/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 rest x y z1 z2)
(RULES
cons(x,cons(x,rest)) -> cons(x,rest)
cons(x,cons(y,rest)) -> cons(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
orient(s(x),0) -> pair(0,s(x))
orient(s(x),s(y)) -> pair(s(z1),s(z2)) | orient(x,y) -> pair(z1,z2)
)

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

Problem 1: 

Dependency Pairs Processor:

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

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 CONS(x,cons(y,rest)) -> CONS(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
 CONS(x,cons(y,rest)) -> CONS(z2,rest) | orient(x,y) -> pair(z1,z2)
-> QPairs:
 Empty
-> Rules:
 cons(x,cons(x,rest)) -> cons(x,rest)
 cons(x,cons(y,rest)) -> cons(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
 orient(s(x),0) -> pair(0,s(x))
 orient(s(x),s(y)) -> pair(s(z1),s(z2)) | orient(x,y) -> pair(z1,z2)
-> Usable rules:
 cons(x,cons(x,rest)) -> cons(x,rest)
 cons(x,cons(y,rest)) -> cons(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
 orient(s(x),0) -> pair(0,s(x))
 orient(s(x),s(y)) -> pair(s(z1),s(z2)) | orient(x,y) -> pair(z1,z2)
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (x_1 >= 0))
->Interpretation:
 
[delta] = 1
[cons](x_1,x_2) = 1+x_2
[orient](x_1,x_2) = 0
[0] = 0
[pair](x_1,x_2) = x_1
[s](x_1) = 0
[CONS](x_1,x_2) = x_2

Problem 1.1: 

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

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 CONS(x,cons(y,rest)) -> CONS(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
-> QPairs:
 Empty
-> Rules:
 cons(x,cons(x,rest)) -> cons(x,rest)
 cons(x,cons(y,rest)) -> cons(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
 orient(s(x),0) -> pair(0,s(x))
 orient(s(x),s(y)) -> pair(s(z1),s(z2)) | orient(x,y) -> pair(z1,z2)
-> Usable rules:
 cons(x,cons(x,rest)) -> cons(x,rest)
 cons(x,cons(y,rest)) -> cons(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
 orient(s(x),0) -> pair(0,s(x))
 orient(s(x),s(y)) -> pair(s(z1),s(z2)) | orient(x,y) -> pair(z1,z2)
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((0 >= 0) /\ (1+x_1 >= 0))
->Interpretation:
 
[delta] = 1
[cons](x_1,x_2) = 0
[orient](x_1,x_2) = x_1
[0] = 0
[pair](x_1,x_2) = 1+x_1
[s](x_1) = 2+x_1
[CONS](x_1,x_2) = 1+x_1+x_2

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

Reduction Triple Processor:
-> Pairs:
 ORIENT(s(x),s(y)) -> ORIENT(x,y)
-> QPairs:
 CONS(x,cons(y,rest)) -> CONS(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
 CONS(x,cons(y,rest)) -> CONS(z2,rest) | orient(x,y) -> pair(z1,z2)
-> Rules:
 cons(x,cons(x,rest)) -> cons(x,rest)
 cons(x,cons(y,rest)) -> cons(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
 orient(s(x),0) -> pair(0,s(x))
 orient(s(x),s(y)) -> pair(s(z1),s(z2)) | orient(x,y) -> pair(z1,z2)
-> Usable rules:
 cons(x,cons(x,rest)) -> cons(x,rest)
 cons(x,cons(y,rest)) -> cons(z1,cons(z2,rest)) | orient(x,y) -> pair(z1,z2)
 orient(s(x),0) -> pair(0,s(x))
 orient(s(x),s(y)) -> pair(s(z1),s(z2)) | orient(x,y) -> pair(z1,z2)
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((0 >= 0) /\ (-1+x_1 >= 0))
->Interpretation:
 
[delta] = 1
[cons](x_1,x_2) = x_1+x_2
[orient](x_1,x_2) = 1+x_1+x_2
[0] = 1
[pair](x_1,x_2) = 1+x_1+x_2
[s](x_1) = x_1
[CONS](x_1,x_2) = x_1+x_2
[ORIENT](x_1,x_2) = 1

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.