/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 M N X X1 X2 X3)
(RULES
U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
active(U11(tt,M,N)) -> mark(U12(tt,M,N))
active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
active(U12(tt,M,N)) -> mark(s(plus(N,M)))
active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
active(plus(N,s(M))) -> mark(U11(tt,M,N))
active(plus(N,0)) -> mark(N)
active(plus(X1,X2)) -> plus(active(X1),X2)
active(plus(X1,X2)) -> plus(X1,active(X2))
active(s(X)) -> s(active(X))
plus(mark(X1),X2) -> mark(plus(X1,X2))
plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
plus(X1,mark(X2)) -> mark(plus(X1,X2))
proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(tt) -> ok(tt)
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
 U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3)
 ACTIVE(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
 ACTIVE(U11(X1,X2,X3)) -> ACTIVE(X1)
 ACTIVE(U12(tt,M,N)) -> PLUS(N,M)
 ACTIVE(U12(tt,M,N)) -> S(plus(N,M))
 ACTIVE(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3)
 ACTIVE(U12(X1,X2,X3)) -> ACTIVE(X1)
 ACTIVE(plus(X1,X2)) -> ACTIVE(X1)
 ACTIVE(plus(X1,X2)) -> ACTIVE(X2)
 ACTIVE(plus(X1,X2)) -> PLUS(active(X1),X2)
 ACTIVE(plus(X1,X2)) -> PLUS(X1,active(X2))
 ACTIVE(s(X)) -> ACTIVE(X)
 ACTIVE(s(X)) -> S(active(X))
 PLUS(mark(X1),X2) -> PLUS(X1,X2)
 PLUS(ok(X1),ok(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
 PROPER(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
 PROPER(U11(X1,X2,X3)) -> PROPER(X1)
 PROPER(U11(X1,X2,X3)) -> PROPER(X2)
 PROPER(U11(X1,X2,X3)) -> PROPER(X3)
 PROPER(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3))
 PROPER(U12(X1,X2,X3)) -> PROPER(X1)
 PROPER(U12(X1,X2,X3)) -> PROPER(X2)
 PROPER(U12(X1,X2,X3)) -> PROPER(X3)
 PROPER(plus(X1,X2)) -> PLUS(proper(X1),proper(X2))
 PROPER(plus(X1,X2)) -> PROPER(X1)
 PROPER(plus(X1,X2)) -> PROPER(X2)
 PROPER(s(X)) -> PROPER(X)
 PROPER(s(X)) -> S(proper(X))
 S(mark(X)) -> S(X)
 S(ok(X)) -> S(X)
 TOP(mark(X)) -> PROPER(X)
 TOP(mark(X)) -> TOP(proper(X))
 TOP(ok(X)) -> ACTIVE(X)
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
 U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3)
 ACTIVE(U11(X1,X2,X3)) -> U11#(active(X1),X2,X3)
 ACTIVE(U11(X1,X2,X3)) -> ACTIVE(X1)
 ACTIVE(U12(tt,M,N)) -> PLUS(N,M)
 ACTIVE(U12(tt,M,N)) -> S(plus(N,M))
 ACTIVE(U12(X1,X2,X3)) -> U12#(active(X1),X2,X3)
 ACTIVE(U12(X1,X2,X3)) -> ACTIVE(X1)
 ACTIVE(plus(X1,X2)) -> ACTIVE(X1)
 ACTIVE(plus(X1,X2)) -> ACTIVE(X2)
 ACTIVE(plus(X1,X2)) -> PLUS(active(X1),X2)
 ACTIVE(plus(X1,X2)) -> PLUS(X1,active(X2))
 ACTIVE(s(X)) -> ACTIVE(X)
 ACTIVE(s(X)) -> S(active(X))
 PLUS(mark(X1),X2) -> PLUS(X1,X2)
 PLUS(ok(X1),ok(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
 PROPER(U11(X1,X2,X3)) -> U11#(proper(X1),proper(X2),proper(X3))
 PROPER(U11(X1,X2,X3)) -> PROPER(X1)
 PROPER(U11(X1,X2,X3)) -> PROPER(X2)
 PROPER(U11(X1,X2,X3)) -> PROPER(X3)
 PROPER(U12(X1,X2,X3)) -> U12#(proper(X1),proper(X2),proper(X3))
 PROPER(U12(X1,X2,X3)) -> PROPER(X1)
 PROPER(U12(X1,X2,X3)) -> PROPER(X2)
 PROPER(U12(X1,X2,X3)) -> PROPER(X3)
 PROPER(plus(X1,X2)) -> PLUS(proper(X1),proper(X2))
 PROPER(plus(X1,X2)) -> PROPER(X1)
 PROPER(plus(X1,X2)) -> PROPER(X2)
 PROPER(s(X)) -> PROPER(X)
 PROPER(s(X)) -> S(proper(X))
 S(mark(X)) -> S(X)
 S(ok(X)) -> S(X)
 TOP(mark(X)) -> PROPER(X)
 TOP(mark(X)) -> TOP(proper(X))
 TOP(ok(X)) -> ACTIVE(X)
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 S(mark(X)) -> S(X)
 S(ok(X)) -> S(X)
->->-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 PLUS(mark(X1),X2) -> PLUS(X1,X2)
 PLUS(ok(X1),ok(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
->->-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3)
->->-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
->->-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(X1,X2,X3)) -> ACTIVE(X1)
 ACTIVE(U12(X1,X2,X3)) -> ACTIVE(X1)
 ACTIVE(plus(X1,X2)) -> ACTIVE(X1)
 ACTIVE(plus(X1,X2)) -> ACTIVE(X2)
 ACTIVE(s(X)) -> ACTIVE(X)
->->-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 PROPER(U11(X1,X2,X3)) -> PROPER(X1)
 PROPER(U11(X1,X2,X3)) -> PROPER(X2)
 PROPER(U11(X1,X2,X3)) -> PROPER(X3)
 PROPER(U12(X1,X2,X3)) -> PROPER(X1)
 PROPER(U12(X1,X2,X3)) -> PROPER(X2)
 PROPER(U12(X1,X2,X3)) -> PROPER(X3)
 PROPER(plus(X1,X2)) -> PROPER(X1)
 PROPER(plus(X1,X2)) -> PROPER(X2)
 PROPER(s(X)) -> PROPER(X)
->->-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->->Cycle:
->->-> Pairs:
 TOP(mark(X)) -> TOP(proper(X))
 TOP(ok(X)) -> TOP(active(X))
->->-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))


The problem is decomposed in 7 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 S(mark(X)) -> S(X)
 S(ok(X)) -> S(X)
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(S) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 PLUS(mark(X1),X2) -> PLUS(X1,X2)
 PLUS(ok(X1),ok(X2)) -> PLUS(X1,X2)
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(PLUS) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
->->-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Problem 1.2: 

Subterm Processor:
-> Pairs:
 PLUS(X1,mark(X2)) -> PLUS(X1,X2)
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(PLUS) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3)
 U12#(ok(X1),ok(X2),ok(X3)) -> U12#(X1,X2,X3)
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(U12#) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3)
 U11#(ok(X1),ok(X2),ok(X3)) -> U11#(X1,X2,X3)
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(U11#) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 ACTIVE(U11(X1,X2,X3)) -> ACTIVE(X1)
 ACTIVE(U12(X1,X2,X3)) -> ACTIVE(X1)
 ACTIVE(plus(X1,X2)) -> ACTIVE(X1)
 ACTIVE(plus(X1,X2)) -> ACTIVE(X2)
 ACTIVE(s(X)) -> ACTIVE(X)
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(ACTIVE) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Subterm Processor:
-> Pairs:
 PROPER(U11(X1,X2,X3)) -> PROPER(X1)
 PROPER(U11(X1,X2,X3)) -> PROPER(X2)
 PROPER(U11(X1,X2,X3)) -> PROPER(X3)
 PROPER(U12(X1,X2,X3)) -> PROPER(X1)
 PROPER(U12(X1,X2,X3)) -> PROPER(X2)
 PROPER(U12(X1,X2,X3)) -> PROPER(X3)
 PROPER(plus(X1,X2)) -> PROPER(X1)
 PROPER(plus(X1,X2)) -> PROPER(X2)
 PROPER(s(X)) -> PROPER(X)
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Projection:
 pi(PROPER) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Reduction Pair Processor:
-> Pairs:
 TOP(mark(X)) -> TOP(proper(X))
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
-> Usable rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1
[active](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1/2
[proper](X) = X
[s](X) = X + 2
[0] = 0
[mark](X) = X + 1/2
[ok](X) = X
[tt] = 1
[TOP](X) = 1/2.X

Problem 1.7: 

SCC Processor:
-> Pairs:
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TOP(ok(X)) -> TOP(active(X))
->->-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Problem 1.7: 

Reduction Pair Processor:
-> Pairs:
 TOP(ok(X)) -> TOP(active(X))
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
-> Usable rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
[active](X) = 2.X + 1
[plus](X1,X2) = X1 + 2.X2 + 2
[s](X) = 2.X + 2
[0] = 1
[mark](X) = X + 2
[ok](X) = 2.X + 2
[tt] = 1
[TOP](X) = 2.X

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
 active(U11(tt,M,N)) -> mark(U12(tt,M,N))
 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
 active(U12(tt,M,N)) -> mark(s(plus(N,M)))
 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
 active(plus(N,s(M))) -> mark(U11(tt,M,N))
 active(plus(N,0)) -> mark(N)
 active(plus(X1,X2)) -> plus(active(X1),X2)
 active(plus(X1,X2)) -> plus(X1,active(X2))
 active(s(X)) -> s(active(X))
 plus(mark(X1),X2) -> mark(plus(X1,X2))
 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
 plus(X1,mark(X2)) -> mark(plus(X1,X2))
 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
 proper(s(X)) -> s(proper(X))
 proper(0) -> ok(0)
 proper(tt) -> ok(tt)
 s(mark(X)) -> mark(s(X))
 s(ok(X)) -> ok(s(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.