confluence (EQUIVALENT)
5.06/2.19	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(4)
5.06/2.19	Obligation:
5.06/2.19	Q restricted rewrite system:
5.06/2.19	The TRS R consists of the following rules:
5.06/2.19	
5.06/2.19	   f(x) -> U1(f(x), x)
5.06/2.19	   U1(z, x) -> U2(x)
5.06/2.19	   U2(z) -> b
5.06/2.19	   a -> f(a)
5.06/2.19	
5.06/2.19	The set Q consists of the following terms:
5.06/2.19	
5.06/2.19	   f(x0)
5.06/2.19	   U1(x0, x1)
5.06/2.19	   U2(x0)
5.06/2.19	   a
5.06/2.19	
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(5) DependencyPairsProof (EQUIVALENT)
5.06/2.19	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(6)
5.06/2.19	Obligation:
5.06/2.19	Q DP problem:
5.06/2.19	The TRS P consists of the following rules:
5.06/2.19	
5.06/2.19	   F(x) -> U1^1(f(x), x)
5.06/2.19	   F(x) -> F(x)
5.06/2.19	   U1^1(z, x) -> U2^1(x)
5.06/2.19	   A -> F(a)
5.06/2.19	   A -> A
5.06/2.19	
5.06/2.19	The TRS R consists of the following rules:
5.06/2.19	
5.06/2.19	   f(x) -> U1(f(x), x)
5.06/2.19	   U1(z, x) -> U2(x)
5.06/2.19	   U2(z) -> b
5.06/2.19	   a -> f(a)
5.06/2.19	
5.06/2.19	The set Q consists of the following terms:
5.06/2.19	
5.06/2.19	   f(x0)
5.06/2.19	   U1(x0, x1)
5.06/2.19	   U2(x0)
5.06/2.19	   a
5.06/2.19	
5.06/2.19	We have to consider all minimal (P,Q,R)-chains.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(7) DependencyGraphProof (EQUIVALENT)
5.06/2.19	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(8)
5.06/2.19	Complex Obligation (AND)
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(9)
5.06/2.19	Obligation:
5.06/2.19	Q DP problem:
5.06/2.19	The TRS P consists of the following rules:
5.06/2.19	
5.06/2.19	   F(x) -> F(x)
5.06/2.19	
5.06/2.19	The TRS R consists of the following rules:
5.06/2.19	
5.06/2.19	   f(x) -> U1(f(x), x)
5.06/2.19	   U1(z, x) -> U2(x)
5.06/2.19	   U2(z) -> b
5.06/2.19	   a -> f(a)
5.06/2.19	
5.06/2.19	The set Q consists of the following terms:
5.06/2.19	
5.06/2.19	   f(x0)
5.06/2.19	   U1(x0, x1)
5.06/2.19	   U2(x0)
5.06/2.19	   a
5.06/2.19	
5.06/2.19	We have to consider all minimal (P,Q,R)-chains.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(10) UsableRulesProof (EQUIVALENT)
5.06/2.19	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.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(11)
5.06/2.19	Obligation:
5.06/2.19	Q DP problem:
5.06/2.19	The TRS P consists of the following rules:
5.06/2.19	
5.06/2.19	   F(x) -> F(x)
5.06/2.19	
5.06/2.19	R is empty.
5.06/2.19	The set Q consists of the following terms:
5.06/2.19	
5.06/2.19	   f(x0)
5.06/2.19	   U1(x0, x1)
5.06/2.19	   U2(x0)
5.06/2.19	   a
5.06/2.19	
5.06/2.19	We have to consider all minimal (P,Q,R)-chains.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(12) QReductionProof (EQUIVALENT)
5.06/2.19	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
5.06/2.19	
5.06/2.19	   f(x0)
5.06/2.19	   U1(x0, x1)
5.06/2.19	   U2(x0)
5.06/2.19	   a
5.06/2.19	
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(13)
5.06/2.19	Obligation:
5.06/2.19	Q DP problem:
5.06/2.19	The TRS P consists of the following rules:
5.06/2.19	
5.06/2.19	   F(x) -> F(x)
5.06/2.19	
5.06/2.19	R is empty.
5.06/2.19	Q is empty.
5.06/2.19	We have to consider all minimal (P,Q,R)-chains.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(14) NonTerminationLoopProof (COMPLETE)
5.06/2.19	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
5.06/2.19	Found a loop by semiunifying a rule from P directly.
5.06/2.19	
5.06/2.19	s = F(x) evaluates to  t =F(x)
5.06/2.19	
5.06/2.19	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
5.06/2.19	* Matcher: [ ]
5.06/2.19	* Semiunifier: [ ]
5.06/2.19	
5.06/2.19	--------------------------------------------------------------------------------
5.06/2.19	Rewriting sequence
5.06/2.19	
5.06/2.19	The DP semiunifies directly so there is only one rewrite step from F(x) to F(x).
5.06/2.19	
5.06/2.19	
5.06/2.19	
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(15)
5.06/2.19	NO
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(16)
5.06/2.19	Obligation:
5.06/2.19	Q DP problem:
5.06/2.19	The TRS P consists of the following rules:
5.06/2.19	
5.06/2.19	   A -> A
5.06/2.19	
5.06/2.19	The TRS R consists of the following rules:
5.06/2.19	
5.06/2.19	   f(x) -> U1(f(x), x)
5.06/2.19	   U1(z, x) -> U2(x)
5.06/2.19	   U2(z) -> b
5.06/2.19	   a -> f(a)
5.06/2.19	
5.06/2.19	The set Q consists of the following terms:
5.06/2.19	
5.06/2.19	   f(x0)
5.06/2.19	   U1(x0, x1)
5.06/2.19	   U2(x0)
5.06/2.19	   a
5.06/2.19	
5.06/2.19	We have to consider all minimal (P,Q,R)-chains.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(17) UsableRulesProof (EQUIVALENT)
5.06/2.19	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.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(18)
5.06/2.19	Obligation:
5.06/2.19	Q DP problem:
5.06/2.19	The TRS P consists of the following rules:
5.06/2.19	
5.06/2.19	   A -> A
5.06/2.19	
5.06/2.19	R is empty.
5.06/2.19	The set Q consists of the following terms:
5.06/2.19	
5.06/2.19	   f(x0)
5.06/2.19	   U1(x0, x1)
5.06/2.19	   U2(x0)
5.06/2.19	   a
5.06/2.19	
5.06/2.19	We have to consider all minimal (P,Q,R)-chains.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(19) QReductionProof (EQUIVALENT)
5.06/2.19	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
5.06/2.19	
5.06/2.19	   f(x0)
5.06/2.19	   U1(x0, x1)
5.06/2.19	   U2(x0)
5.06/2.19	   a
5.06/2.19	
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(20)
5.06/2.19	Obligation:
5.06/2.19	Q DP problem:
5.06/2.19	The TRS P consists of the following rules:
5.06/2.19	
5.06/2.19	   A -> A
5.06/2.19	
5.06/2.19	R is empty.
5.06/2.19	Q is empty.
5.06/2.19	We have to consider all minimal (P,Q,R)-chains.
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(21) NonTerminationLoopProof (COMPLETE)
5.06/2.19	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
5.06/2.19	Found a loop by semiunifying a rule from P directly.
5.06/2.19	
5.06/2.19	s = A evaluates to  t =A
5.06/2.19	
5.06/2.19	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
5.06/2.19	* Matcher: [ ]
5.06/2.19	* Semiunifier: [ ]
5.06/2.19	
5.06/2.19	--------------------------------------------------------------------------------
5.06/2.19	Rewriting sequence
5.06/2.19	
5.06/2.19	The DP semiunifies directly so there is only one rewrite step from A to A.
5.06/2.19	
5.06/2.19	
5.06/2.19	
5.06/2.19	
5.06/2.19	----------------------------------------
5.06/2.19	
5.06/2.19	(22)
5.06/2.19	NO
5.32/2.23	EOF