/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 X)
(STRATEGY CONTEXTSENSITIVE
(U11 1)
(U12 1)
(U13 1)
(U21 1)
(U22 1)
(U31 1)
(U41 1)
(and 1)
(isNat)
(isNatKind)
(plus 1 2)
(0)
(s 1)
(tt)
)
(RULES
U11(tt,V1,V2) -> U12(isNat(V1),V2)
U12(tt,V2) -> U13(isNat(V2))
U13(tt) -> tt
U21(tt,V1) -> U22(isNat(V1))
U22(tt) -> tt
U31(tt,N) -> N
U41(tt,M,N) -> s(plus(N,M))
and(tt,X) -> X
isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
isNat(0) -> tt
isNat(s(V1)) -> U21(isNatKind(V1),V1)
isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
isNatKind(0) -> tt
isNatKind(s(V1)) -> isNatKind(V1)
plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
 U11#(tt,V1,V2) -> ISNAT(V1)
 U12#(tt,V2) -> U13#(isNat(V2))
 U12#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> U22#(isNat(V1))
 U21#(tt,V1) -> ISNAT(V1)
 U31#(tt,N) -> N
 U41#(tt,M,N) -> PLUS(N,M)
 U41#(tt,M,N) -> M
 U41#(tt,M,N) -> N
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
 PLUS(N,0) -> U31#(and(isNat(N),isNatKind(N)),N)
 PLUS(N,0) -> AND(isNat(N),isNatKind(N))
 PLUS(N,0) -> ISNAT(N)
 PLUS(N,s(M)) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
 PLUS(N,s(M)) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
 PLUS(N,s(M)) -> AND(isNat(M),isNatKind(M))
 PLUS(N,s(M)) -> ISNAT(M)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding Rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
 U11#(tt,V1,V2) -> ISNAT(V1)
 U12#(tt,V2) -> U13#(isNat(V2))
 U12#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> U22#(isNat(V1))
 U21#(tt,V1) -> ISNAT(V1)
 U31#(tt,N) -> N
 U41#(tt,M,N) -> PLUS(N,M)
 U41#(tt,M,N) -> M
 U41#(tt,M,N) -> N
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
 PLUS(N,0) -> U31#(and(isNat(N),isNatKind(N)),N)
 PLUS(N,0) -> AND(isNat(N),isNatKind(N))
 PLUS(N,0) -> ISNAT(N)
 PLUS(N,s(M)) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
 PLUS(N,s(M)) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
 PLUS(N,s(M)) -> AND(isNat(M),isNatKind(M))
 PLUS(N,s(M)) -> ISNAT(M)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
 U11#(tt,V1,V2) -> ISNAT(V1)
 U12#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> ISNAT(V1)
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
->->-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->->-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
->->Cycle:
->->-> Pairs:
 U41#(tt,M,N) -> PLUS(N,M)
 PLUS(N,s(M)) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->->-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->->-> Unhiding rules:
 Empty


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
 U11#(tt,V1,V2) -> ISNAT(V1)
 U12#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> ISNAT(V1)
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
-> Usable rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2
[U12](X1,X2) = X1
[U13](X) = 2
[U21](X1,X2) = 2
[U22](X) = 2
[U31](X1,X2) = 2.X2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[and](X1,X2) = X2 + 2
[isNat](X) = 2
[isNatKind](X) = 2.X + 2
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[0] = 2
[s](X) = X + 1
[tt] = 2
[U11#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U12#](X1,X2) = 2.X2 + 1
[U21#](X1,X2) = 2.X2
[AND](X1,X2) = X2
[ISNAT](X) = 2.X
[ISNATKIND](X) = 2.X + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 U11#(tt,V1,V2) -> ISNAT(V1)
 U12#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> ISNAT(V1)
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(tt,V1,V2) -> ISNAT(V1)
 U21#(tt,V1) -> ISNAT(V1)
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
->->-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->->-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 U11#(tt,V1,V2) -> ISNAT(V1)
 U21#(tt,V1) -> ISNAT(V1)
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
-> Usable rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1 + 2.X2 + 2
[U12](X1,X2) = X1 + 2
[U13](X) = 2
[U21](X1,X2) = 2.X2 + 2
[U22](X) = X
[U31](X1,X2) = X2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[and](X1,X2) = X1 + 2.X2 + 1
[isNat](X) = 2.X + 2
[isNatKind](X) = 2.X + 1
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 2
[s](X) = X
[tt] = 2
[U11#](X1,X2,X3) = X1 + 2.X2
[U21#](X1,X2) = 2.X2 + 1
[AND](X1,X2) = 2.X1 + X2
[ISNAT](X) = 2.X + 1
[ISNATKIND](X) = 2.X + 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 U21#(tt,V1) -> ISNAT(V1)
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U21#(tt,V1) -> ISNAT(V1)
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
->->-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->->-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 U21#(tt,V1) -> ISNAT(V1)
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
-> Usable rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U12](X1,X2) = 2
[U13](X) = 2
[U21](X1,X2) = X1 + 2.X2 + 2
[U22](X) = X + 1
[U31](X1,X2) = X2 + 1
[U41](X1,X2,X3) = 2.X2 + X3 + 2
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X + 1
[isNatKind](X) = 0
[plus](X1,X2) = X1 + 2.X2 + 1
[0] = 0
[s](X) = X + 1
[tt] = 0
[U21#](X1,X2) = 2.X2 + 2
[AND](X1,X2) = X2
[ISNAT](X) = 2.X + 1
[ISNATKIND](X) = 0

Problem 1.1: 

SCC Processor:
-> Pairs:
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
->->-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->->-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 AND(tt,X) -> X
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
-> Usable rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1 + 2.X2
[U12](X1,X2) = X1
[U13](X) = 2
[U21](X1,X2) = 2
[U22](X) = 2
[U31](X1,X2) = 2.X2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[and](X1,X2) = X2 + 1
[isNat](X) = X + 2
[isNatKind](X) = 2.X + 2
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[0] = 1
[s](X) = X + 1
[tt] = 2
[AND](X1,X2) = X2 + 1
[ISNAT](X) = 2.X + 1
[ISNATKIND](X) = 2.X + 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
 and(isNat(N),isNatKind(N)) -> ISNAT(N)
 isNatKind(M) -> ISNATKIND(M)
 isNatKind(N) -> ISNATKIND(N)
 isNatKind(V2) -> ISNATKIND(V2)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
->->-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
->->-> Unhiding rules:
 Empty

Problem 1.1: 

SubNColl Processor:
-> Pairs:
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 Empty
->Projection:
 pi(ISNATKIND) = 1

Problem 1.1: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 Empty
-> Result:
 Set P is empty

The problem is finite.

Problem 1.2: 

SubNColl Processor:
-> Pairs:
 U41#(tt,M,N) -> PLUS(N,M)
 PLUS(N,s(M)) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 Empty
->Projection:
 pi(U41#) = 2
 pi(PLUS) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 U41#(tt,M,N) -> PLUS(N,M)
-> Rules:
 U11(tt,V1,V2) -> U12(isNat(V1),V2)
 U12(tt,V2) -> U13(isNat(V2))
 U13(tt) -> tt
 U21(tt,V1) -> U22(isNat(V1))
 U22(tt) -> tt
 U31(tt,N) -> N
 U41(tt,M,N) -> s(plus(N,M))
 and(tt,X) -> X
 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> isNatKind(V1)
 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.