/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 x1)
(RULES
0(ql(x1)) -> ql(0(x1))
0(qr(x1)) -> qr(0(x1))
1(ql(x1)) -> ql(1(x1))
1(qr(x1)) -> qr(1(x1))
b(ql(0(x1))) -> 0(b(r0(x1)))
b(ql(1(x1))) -> 1(b(r1(x1)))
m(qr(x1)) -> ql(m(x1))
r0(0(x1)) -> 0(r0(x1))
r0(1(x1)) -> 1(r0(x1))
r0(b(x1)) -> qr(0(b(x1)))
r0(m(x1)) -> m(r0(x1))
r1(0(x1)) -> 0(r1(x1))
r1(1(x1)) -> 1(r1(x1))
r1(b(x1)) -> qr(1(b(x1)))
r1(m(x1)) -> m(r1(x1))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 0#(ql(x1)) -> 0#(x1)
 0#(qr(x1)) -> 0#(x1)
 1#(ql(x1)) -> 1#(x1)
 1#(qr(x1)) -> 1#(x1)
 B(ql(0(x1))) -> 0#(b(r0(x1)))
 B(ql(0(x1))) -> B(r0(x1))
 B(ql(0(x1))) -> R0(x1)
 B(ql(1(x1))) -> 1#(b(r1(x1)))
 B(ql(1(x1))) -> B(r1(x1))
 B(ql(1(x1))) -> R1(x1)
 M(qr(x1)) -> M(x1)
 R0(0(x1)) -> 0#(r0(x1))
 R0(0(x1)) -> R0(x1)
 R0(1(x1)) -> 1#(r0(x1))
 R0(1(x1)) -> R0(x1)
 R0(b(x1)) -> 0#(b(x1))
 R0(m(x1)) -> M(r0(x1))
 R0(m(x1)) -> R0(x1)
 R1(0(x1)) -> 0#(r1(x1))
 R1(0(x1)) -> R1(x1)
 R1(1(x1)) -> 1#(r1(x1))
 R1(1(x1)) -> R1(x1)
 R1(b(x1)) -> 1#(b(x1))
 R1(m(x1)) -> M(r1(x1))
 R1(m(x1)) -> R1(x1)
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))

Problem 1: 

SCC Processor:
-> Pairs:
 0#(ql(x1)) -> 0#(x1)
 0#(qr(x1)) -> 0#(x1)
 1#(ql(x1)) -> 1#(x1)
 1#(qr(x1)) -> 1#(x1)
 B(ql(0(x1))) -> 0#(b(r0(x1)))
 B(ql(0(x1))) -> B(r0(x1))
 B(ql(0(x1))) -> R0(x1)
 B(ql(1(x1))) -> 1#(b(r1(x1)))
 B(ql(1(x1))) -> B(r1(x1))
 B(ql(1(x1))) -> R1(x1)
 M(qr(x1)) -> M(x1)
 R0(0(x1)) -> 0#(r0(x1))
 R0(0(x1)) -> R0(x1)
 R0(1(x1)) -> 1#(r0(x1))
 R0(1(x1)) -> R0(x1)
 R0(b(x1)) -> 0#(b(x1))
 R0(m(x1)) -> M(r0(x1))
 R0(m(x1)) -> R0(x1)
 R1(0(x1)) -> 0#(r1(x1))
 R1(0(x1)) -> R1(x1)
 R1(1(x1)) -> 1#(r1(x1))
 R1(1(x1)) -> R1(x1)
 R1(b(x1)) -> 1#(b(x1))
 R1(m(x1)) -> M(r1(x1))
 R1(m(x1)) -> R1(x1)
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 M(qr(x1)) -> M(x1)
->->-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->->Cycle:
->->-> Pairs:
 1#(ql(x1)) -> 1#(x1)
 1#(qr(x1)) -> 1#(x1)
->->-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->->Cycle:
->->-> Pairs:
 0#(ql(x1)) -> 0#(x1)
 0#(qr(x1)) -> 0#(x1)
->->-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->->Cycle:
->->-> Pairs:
 R0(0(x1)) -> R0(x1)
 R0(1(x1)) -> R0(x1)
 R0(m(x1)) -> R0(x1)
->->-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->->Cycle:
->->-> Pairs:
 R1(0(x1)) -> R1(x1)
 R1(1(x1)) -> R1(x1)
 R1(m(x1)) -> R1(x1)
->->-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->->Cycle:
->->-> Pairs:
 B(ql(0(x1))) -> B(r0(x1))
 B(ql(1(x1))) -> B(r1(x1))
->->-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))


The problem is decomposed in 6 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 M(qr(x1)) -> M(x1)
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Projection:
 pi(M) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 1#(ql(x1)) -> 1#(x1)
 1#(qr(x1)) -> 1#(x1)
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Projection:
 pi(1#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 0#(ql(x1)) -> 0#(x1)
 0#(qr(x1)) -> 0#(x1)
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Projection:
 pi(0#) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 R0(0(x1)) -> R0(x1)
 R0(1(x1)) -> R0(x1)
 R0(m(x1)) -> R0(x1)
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Projection:
 pi(R0) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 R1(0(x1)) -> R1(x1)
 R1(1(x1)) -> R1(x1)
 R1(m(x1)) -> R1(x1)
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Projection:
 pi(R1) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Reduction Pair Processor:
-> Pairs:
 B(ql(0(x1))) -> B(r0(x1))
 B(ql(1(x1))) -> B(r1(x1))
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
-> Usable rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2.X + 2
[1](X) = 2.X + 2
[b](X) = 2.X + 2
[m](X) = 2
[r0](X) = X
[r1](X) = X
[ql](X) = X
[qr](X) = 0
[B](X) = 2.X

Problem 1.6: 

SCC Processor:
-> Pairs:
 B(ql(1(x1))) -> B(r1(x1))
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 B(ql(1(x1))) -> B(r1(x1))
->->-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))

Problem 1.6: 

Reduction Pair Processor:
-> Pairs:
 B(ql(1(x1))) -> B(r1(x1))
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
-> Usable rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2
[1](X) = 2.X + 2
[b](X) = 2.X + 2
[m](X) = 2
[r0](X) = 2.X + 2
[r1](X) = X
[ql](X) = X
[qr](X) = 2
[B](X) = 2.X

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0(ql(x1)) -> ql(0(x1))
 0(qr(x1)) -> qr(0(x1))
 1(ql(x1)) -> ql(1(x1))
 1(qr(x1)) -> qr(1(x1))
 b(ql(0(x1))) -> 0(b(r0(x1)))
 b(ql(1(x1))) -> 1(b(r1(x1)))
 m(qr(x1)) -> ql(m(x1))
 r0(0(x1)) -> 0(r0(x1))
 r0(1(x1)) -> 1(r0(x1))
 r0(b(x1)) -> qr(0(b(x1)))
 r0(m(x1)) -> m(r0(x1))
 r1(0(x1)) -> 0(r1(x1))
 r1(1(x1)) -> 1(r1(x1))
 r1(b(x1)) -> qr(1(b(x1)))
 r1(m(x1)) -> m(r1(x1))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.