/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem 1: 

(VAR v_NonEmpty:S rest:S x:S y:S z1:S z2:S)
(RULES
cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
orient(s(x:S),0) -> pair(0,s(x:S))
orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z2:S,rest:S) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> QPairs:
 Empty
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)

Conditional Termination Problem 2:
-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> ORIENT(x:S,y:S)
 ORIENT(s(x:S),s(y:S)) -> ORIENT(x:S,y:S)
-> QPairs:
 Empty
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z2:S,rest:S) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> QPairs:
 Empty
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z2:S,rest:S) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> QPairs:
 Empty
->->-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z2:S,rest:S) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> QPairs:
 Empty
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> Usable rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons](X1,X2) = 2.X2 + 2
[orient](X1,X2) = 2.X1 + 2.X2 + 1
[0] = 2
[fSNonEmpty] = 0
[pair](X1,X2) = 1
[s](X) = 2.X + 2
[CONS](X1,X2) = 2.X2
[ORIENT](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> QPairs:
 Empty
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> QPairs:
 Empty
->->-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)

Problem 1.1: 

Ohlebusch Transformation Processor:
-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> CONS(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> QPairs:
 Empty
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
-> New Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> U14(orient(x:S,y:S),rest:S,x:S,y:S)
 U14(pair(z1:S,z2:S),rest:S,x:S,y:S) -> CONS(z1:S,cons(z2:S,rest:S))
-> New Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> U12(orient(x:S,y:S),rest:S,x:S,y:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> U13(orient(x:S,y:S),x:S,y:S)
 U12(pair(z1:S,z2:S),rest:S,x:S,y:S) -> cons(z1:S,cons(z2:S,rest:S))
 U13(pair(z1:S,z2:S),x:S,y:S) -> pair(s(z1:S),s(z2:S))

Problem 1.1: 

SCC Processor:
-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> U14(orient(x:S,y:S),rest:S,x:S,y:S)
 U14(pair(z1:S,z2:S),rest:S,x:S,y:S) -> CONS(z1:S,cons(z2:S,rest:S))
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> U12(orient(x:S,y:S),rest:S,x:S,y:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> U13(orient(x:S,y:S),x:S,y:S)
 U12(pair(z1:S,z2:S),rest:S,x:S,y:S) -> cons(z1:S,cons(z2:S,rest:S))
 U13(pair(z1:S,z2:S),x:S,y:S) -> pair(s(z1:S),s(z2:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> U14(orient(x:S,y:S),rest:S,x:S,y:S)
 U14(pair(z1:S,z2:S),rest:S,x:S,y:S) -> CONS(z1:S,cons(z2:S,rest:S))
->->-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> U12(orient(x:S,y:S),rest:S,x:S,y:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> U13(orient(x:S,y:S),x:S,y:S)
 U12(pair(z1:S,z2:S),rest:S,x:S,y:S) -> cons(z1:S,cons(z2:S,rest:S))
 U13(pair(z1:S,z2:S),x:S,y:S) -> pair(s(z1:S),s(z2:S))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> U14(orient(x:S,y:S),rest:S,x:S,y:S)
 U14(pair(z1:S,z2:S),rest:S,x:S,y:S) -> CONS(z1:S,cons(z2:S,rest:S))
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> U12(orient(x:S,y:S),rest:S,x:S,y:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> U13(orient(x:S,y:S),x:S,y:S)
 U12(pair(z1:S,z2:S),rest:S,x:S,y:S) -> cons(z1:S,cons(z2:S,rest:S))
 U13(pair(z1:S,z2:S),x:S,y:S) -> pair(s(z1:S),s(z2:S))
-> Usable rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> U12(orient(x:S,y:S),rest:S,x:S,y:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> U13(orient(x:S,y:S),x:S,y:S)
 U12(pair(z1:S,z2:S),rest:S,x:S,y:S) -> cons(z1:S,cons(z2:S,rest:S))
 U13(pair(z1:S,z2:S),x:S,y:S) -> pair(s(z1:S),s(z2:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons](X1,X2) = 2.X1 + 2
[orient](X1,X2) = 2.X1 + 1
[0] = 0
[pair](X1,X2) = 2.X1 + 2
[s](X) = 2.X + 2
[CONS](X1,X2) = 2.X1 + 2
[U12](X1,X2,X3,X4) = X1
[U13](X1,X2,X3) = 2.X1 + 2
[U14](X1,X2,X3,X4) = X1

Problem 1.1: 

SCC Processor:
-> Pairs:
 U14(pair(z1:S,z2:S),rest:S,x:S,y:S) -> CONS(z1:S,cons(z2:S,rest:S))
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> U12(orient(x:S,y:S),rest:S,x:S,y:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> U13(orient(x:S,y:S),x:S,y:S)
 U12(pair(z1:S,z2:S),rest:S,x:S,y:S) -> cons(z1:S,cons(z2:S,rest:S))
 U13(pair(z1:S,z2:S),x:S,y:S) -> pair(s(z1:S),s(z2:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 CONS(x:S,cons(y:S,rest:S)) -> ORIENT(x:S,y:S)
 ORIENT(s(x:S),s(y:S)) -> ORIENT(x:S,y:S)
-> QPairs:
 Empty
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ORIENT(s(x:S),s(y:S)) -> ORIENT(x:S,y:S)
-> QPairs:
 Empty
->->-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)

Problem 1.2: 

Conditional Subterm Processor:
-> Pairs:
 ORIENT(s(x:S),s(y:S)) -> ORIENT(x:S,y:S)
-> QPairs:
 Empty
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
->Projection:
 pi(ORIENT) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 cons(x:S,cons(x:S,rest:S)) -> cons(x:S,rest:S)
 cons(x:S,cons(y:S,rest:S)) -> cons(z1:S,cons(z2:S,rest:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
 orient(s(x:S),0) -> pair(0,s(x:S))
 orient(s(x:S),s(y:S)) -> pair(s(z1:S),s(z2:S)) | orient(x:S,y:S) ->* pair(z1:S,z2:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.