/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 M N V1 V2)
(STRATEGY CONTEXTSENSITIVE
(U11 1)
(U12 1)
(U21 1)
(U31 1)
(U32 1)
(U41 1)
(U51 1)
(U52 1)
(U61 1)
(U71 1)
(U72 1)
(isNat)
(plus 1 2)
(x 1 2)
(0)
(s 1)
(tt)
)
(RULES
U11(tt,V2) -> U12(isNat(V2))
U12(tt) -> tt
U21(tt) -> tt
U31(tt,V2) -> U32(isNat(V2))
U32(tt) -> tt
U41(tt,N) -> N
U51(tt,M,N) -> U52(isNat(N),M,N)
U52(tt,M,N) -> s(plus(N,M))
U61(tt) -> 0
U71(tt,M,N) -> U72(isNat(N),M,N)
U72(tt,M,N) -> plus(x(N,M),N)
isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
isNat(x(V1,V2)) -> U31(isNat(V1),V2)
isNat(0) -> tt
isNat(s(V1)) -> U21(isNat(V1))
plus(N,0) -> U41(isNat(N),N)
plus(N,s(M)) -> U51(isNat(M),M,N)
x(N,0) -> U61(isNat(N))
x(N,s(M)) -> U71(isNat(M),M,N)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U11#(tt,V2) -> U12#(isNat(V2))
 U11#(tt,V2) -> ISNAT(V2)
 U31#(tt,V2) -> U32#(isNat(V2))
 U31#(tt,V2) -> ISNAT(V2)
 U41#(tt,N) -> N
 U51#(tt,M,N) -> U52#(isNat(N),M,N)
 U51#(tt,M,N) -> ISNAT(N)
 U52#(tt,M,N) -> PLUS(N,M)
 U52#(tt,M,N) -> M
 U52#(tt,M,N) -> N
 U71#(tt,M,N) -> U72#(isNat(N),M,N)
 U71#(tt,M,N) -> ISNAT(N)
 U72#(tt,M,N) -> PLUS(x(N,M),N)
 U72#(tt,M,N) -> X(N,M)
 U72#(tt,M,N) -> M
 U72#(tt,M,N) -> N
 ISNAT(plus(V1,V2)) -> U11#(isNat(V1),V2)
 ISNAT(plus(V1,V2)) -> ISNAT(V1)
 ISNAT(x(V1,V2)) -> U31#(isNat(V1),V2)
 ISNAT(x(V1,V2)) -> ISNAT(V1)
 ISNAT(s(V1)) -> U21#(isNat(V1))
 ISNAT(s(V1)) -> ISNAT(V1)
 PLUS(N,0) -> U41#(isNat(N),N)
 PLUS(N,0) -> ISNAT(N)
 PLUS(N,s(M)) -> U51#(isNat(M),M,N)
 PLUS(N,s(M)) -> ISNAT(M)
 X(N,0) -> U61#(isNat(N))
 X(N,0) -> ISNAT(N)
 X(N,s(M)) -> U71#(isNat(M),M,N)
 X(N,s(M)) -> ISNAT(M)
-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
-> Unhiding Rules:
 Empty

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(tt,V2) -> U12#(isNat(V2))
 U11#(tt,V2) -> ISNAT(V2)
 U31#(tt,V2) -> U32#(isNat(V2))
 U31#(tt,V2) -> ISNAT(V2)
 U41#(tt,N) -> N
 U51#(tt,M,N) -> U52#(isNat(N),M,N)
 U51#(tt,M,N) -> ISNAT(N)
 U52#(tt,M,N) -> PLUS(N,M)
 U52#(tt,M,N) -> M
 U52#(tt,M,N) -> N
 U71#(tt,M,N) -> U72#(isNat(N),M,N)
 U71#(tt,M,N) -> ISNAT(N)
 U72#(tt,M,N) -> PLUS(x(N,M),N)
 U72#(tt,M,N) -> X(N,M)
 U72#(tt,M,N) -> M
 U72#(tt,M,N) -> N
 ISNAT(plus(V1,V2)) -> U11#(isNat(V1),V2)
 ISNAT(plus(V1,V2)) -> ISNAT(V1)
 ISNAT(x(V1,V2)) -> U31#(isNat(V1),V2)
 ISNAT(x(V1,V2)) -> ISNAT(V1)
 ISNAT(s(V1)) -> U21#(isNat(V1))
 ISNAT(s(V1)) -> ISNAT(V1)
 PLUS(N,0) -> U41#(isNat(N),N)
 PLUS(N,0) -> ISNAT(N)
 PLUS(N,s(M)) -> U51#(isNat(M),M,N)
 PLUS(N,s(M)) -> ISNAT(M)
 X(N,0) -> U61#(isNat(N))
 X(N,0) -> ISNAT(N)
 X(N,s(M)) -> U71#(isNat(M),M,N)
 X(N,s(M)) -> ISNAT(M)
-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(tt,V2) -> ISNAT(V2)
 U31#(tt,V2) -> ISNAT(V2)
 ISNAT(plus(V1,V2)) -> U11#(isNat(V1),V2)
 ISNAT(plus(V1,V2)) -> ISNAT(V1)
 ISNAT(x(V1,V2)) -> U31#(isNat(V1),V2)
 ISNAT(x(V1,V2)) -> ISNAT(V1)
 ISNAT(s(V1)) -> ISNAT(V1)
->->-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
->->-> Unhiding rules:
 Empty
->->Cycle:
->->-> Pairs:
 U51#(tt,M,N) -> U52#(isNat(N),M,N)
 U52#(tt,M,N) -> PLUS(N,M)
 PLUS(N,s(M)) -> U51#(isNat(M),M,N)
->->-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
->->-> Unhiding rules:
 Empty
->->Cycle:
->->-> Pairs:
 U71#(tt,M,N) -> U72#(isNat(N),M,N)
 U72#(tt,M,N) -> X(N,M)
 X(N,s(M)) -> U71#(isNat(M),M,N)
->->-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
->->-> Unhiding rules:
 Empty


The problem is decomposed in 3 subproblems.

Problem 1.1: 

SubNColl Processor:
-> Pairs:
 U11#(tt,V2) -> ISNAT(V2)
 U31#(tt,V2) -> ISNAT(V2)
 ISNAT(plus(V1,V2)) -> U11#(isNat(V1),V2)
 ISNAT(plus(V1,V2)) -> ISNAT(V1)
 ISNAT(x(V1,V2)) -> U31#(isNat(V1),V2)
 ISNAT(x(V1,V2)) -> ISNAT(V1)
 ISNAT(s(V1)) -> ISNAT(V1)
-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Projection:
 pi(U11#) = 2
 pi(U31#) = 2
 pi(ISNAT) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 U11#(tt,V2) -> ISNAT(V2)
 U31#(tt,V2) -> ISNAT(V2)
-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SubNColl Processor:
-> Pairs:
 U51#(tt,M,N) -> U52#(isNat(N),M,N)
 U52#(tt,M,N) -> PLUS(N,M)
 PLUS(N,s(M)) -> U51#(isNat(M),M,N)
-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Projection:
 pi(U51#) = 2
 pi(U52#) = 2
 pi(PLUS) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 U51#(tt,M,N) -> U52#(isNat(N),M,N)
 U52#(tt,M,N) -> PLUS(N,M)
-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

SubNColl Processor:
-> Pairs:
 U71#(tt,M,N) -> U72#(isNat(N),M,N)
 U72#(tt,M,N) -> X(N,M)
 X(N,s(M)) -> U71#(isNat(M),M,N)
-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Projection:
 pi(U71#) = 2
 pi(U72#) = 2
 pi(X) = 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 U71#(tt,M,N) -> U72#(isNat(N),M,N)
 U72#(tt,M,N) -> X(N,M)
-> Rules:
 U11(tt,V2) -> U12(isNat(V2))
 U12(tt) -> tt
 U21(tt) -> tt
 U31(tt,V2) -> U32(isNat(V2))
 U32(tt) -> tt
 U41(tt,N) -> N
 U51(tt,M,N) -> U52(isNat(N),M,N)
 U52(tt,M,N) -> s(plus(N,M))
 U61(tt) -> 0
 U71(tt,M,N) -> U72(isNat(N),M,N)
 U72(tt,M,N) -> plus(x(N,M),N)
 isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
 isNat(x(V1,V2)) -> U31(isNat(V1),V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNat(V1))
 plus(N,0) -> U41(isNat(N),N)
 plus(N,s(M)) -> U51(isNat(M),M,N)
 x(N,0) -> U61(isNat(N))
 x(N,s(M)) -> U71(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.