/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
double(0) -> 0
double(s(x:S)) -> s(s(double(x:S)))
minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
minus(x:S,0) -> x:S
plus(0,y:S) -> y:S
plus(s(x:S),y:S) -> plus(x:S,s(y:S))
plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
plus(s(x:S),y:S) -> s(plus(x:S,y:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 DOUBLE(s(x:S)) -> DOUBLE(x:S)
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 PLUS(s(x:S),y:S) -> DOUBLE(y:S)
 PLUS(s(x:S),y:S) -> MINUS(x:S,y:S)
 PLUS(s(x:S),y:S) -> PLUS(minus(x:S,y:S),double(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,s(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))

Problem 1: 

SCC Processor:
-> Pairs:
 DOUBLE(s(x:S)) -> DOUBLE(x:S)
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
 PLUS(s(x:S),y:S) -> DOUBLE(y:S)
 PLUS(s(x:S),y:S) -> MINUS(x:S,y:S)
 PLUS(s(x:S),y:S) -> PLUS(minus(x:S,y:S),double(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,s(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
->->-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->->Cycle:
->->-> Pairs:
 DOUBLE(s(x:S)) -> DOUBLE(x:S)
->->-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->->Cycle:
->->-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(minus(x:S,y:S),double(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,s(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
->->-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS(s(x:S),s(y:S)) -> MINUS(x:S,y:S)
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->Projection:
 pi(MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 DOUBLE(s(x:S)) -> DOUBLE(x:S)
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->Projection:
 pi(DOUBLE) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(minus(x:S,y:S),double(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,s(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
-> Usable rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[double](X) = 2.X + 2
[minus](X1,X2) = X1 + 1
[0] = 2
[s](X) = X + 2
[PLUS](X1,X2) = 2.X1

Problem 1.3: 

SCC Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,s(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,s(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
->->-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))

Problem 1.3: 

Subterm Processor:
-> Pairs:
 PLUS(s(x:S),y:S) -> PLUS(x:S,s(y:S))
 PLUS(s(x:S),y:S) -> PLUS(x:S,y:S)
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->Projection:
 pi(PLUS) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 minus(s(x:S),s(y:S)) -> minus(x:S,y:S)
 minus(x:S,0) -> x:S
 plus(0,y:S) -> y:S
 plus(s(x:S),y:S) -> plus(x:S,s(y:S))
 plus(s(x:S),y:S) -> s(plus(minus(x:S,y:S),double(y:S)))
 plus(s(x:S),y:S) -> s(plus(x:S,y:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.