/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 x y z)
(RULES
minus(s(x),s(y)) -> minus(x,y)
minus(x,0) -> x
plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
plus(0,y) -> y
plus(s(x),y) -> s(plus(x,y))
quot(0,s(y)) -> 0
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
 PLUS(minus(x,s(0)),minus(y,s(s(z)))) -> PLUS(minus(y,s(s(z))),minus(x,s(0)))
 PLUS(plus(x,s(0)),plus(y,s(s(z)))) -> PLUS(plus(y,s(s(z))),plus(x,s(0)))
 PLUS(s(x),y) -> PLUS(x,y)
 QUOT(s(x),s(y)) -> MINUS(x,y)
 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))
-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))

Problem 1: 

SCC Processor:
-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
 PLUS(minus(x,s(0)),minus(y,s(s(z)))) -> PLUS(minus(y,s(s(z))),minus(x,s(0)))
 PLUS(plus(x,s(0)),plus(y,s(s(z)))) -> PLUS(plus(y,s(s(z))),plus(x,s(0)))
 PLUS(s(x),y) -> PLUS(x,y)
 QUOT(s(x),s(y)) -> MINUS(x,y)
 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))
-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(minus(x,s(0)),minus(y,s(s(z)))) -> PLUS(minus(y,s(s(z))),minus(x,s(0)))
 PLUS(plus(x,s(0)),plus(y,s(s(z)))) -> PLUS(plus(y,s(s(z))),plus(x,s(0)))
 PLUS(s(x),y) -> PLUS(x,y)
->->-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
->->Cycle:
->->-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
->->-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
->->Cycle:
->->-> Pairs:
 QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))
->->-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

Narrowing Processor:
-> Pairs:
 PLUS(minus(x,s(0)),minus(y,s(s(z)))) -> PLUS(minus(y,s(s(z))),minus(x,s(0)))
 PLUS(plus(x,s(0)),plus(y,s(s(z)))) -> PLUS(plus(y,s(s(z))),plus(x,s(0)))
-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
->Narrowed Pairs:
->->Original Pair:
 PLUS(minus(x,s(0)),minus(y,s(s(z)))) -> PLUS(minus(y,s(s(z))),minus(x,s(0)))
->-> Narrowed pairs:
 PLUS(minus(s(x),s(0)),minus(x4,s(s(x5)))) -> PLUS(minus(x4,s(s(x5))),minus(x,0))
 PLUS(minus(x3,s(0)),minus(s(x),s(s(x5)))) -> PLUS(minus(x,s(x5)),minus(x3,s(0)))
->->Original Pair:
 PLUS(plus(x,s(0)),plus(y,s(s(z)))) -> PLUS(plus(y,s(s(z))),plus(x,s(0)))
->-> Narrowed pairs:
 PLUS(plus(0,s(0)),plus(x7,s(s(x8)))) -> PLUS(plus(x7,s(s(x8))),s(0))
 PLUS(plus(s(x),s(0)),plus(x7,s(s(x8)))) -> PLUS(plus(x7,s(s(x8))),s(plus(x,s(0))))
 PLUS(plus(x6,s(0)),plus(0,s(s(x8)))) -> PLUS(s(s(x8)),plus(x6,s(0)))
 PLUS(plus(x6,s(0)),plus(s(x),s(s(x8)))) -> PLUS(s(plus(x,s(s(x8)))),plus(x6,s(0)))

Problem 1.1: 

SCC Processor:
-> Pairs:
 PLUS(minus(s(x),s(0)),minus(x4,s(s(x5)))) -> PLUS(minus(x4,s(s(x5))),minus(x,0))
 PLUS(minus(x3,s(0)),minus(s(x),s(s(x5)))) -> PLUS(minus(x,s(x5)),minus(x3,s(0)))
 PLUS(plus(0,s(0)),plus(x7,s(s(x8)))) -> PLUS(plus(x7,s(s(x8))),s(0))
 PLUS(plus(s(x),s(0)),plus(x7,s(s(x8)))) -> PLUS(plus(x7,s(s(x8))),s(plus(x,s(0))))
 PLUS(plus(x6,s(0)),plus(0,s(s(x8)))) -> PLUS(s(s(x8)),plus(x6,s(0)))
 PLUS(plus(x6,s(0)),plus(s(x),s(s(x8)))) -> PLUS(s(plus(x,s(s(x8)))),plus(x6,s(0)))
-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(minus(s(x),s(0)),minus(x4,s(s(x5)))) -> PLUS(minus(x4,s(s(x5))),minus(x,0))
 PLUS(minus(x3,s(0)),minus(s(x),s(s(x5)))) -> PLUS(minus(x,s(x5)),minus(x3,s(0)))
->->-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 PLUS(minus(s(x),s(0)),minus(x4,s(s(x5)))) -> PLUS(minus(x4,s(s(x5))),minus(x,0))
 PLUS(minus(x3,s(0)),minus(s(x),s(s(x5)))) -> PLUS(minus(x,s(x5)),minus(x3,s(0)))
-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
-> Usable rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[minus](X1,X2) = 2.X1 + X2
[0] = 2
[s](X) = 2.X
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1.1: 

SCC Processor:
-> Pairs:
 PLUS(minus(x3,s(0)),minus(s(x),s(s(x5)))) -> PLUS(minus(x,s(x5)),minus(x3,s(0)))
-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(minus(x3,s(0)),minus(s(x),s(s(x5)))) -> PLUS(minus(x,s(x5)),minus(x3,s(0)))
->->-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 PLUS(minus(x3,s(0)),minus(s(x),s(s(x5)))) -> PLUS(minus(x,s(x5)),minus(x3,s(0)))
-> Rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
 plus(plus(x,s(0)),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0)))
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 quot(0,s(y)) -> 0
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
-> Usable rules:
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[minus](X1,X2) = 2.X1
[0] = 0
[s](X) = X + 2
[PLUS](X1,X2) = X1 + X2

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.