0.00/0.34	MAYBE
0.00/0.34	(ignored inputs)COMMENT doi:10.1007/11805618_3 [69] p. 27 submitted by: Thomas Sternagel and Aart Middeldorp
0.00/0.34	Conditional Rewrite Rules:
0.00/0.34	   [ lt(?x,0) -> false,
0.00/0.34	     lt(0,s(?y)) -> true,
0.00/0.34	     lt(s(?x),s(?y)) -> lt(?x,?y),
0.00/0.34	     cons(?x,cons(?y,?ys)) -> cons(?y,cons(?x,?ys)) | lt(?x,?y) == true ]
0.00/0.34	Check whether all rules are type 3
0.00/0.34	OK
0.00/0.34	Check whether the input is deterministic
0.00/0.34	OK
0.00/0.34	Result of unraveling:
0.00/0.34	   [ lt(?x,0) -> false,
0.00/0.34	     lt(0,s(?y)) -> true,
0.00/0.34	     lt(s(?x),s(?y)) -> lt(?x,?y),
0.00/0.34	     cons(?x,cons(?y,?ys)) -> U0(lt(?x,?y),?x,?y,?ys),
0.00/0.34	     U0(true,?x,?y,?ys) -> cons(?y,cons(?x,?ys)) ]
0.00/0.34	Check whether U(R) is terminating
0.00/0.34	failed to show termination
0.00/0.34	Check whether the input is weakly left-linear
0.00/0.34	OK
0.00/0.34	Conditional critical pairs (CCPs):
0.00/0.34	   [ cons(?x_1,cons(?y,cons(?x,?ys))) = cons(?x,cons(?x_1,cons(?y,?ys))) | lt(?x,?y) == true,lt(?x_1,?x) == true ]
0.00/0.34	Check whether the input is almost orthogonale
0.00/0.34	not almost orthogonal
0.00/0.34	OK
0.00/0.34	Check U(R) is confluent
0.00/0.34	Rewrite Rules:
0.00/0.34	   [ lt(?x,0) -> false,
0.00/0.34	     lt(0,s(?y)) -> true,
0.00/0.34	     lt(s(?x),s(?y)) -> lt(?x,?y),
0.00/0.34	     cons(?x,cons(?y,?ys)) -> U0(lt(?x,?y),?x,?y,?ys),
0.00/0.34	     U0(true,?x,?y,?ys) -> cons(?y,cons(?x,?ys)) ]
0.00/0.34	Apply Direct Methods...
0.00/0.34	Inner CPs:
0.00/0.34	   [ cons(?x_1,U0(lt(?x,?y),?x,?y,?ys)) = U0(lt(?x_1,?x),?x_1,?x,cons(?y,?ys)) ]
0.00/0.34	Outer CPs:
0.00/0.34	   [  ]
0.00/0.34	not Overlay, check Termination...
0.00/0.34	unknown/not Terminating
0.00/0.34	unknown Knuth & Bendix
0.00/0.34	Left-Linear, not Right-Linear
0.00/0.34	unknown Development Closed
0.00/0.34	unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
0.00/0.34	inner CP cond (upside-parallel)
0.00/0.34	innter CP Cond (outside)
0.00/0.34	unknown Upside-Parallel-Closed/Outside-Closed
0.00/0.34	(inner) Parallel CPs: (not computed)
0.00/0.34	unknown Toyama (Parallel CPs)
0.00/0.34	Simultaneous CPs:
0.00/0.34	   [ cons(?x,U0(lt(?y,?y_1),?y,?y_1,?ys_1)) = U0(lt(?x,?y),?x,?y,cons(?y_1,?ys_1)),
0.00/0.34	     U0(lt(?x_1,?x),?x_1,?x,U0(lt(?y,?y_2),?y,?y_2,?ys_2)) = cons(?x_1,U0(lt(?x,?y),?x,?y,cons(?y_2,?ys_2))),
0.00/0.34	     U0(lt(?x_1,?x),?x_1,?x,cons(?y,?ys)) = cons(?x_1,U0(lt(?x,?y),?x,?y,?ys)) ]
0.00/0.34	unknown Okui (Simultaneous CPs)
0.00/0.34	unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
0.00/0.34	unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
0.00/0.34	check Locally Decreasing Diagrams by Rule Labelling...
0.00/0.34	Critical Pair <cons(?x_1,U0(lt(?x,?y),?x,?y,?ys)), U0(lt(?x_1,?x),?x_1,?x,cons(?y,?ys))> by Rules <3, 3> preceded by [(cons,2)]
0.00/0.34	unknown Diagram Decreasing
0.00/0.34	   [ lt(?x,0) -> false,
0.00/0.34	     lt(0,s(?y_1)) -> true,
0.00/0.34	     lt(s(?x_2),s(?y_2)) -> lt(?x_2,?y_2),
0.00/0.34	     cons(?x_3,cons(?y_3,?ys_3)) -> U0(lt(?x_3,?y_3),?x_3,?y_3,?ys_3),
0.00/0.34	     U0(true,?x_4,?y_4,?ys_4) -> cons(?y_4,cons(?x_4,?ys_4)) ]
0.00/0.34	Sort Assignment:
0.00/0.34	 0 : =>16
0.00/0.34	 s : 16=>16
0.00/0.34	 U0 : 14*16*16*27=>27
0.00/0.34	 lt : 16*16=>14
0.00/0.34	 cons : 16*27=>27
0.00/0.34	 true : =>14
0.00/0.34	 false : =>14
0.00/0.34	non-linear variables: {?x_3,?y_3}
0.00/0.34	non-linear types: {16}
0.00/0.34	types leq non-linear types: {16}
0.00/0.34	rules applicable to terms of non-linear types:
0.00/0.34	   [ ]
0.00/0.34	Rnl:
0.00/0.34	0: {}
0.00/0.34	1: {}
0.00/0.34	2: {}
0.00/0.34	3: {}
0.00/0.34	4: {}
0.00/0.34	terms of non-linear types are innermost terminating
0.00/0.34	Critical Pair <cons(?x_1,U0(lt(?x,?y),?x,?y,?ys)), U0(lt(?x_1,?x),?x_1,?x,cons(?y,?ys))> by Rules <3, 3>
0.00/0.34	no joinable sequence for some critical pairs
0.00/0.34	unknown Quasi-Linear & Linearized-Decreasing
0.00/0.34	   [ lt(?x,0) -> false,
0.00/0.34	     lt(0,s(?y_1)) -> true,
0.00/0.34	     lt(s(?x_2),s(?y_2)) -> lt(?x_2,?y_2),
0.00/0.34	     cons(?x_3,cons(?y_3,?ys_3)) -> U0(lt(?x_3,?y_3),?x_3,?y_3,?ys_3),
0.00/0.34	     U0(true,?x_4,?y_4,?ys_4) -> cons(?y_4,cons(?x_4,?ys_4)) ]
0.00/0.34	Sort Assignment:
0.00/0.34	 0 : =>16
0.00/0.34	 s : 16=>16
0.00/0.34	 U0 : 14*16*16*27=>27
0.00/0.34	 lt : 16*16=>14
0.00/0.34	 cons : 16*27=>27
0.00/0.34	 true : =>14
0.00/0.34	 false : =>14
0.00/0.34	non-linear variables: {?x_3,?y_3}
0.00/0.34	non-linear types: {16}
0.00/0.34	types leq non-linear types: {16}
0.00/0.34	rules applicable to terms of non-linear types:
0.00/0.34	   [ ]
0.00/0.34	terms of non-linear types are terminating
0.00/0.34	Check Joinablility of CP from NLR:
0.00/0.34	done.
0.00/0.34	Critical Pair <cons(?x_1,U0(lt(?x,?y),?x,?y,?ys)), U0(lt(?x_1,?x),?x_1,?x,cons(?y,?ys))> by Rules <3, 3>
0.00/0.34	no joinable sequence for some critical pairs
0.00/0.34	unknown Strongly Quasi-Linear & Hierarchically Decreasing
0.00/0.34	unknown Huet (modulo AC)
0.00/0.34	check by Reduction-Preserving Completion...
0.00/0.34	failure(empty P)
0.00/0.34	unknown Reduction-Preserving Completion
0.00/0.34	check by Ordered Rewriting...
0.00/0.34	remove redundants rules and split
0.00/0.34	R-part:
0.00/0.34	   [ lt(?x,0) -> false,
0.00/0.34	     lt(0,s(?y)) -> true,
0.00/0.34	     lt(s(?x),s(?y)) -> lt(?x,?y),
0.00/0.34	     cons(?x,cons(?y,?ys)) -> U0(lt(?x,?y),?x,?y,?ys),
0.00/0.34	     U0(true,?x,?y,?ys) -> cons(?y,cons(?x,?ys)) ]
0.00/0.34	E-part:
0.00/0.34	   [ ]
0.00/0.34	...failed to find a suitable LPO.
0.00/0.34	unknown Confluence by Ordered Rewriting
0.00/0.34	Direct Methods: Can't judge
0.00/0.34	
0.00/0.34	Try Persistent Decomposition for...
0.00/0.34	   [ lt(?x,0) -> false,
0.00/0.34	     lt(0,s(?y)) -> true,
0.00/0.34	     lt(s(?x),s(?y)) -> lt(?x,?y),
0.00/0.34	     cons(?x,cons(?y,?ys)) -> U0(lt(?x,?y),?x,?y,?ys),
0.00/0.34	     U0(true,?x,?y,?ys) -> cons(?y,cons(?x,?ys)) ]
0.00/0.34	Sort Assignment:
0.00/0.34	 0 : =>16
0.00/0.34	 s : 16=>16
0.00/0.34	 U0 : 14*16*16*27=>27
0.00/0.34	 lt : 16*16=>14
0.00/0.34	 cons : 16*27=>27
0.00/0.34	 true : =>14
0.00/0.34	 false : =>14
0.00/0.34	maximal types: {14,16,27}
0.00/0.34	Persistent Decomposition failed: Can't judge
0.00/0.34	
0.00/0.34	Try Layer Preserving Decomposition for...
0.00/0.34	   [ lt(?x,0) -> false,
0.00/0.34	     lt(0,s(?y)) -> true,
0.00/0.34	     lt(s(?x),s(?y)) -> lt(?x,?y),
0.00/0.34	     cons(?x,cons(?y,?ys)) -> U0(lt(?x,?y),?x,?y,?ys),
0.00/0.34	     U0(true,?x,?y,?ys) -> cons(?y,cons(?x,?ys)) ]
0.00/0.34	Layer Preserving Decomposition failed: Can't judge
0.00/0.34	
0.00/0.34	Try Commutative Decomposition for...
0.00/0.34	   [ lt(?x,0) -> false,
0.00/0.34	     lt(0,s(?y)) -> true,
0.00/0.34	     lt(s(?x),s(?y)) -> lt(?x,?y),
0.00/0.34	     cons(?x,cons(?y,?ys)) -> U0(lt(?x,?y),?x,?y,?ys),
0.00/0.34	     U0(true,?x,?y,?ys) -> cons(?y,cons(?x,?ys)) ]
0.00/0.34	Try A Minimal Decomposition {0}{1}{2}{3}{4}
0.00/0.34	{0}
0.00/0.34	(cm)Rewrite Rules:
0.00/0.34	   [ lt(?x,0) -> false ]
0.00/0.34	Apply Direct Methods...
0.00/0.34	Inner CPs:
0.00/0.34	   [  ]
0.00/0.34	Outer CPs:
0.00/0.34	   [  ]
0.00/0.34	Overlay, check Innermost Termination...
0.00/0.34	Innermost Terminating (hence Terminating), WCR
0.00/0.34	Knuth & Bendix
0.00/0.34	Direct Methods: CR
0.00/0.34	{1}
0.00/0.34	(cm)Rewrite Rules:
0.00/0.34	   [ lt(0,s(?y)) -> true ]
0.00/0.34	Apply Direct Methods...
0.00/0.34	Inner CPs:
0.00/0.34	   [  ]
0.00/0.34	Outer CPs:
0.00/0.34	   [  ]
0.00/0.34	Overlay, check Innermost Termination...
0.00/0.34	Innermost Terminating (hence Terminating), WCR
0.00/0.34	Knuth & Bendix
0.00/0.34	Direct Methods: CR
0.00/0.34	{2}
0.00/0.34	(cm)Rewrite Rules:
0.00/0.34	   [ lt(s(?x),s(?y)) -> lt(?x,?y) ]
0.00/0.34	Apply Direct Methods...
0.00/0.34	Inner CPs:
0.00/0.34	   [  ]
0.00/0.34	Outer CPs:
0.00/0.34	   [  ]
0.00/0.34	Overlay, check Innermost Termination...
0.00/0.34	Innermost Terminating (hence Terminating), WCR
0.00/0.34	Knuth & Bendix
0.00/0.34	Direct Methods: CR
0.00/0.34	{3}
0.00/0.34	(cm)Rewrite Rules:
0.00/0.34	   [ cons(?x,cons(?y,?ys)) -> U0(lt(?x,?y),?x,?y,?ys) ]
0.00/0.34	Apply Direct Methods...
0.00/0.34	Inner CPs:
0.00/0.34	   [ cons(?x_1,U0(lt(?x,?y),?x,?y,?ys)) = U0(lt(?x_1,?x),?x_1,?x,cons(?y,?ys)) ]
0.00/0.34	Outer CPs:
0.00/0.34	   [  ]
0.00/0.34	not Overlay, check Termination...
0.00/0.34	Terminating, not WCR
0.00/0.34	Knuth & Bendix
0.00/0.34	Direct Methods: not CR
0.00/0.34	{4}
0.00/0.34	(cm)Rewrite Rules:
0.00/0.34	   [ U0(true,?x,?y,?ys) -> cons(?y,cons(?x,?ys)) ]
0.00/0.34	Apply Direct Methods...
0.00/0.34	Inner CPs:
0.00/0.34	   [  ]
0.00/0.34	Outer CPs:
0.00/0.34	   [  ]
0.00/0.34	Overlay, check Innermost Termination...
0.00/0.34	Innermost Terminating (hence Terminating), WCR
0.00/0.34	Knuth & Bendix
0.00/0.34	Direct Methods: CR
0.00/0.34	Commutative Decomposition failed: Can't judge
0.00/0.34	No further decomposition possible
0.00/0.34	
0.00/0.34	
0.00/0.34	Combined result: Can't judge
0.00/0.34	failed to show confluence of U(R)
0.00/0.34	/export/starexec/sandbox/benchmark/theBenchmark.trs: Failure(unknown CR)
0.00/0.34	(180 msec.)
0.00/0.34	EOF