/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
ack(0,y:S) -> s(y:S)
ack(s(x:S),0) -> ack(x:S,s(0))
ack(s(x:S),s(y:S)) -> ack(x:S,plus(y:S,ack(s(x:S),y:S)))
plus(0,y:S) -> y:S
plus(s(0),y:S) -> s(y:S)
plus(s(s(x:S)),y:S) -> s(plus(x:S,s(y:S)))
plus(x:S,s(s(y:S))) -> s(plus(s(x:S),y:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACK(s(x:S),0) -> ACK(x:S,s(0))
 ACK(s(x:S),s(y:S)) -> ACK(s(x:S),y:S)
 ACK(s(x:S),s(y:S)) -> ACK(x:S,plus(y:S,ack(s(x:S),y:S)))
 ACK(s(x:S),s(y:S)) -> PLUS(y:S,ack(s(x:S),y:S))
 PLUS(s(s(x:S)),y:S) -> PLUS(x:S,s(y:S))
 PLUS(x:S,s(s(y:S))) -> PLUS(s(x:S),y:S)
-> Rules:
 ack(0,y:S) -> s(y:S)
 ack(s(x:S),0) -> ack(x:S,s(0))
 ack(s(x:S),s(y:S)) -> ack(x:S,plus(y:S,ack(s(x:S),y:S)))
 plus(0,y:S) -> y:S
 plus(s(0),y:S) -> s(y:S)
 plus(s(s(x:S)),y:S) -> s(plus(x:S,s(y:S)))
 plus(x:S,s(s(y:S))) -> s(plus(s(x:S),y:S))

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(x:S,s(s(y:S))) -> PLUS(s(x:S),y:S)
-> Rules:
 ack(0,y:S) -> s(y:S)
 ack(s(x:S),0) -> ack(x:S,s(0))
 ack(s(x:S),s(y:S)) -> ack(x:S,plus(y:S,ack(s(x:S),y:S)))
 plus(0,y:S) -> y:S
 plus(s(0),y:S) -> s(y:S)
 plus(s(s(x:S)),y:S) -> s(plus(x:S,s(y:S)))
 plus(x:S,s(s(y:S))) -> s(plus(s(x:S),y:S))
->Projection:
 pi(PLUS) = 2

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

Subterm Processor:
-> Pairs:
 ACK(s(x:S),s(y:S)) -> ACK(s(x:S),y:S)
-> Rules:
 ack(0,y:S) -> s(y:S)
 ack(s(x:S),0) -> ack(x:S,s(0))
 ack(s(x:S),s(y:S)) -> ack(x:S,plus(y:S,ack(s(x:S),y:S)))
 plus(0,y:S) -> y:S
 plus(s(0),y:S) -> s(y:S)
 plus(s(s(x:S)),y:S) -> s(plus(x:S,s(y:S)))
 plus(x:S,s(s(y:S))) -> s(plus(s(x:S),y:S))
->Projection:
 pi(ACK) = 2

Problem 1.2: 

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

The problem is finite.