/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 N:S X:S X1:S X2:S Y:S Z:S)
(RULES
a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
a(ns(X:S)) -> s(a(X:S))
a(nt(X:S)) -> t(a(X:S))
a(X:S) -> X:S
d(s(X:S)) -> s(s(d(X:S)))
d(0) -> 0
f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
f(0,X:S) -> nil
f(X1:S,X2:S) -> nf(X1:S,X2:S)
p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
p(0,X:S) -> X:S
p(X:S,0) -> X:S
q(s(X:S)) -> s(p(q(X:S),d(X:S)))
q(0) -> 0
s(X:S) -> ns(X:S)
t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
t(X:S) -> nt(X:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(nf(X1:S,X2:S)) -> A(X1:S)
 A(nf(X1:S,X2:S)) -> A(X2:S)
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(ns(X:S)) -> S(a(X:S))
 A(nt(X:S)) -> A(X:S)
 A(nt(X:S)) -> T(a(X:S))
 D(s(X:S)) -> D(X:S)
 D(s(X:S)) -> S(d(X:S))
 D(s(X:S)) -> S(s(d(X:S)))
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
 P(s(X:S),s(Y:S)) -> P(X:S,Y:S)
 P(s(X:S),s(Y:S)) -> S(p(X:S,Y:S))
 P(s(X:S),s(Y:S)) -> S(s(p(X:S,Y:S)))
 Q(s(X:S)) -> D(X:S)
 Q(s(X:S)) -> P(q(X:S),d(X:S))
 Q(s(X:S)) -> Q(X:S)
 Q(s(X:S)) -> S(p(q(X:S),d(X:S)))
 T(N:S) -> Q(N:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 A(nf(X1:S,X2:S)) -> A(X1:S)
 A(nf(X1:S,X2:S)) -> A(X2:S)
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(ns(X:S)) -> S(a(X:S))
 A(nt(X:S)) -> A(X:S)
 A(nt(X:S)) -> T(a(X:S))
 D(s(X:S)) -> D(X:S)
 D(s(X:S)) -> S(d(X:S))
 D(s(X:S)) -> S(s(d(X:S)))
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
 P(s(X:S),s(Y:S)) -> P(X:S,Y:S)
 P(s(X:S),s(Y:S)) -> S(p(X:S,Y:S))
 P(s(X:S),s(Y:S)) -> S(s(p(X:S,Y:S)))
 Q(s(X:S)) -> D(X:S)
 Q(s(X:S)) -> P(q(X:S),d(X:S))
 Q(s(X:S)) -> Q(X:S)
 Q(s(X:S)) -> S(p(q(X:S),d(X:S)))
 T(N:S) -> Q(N:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 P(s(X:S),s(Y:S)) -> P(X:S,Y:S)
->->-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->->Cycle:
->->-> Pairs:
 D(s(X:S)) -> D(X:S)
->->-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->->Cycle:
->->-> Pairs:
 Q(s(X:S)) -> Q(X:S)
->->-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->->Cycle:
->->-> Pairs:
 A(nf(X1:S,X2:S)) -> A(X1:S)
 A(nf(X1:S,X2:S)) -> A(X2:S)
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
->->-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 P(s(X:S),s(Y:S)) -> P(X:S,Y:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Projection:
 pi(P) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 D(s(X:S)) -> D(X:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Projection:
 pi(D) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 Q(s(X:S)) -> Q(X:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Projection:
 pi(Q) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 A(nf(X1:S,X2:S)) -> A(X1:S)
 A(nf(X1:S,X2:S)) -> A(X2:S)
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
-> Usable rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[d](X) = 0
[f](X1,X2) = 2.X1 + 2.X2 + 2
[p](X1,X2) = 2.X1 + X2
[q](X) = 0
[s](X) = X
[t](X) = 2.X
[0] = 0
[cs](X1,X2) = X2
[nf](X1,X2) = 2.X1 + 2.X2 + 2
[nil] = 2
[ns](X) = X
[nt](X) = 2.X
[r](X) = 2.X + 2
[A](X) = X
[F](X1,X2) = X1 + X2

Problem 1.4: 

SCC Processor:
-> Pairs:
 A(nf(X1:S,X2:S)) -> A(X2:S)
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(nf(X1:S,X2:S)) -> A(X2:S)
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
->->-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 A(nf(X1:S,X2:S)) -> A(X2:S)
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
-> Usable rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[d](X) = 0
[f](X1,X2) = 2.X2 + 2
[p](X1,X2) = 2.X1 + 2.X2
[q](X) = 0
[s](X) = X
[t](X) = 2.X + 2
[0] = 0
[cs](X1,X2) = 2.X1 + X2
[nf](X1,X2) = 2.X2 + 2
[nil] = 2
[ns](X) = X
[nt](X) = 2.X + 2
[r](X) = 0
[A](X) = 2.X + 1
[F](X1,X2) = 2.X2 + 2

Problem 1.4: 

SCC Processor:
-> Pairs:
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
->->-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 A(nf(X1:S,X2:S)) -> F(a(X1:S),a(X2:S))
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
-> Usable rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X) = X
[d](X) = 0
[f](X1,X2) = 2.X1 + 2.X2 + 2
[p](X1,X2) = X1 + 2.X2
[q](X) = X + 2
[s](X) = X
[t](X) = 2.X + 2
[0] = 0
[cs](X1,X2) = X2
[nf](X1,X2) = 2.X1 + 2.X2 + 2
[nil] = 1
[ns](X) = X
[nt](X) = 2.X + 2
[r](X) = 2
[A](X) = 2.X + 2
[F](X1,X2) = 2.X2 + 2

Problem 1.4: 

SCC Processor:
-> Pairs:
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
 F(s(X:S),cs(Y:S,Z:S)) -> A(Z:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
->->-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)

Problem 1.4: 

Subterm Processor:
-> Pairs:
 A(ns(X:S)) -> A(X:S)
 A(nt(X:S)) -> A(X:S)
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Projection:
 pi(A) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(nf(X1:S,X2:S)) -> f(a(X1:S),a(X2:S))
 a(ns(X:S)) -> s(a(X:S))
 a(nt(X:S)) -> t(a(X:S))
 a(X:S) -> X:S
 d(s(X:S)) -> s(s(d(X:S)))
 d(0) -> 0
 f(s(X:S),cs(Y:S,Z:S)) -> cs(Y:S,nf(X:S,a(Z:S)))
 f(0,X:S) -> nil
 f(X1:S,X2:S) -> nf(X1:S,X2:S)
 p(s(X:S),s(Y:S)) -> s(s(p(X:S,Y:S)))
 p(0,X:S) -> X:S
 p(X:S,0) -> X:S
 q(s(X:S)) -> s(p(q(X:S),d(X:S)))
 q(0) -> 0
 s(X:S) -> ns(X:S)
 t(N:S) -> cs(r(q(N:S)),nt(ns(N:S)))
 t(X:S) -> nt(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.