16.92/5.25	YES
19.12/5.83	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
19.12/5.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
19.12/5.83	
19.12/5.83	
19.12/5.83	Termination w.r.t. Q of the given QTRS could be proven:
19.12/5.83	
19.12/5.83	(0) QTRS
19.12/5.83	(1) QTRS Reverse [EQUIVALENT, 0 ms]
19.12/5.83	(2) QTRS
19.12/5.83	(3) QTRSRRRProof [EQUIVALENT, 28 ms]
19.12/5.83	(4) QTRS
19.12/5.83	(5) DependencyPairsProof [EQUIVALENT, 2 ms]
19.12/5.83	(6) QDP
19.12/5.83	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
19.12/5.83	(8) AND
19.12/5.83	    (9) QDP
19.12/5.83	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
19.12/5.83	        (11) QDP
19.12/5.83	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
19.12/5.83	        (13) YES
19.12/5.83	    (14) QDP
19.12/5.83	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
19.12/5.83	        (16) QDP
19.12/5.83	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
19.12/5.83	        (18) YES
19.12/5.83	    (19) QDP
19.12/5.83	        (20) UsableRulesProof [EQUIVALENT, 0 ms]
19.12/5.83	        (21) QDP
19.12/5.83	        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
19.12/5.83	        (23) YES
19.12/5.83	    (24) QDP
19.12/5.83	        (25) UsableRulesProof [EQUIVALENT, 0 ms]
19.12/5.83	        (26) QDP
19.12/5.83	        (27) QDPOrderProof [EQUIVALENT, 11 ms]
19.12/5.83	        (28) QDP
19.12/5.83	        (29) PisEmptyProof [EQUIVALENT, 0 ms]
19.12/5.83	        (30) YES
19.12/5.83	
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(0)
19.12/5.83	Obligation:
19.12/5.83	Q restricted rewrite system:
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   0(*(x1)) -> *(1(x1))
19.12/5.83	   1(*(x1)) -> 0(#(x1))
19.12/5.83	   #(0(x1)) -> 0(#(x1))
19.12/5.83	   #(1(x1)) -> 1(#(x1))
19.12/5.83	   #($(x1)) -> *($(x1))
19.12/5.83	   #(#(x1)) -> #(x1)
19.12/5.83	   #(*(x1)) -> *(x1)
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(1) QTRS Reverse (EQUIVALENT)
19.12/5.83	We applied the QTRS Reverse Processor [REVERSE].
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(2)
19.12/5.83	Obligation:
19.12/5.83	Q restricted rewrite system:
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	   $(#(x1)) -> $(*(x1))
19.12/5.83	   #(#(x1)) -> #(x1)
19.12/5.83	   *(#(x1)) -> *(x1)
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(3) QTRSRRRProof (EQUIVALENT)
19.12/5.83	Used ordering:
19.12/5.83	Polynomial interpretation [POLO]:
19.12/5.83	
19.12/5.83	   POL(#(x_1)) = 1 + x_1
19.12/5.83	   POL($(x_1)) = x_1
19.12/5.83	   POL(*(x_1)) = 1 + x_1
19.12/5.83	   POL(0(x_1)) = x_1
19.12/5.83	   POL(1(x_1)) = x_1
19.12/5.83	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
19.12/5.83	
19.12/5.83	   #(#(x1)) -> #(x1)
19.12/5.83	   *(#(x1)) -> *(x1)
19.12/5.83	
19.12/5.83	
19.12/5.83	
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(4)
19.12/5.83	Obligation:
19.12/5.83	Q restricted rewrite system:
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	   $(#(x1)) -> $(*(x1))
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(5) DependencyPairsProof (EQUIVALENT)
19.12/5.83	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(6)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	The TRS P consists of the following rules:
19.12/5.83	
19.12/5.83	   *^1(0(x1)) -> 1^1(*(x1))
19.12/5.83	   *^1(0(x1)) -> *^1(x1)
19.12/5.83	   *^1(1(x1)) -> 0^1(x1)
19.12/5.83	   0^1(#(x1)) -> 0^1(x1)
19.12/5.83	   1^1(#(x1)) -> 1^1(x1)
19.12/5.83	   $^1(#(x1)) -> $^1(*(x1))
19.12/5.83	   $^1(#(x1)) -> *^1(x1)
19.12/5.83	
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	   $(#(x1)) -> $(*(x1))
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(7) DependencyGraphProof (EQUIVALENT)
19.12/5.83	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 3 less nodes.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(8)
19.12/5.83	Complex Obligation (AND)
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(9)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	The TRS P consists of the following rules:
19.12/5.83	
19.12/5.83	   1^1(#(x1)) -> 1^1(x1)
19.12/5.83	
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	   $(#(x1)) -> $(*(x1))
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(10) UsableRulesProof (EQUIVALENT)
19.12/5.83	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.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(11)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	The TRS P consists of the following rules:
19.12/5.83	
19.12/5.83	   1^1(#(x1)) -> 1^1(x1)
19.12/5.83	
19.12/5.83	R is empty.
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(12) QDPSizeChangeProof (EQUIVALENT)
19.12/5.83	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
19.12/5.83	
19.12/5.83	From the DPs we obtained the following set of size-change graphs:
19.12/5.83	*1^1(#(x1)) -> 1^1(x1)
19.12/5.83	The graph contains the following edges 1 > 1
19.12/5.83	
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(13)
19.12/5.83	YES
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(14)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	The TRS P consists of the following rules:
19.12/5.83	
19.12/5.83	   0^1(#(x1)) -> 0^1(x1)
19.12/5.83	
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	   $(#(x1)) -> $(*(x1))
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(15) UsableRulesProof (EQUIVALENT)
19.12/5.83	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.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(16)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	The TRS P consists of the following rules:
19.12/5.83	
19.12/5.83	   0^1(#(x1)) -> 0^1(x1)
19.12/5.83	
19.12/5.83	R is empty.
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(17) QDPSizeChangeProof (EQUIVALENT)
19.12/5.83	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
19.12/5.83	
19.12/5.83	From the DPs we obtained the following set of size-change graphs:
19.12/5.83	*0^1(#(x1)) -> 0^1(x1)
19.12/5.83	The graph contains the following edges 1 > 1
19.12/5.83	
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(18)
19.12/5.83	YES
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(19)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	The TRS P consists of the following rules:
19.12/5.83	
19.12/5.83	   *^1(0(x1)) -> *^1(x1)
19.12/5.83	
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	   $(#(x1)) -> $(*(x1))
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(20) UsableRulesProof (EQUIVALENT)
19.12/5.83	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.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(21)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	The TRS P consists of the following rules:
19.12/5.83	
19.12/5.83	   *^1(0(x1)) -> *^1(x1)
19.12/5.83	
19.12/5.83	R is empty.
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(22) QDPSizeChangeProof (EQUIVALENT)
19.12/5.83	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
19.12/5.83	
19.12/5.83	From the DPs we obtained the following set of size-change graphs:
19.12/5.83	**^1(0(x1)) -> *^1(x1)
19.12/5.83	The graph contains the following edges 1 > 1
19.12/5.83	
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(23)
19.12/5.83	YES
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(24)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	The TRS P consists of the following rules:
19.12/5.83	
19.12/5.83	   $^1(#(x1)) -> $^1(*(x1))
19.12/5.83	
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	   $(#(x1)) -> $(*(x1))
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(25) UsableRulesProof (EQUIVALENT)
19.12/5.83	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.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(26)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	The TRS P consists of the following rules:
19.12/5.83	
19.12/5.83	   $^1(#(x1)) -> $^1(*(x1))
19.12/5.83	
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(27) QDPOrderProof (EQUIVALENT)
19.12/5.83	We use the reduction pair processor [LPAR04,JAR06].
19.12/5.83	
19.12/5.83	
19.12/5.83	The following pairs can be oriented strictly and are deleted.
19.12/5.83	
19.12/5.83	   $^1(#(x1)) -> $^1(*(x1))
19.12/5.83	The remaining pairs can at least be oriented weakly.
19.12/5.83	Used ordering:  Polynomial interpretation [POLO]:
19.12/5.83	
19.12/5.83	   POL(#(x_1)) = 4 + 2*x_1
19.12/5.83	   POL($^1(x_1)) = 2*x_1
19.12/5.83	   POL(*(x_1)) = 2*x_1
19.12/5.83	   POL(0(x_1)) = 2 + 2*x_1
19.12/5.83	   POL(1(x_1)) = 4 + 2*x_1
19.12/5.83	
19.12/5.83	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	
19.12/5.83	
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(28)
19.12/5.83	Obligation:
19.12/5.83	Q DP problem:
19.12/5.83	P is empty.
19.12/5.83	The TRS R consists of the following rules:
19.12/5.83	
19.12/5.83	   *(0(x1)) -> 1(*(x1))
19.12/5.83	   *(1(x1)) -> #(0(x1))
19.12/5.83	   0(#(x1)) -> #(0(x1))
19.12/5.83	   1(#(x1)) -> #(1(x1))
19.12/5.83	
19.12/5.83	Q is empty.
19.12/5.83	We have to consider all minimal (P,Q,R)-chains.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(29) PisEmptyProof (EQUIVALENT)
19.12/5.83	The TRS P is empty. Hence, there is no (P,Q,R) chain.
19.12/5.83	----------------------------------------
19.12/5.83	
19.12/5.83	(30)
19.12/5.83	YES
19.21/8.57	EOF