/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files
--------------------------------------------------------------------------------
YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished
Termination w.r.t. Q of the given QTRS could be proven:
(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) FlatCCProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RootLabelingProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) DependencyPairsProof [EQUIVALENT, 34 ms]
(8) QDP
(9) SemLabProof [SOUND, 245 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 1 ms]
(12) AND
(13) QDP
(14) UsableRulesReductionPairsProof [EQUIVALENT, 4 ms]
(15) QDP
(16) MRRProof [EQUIVALENT, 7 ms]
(17) QDP
(18) PisEmptyProof [EQUIVALENT, 0 ms]
(19) YES
(20) QDP
(21) MRRProof [EQUIVALENT, 31 ms]
(22) QDP
(23) DependencyGraphProof [EQUIVALENT, 0 ms]
(24) QDP
(25) QDPOrderProof [EQUIVALENT, 36 ms]
(26) QDP
(27) PisEmptyProof [EQUIVALENT, 0 ms]
(28) YES
----------------------------------------
(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(a(b(a(b(a(b(b(a(x1)))))))))) -> a(b(a(b(a(b(b(a(a(b(a(b(a(b(x1))))))))))))))
Q is empty.
----------------------------------------
(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------
(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(b(a(b(a(b(a(a(b(x1)))))))))) -> b(a(b(a(b(a(a(b(b(a(b(a(b(a(x1))))))))))))))
Q is empty.
----------------------------------------
(3) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]
As Q is empty the flat context closure was sound AND complete.
----------------------------------------
(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(b(b(a(b(a(b(a(a(b(x1))))))))))) -> a(b(a(b(a(b(a(a(b(b(a(b(a(b(a(x1)))))))))))))))
b(a(b(b(a(b(a(b(a(a(b(x1))))))))))) -> b(b(a(b(a(b(a(a(b(b(a(b(a(b(a(x1)))))))))))))))
Q is empty.
----------------------------------------
(5) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.
----------------------------------------
(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))))))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))))))))
Q is empty.
----------------------------------------
(7) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------
(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{a_1}(x1))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> A_{A_1}(x1)
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(x1))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> B_{A_1}(a_{a_1}(x1))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> A_{A_1}(x1)
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> B_{A_1}(a_{b_1}(x1))
The TRS R consists of the following rules:
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))))))))))))
a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))))))))))))
b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(9) SemLabProof (SOUND)
We found the following model for the rules of the TRSs R and P.
Interpretation over the domain with elements from 0 to 1.
b_{b_1}: 1
b_{a_1}: x0
A_{A_1}: 0
B_{A_1}: 0
a_{b_1}: x0
a_{a_1}: 0
By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
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(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.0(x1))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(x1))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> A_{A_1}.0(x1)
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(x1)
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(x1))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(x1))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.0(x1))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(x1))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> A_{A_1}.0(x1)
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(x1)
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> B_{A_1}.0(a_{b_1}.0(x1))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(x1))
The TRS R consists of the following rules:
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 49 less nodes.
----------------------------------------
(12)
Complex Obligation (AND)
----------------------------------------
(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))
The TRS R consists of the following rules:
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(14) UsableRulesReductionPairsProof (EQUIVALENT)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
No dependency pairs are removed.
The following rules are removed from R:
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(A_{A_1}.1(x_1)) = x_1
POL(a_{a_1}.0(x_1)) = x_1
POL(a_{b_1}.0(x_1)) = x_1
POL(a_{b_1}.1(x_1)) = x_1
POL(b_{a_1}.0(x_1)) = x_1
POL(b_{b_1}.0(x_1)) = x_1
----------------------------------------
(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(16) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))
Used ordering: Polynomial interpretation [POLO]:
POL(A_{A_1}.1(x_1)) = x_1
POL(a_{a_1}.0(x_1)) = x_1
POL(a_{b_1}.0(x_1)) = 1 + x_1
POL(a_{b_1}.1(x_1)) = x_1
POL(b_{a_1}.0(x_1)) = x_1
POL(b_{b_1}.0(x_1)) = x_1
----------------------------------------
(17)
Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(18) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------
(19)
YES
----------------------------------------
(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(x1)
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(x1))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(x1))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(x1)
The TRS R consists of the following rules:
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(21) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(x1))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))
B_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(x1)
Used ordering: Polynomial interpretation [POLO]:
POL(A_{A_1}.1(x_1)) = x_1
POL(B_{A_1}.1(x_1)) = 1 + x_1
POL(a_{a_1}.0(x_1)) = x_1
POL(a_{a_1}.1(x_1)) = x_1
POL(a_{b_1}.0(x_1)) = x_1
POL(a_{b_1}.1(x_1)) = x_1
POL(b_{a_1}.0(x_1)) = x_1
POL(b_{a_1}.1(x_1)) = x_1
POL(b_{b_1}.0(x_1)) = x_1
POL(b_{b_1}.1(x_1)) = 1 + x_1
----------------------------------------
(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(x1)
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> B_{A_1}.1(a_{b_1}.1(x1))
The TRS R consists of the following rules:
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------
(23) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
----------------------------------------
(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(x1)
The TRS R consists of the following rules:
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(25) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))
A_{A_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> A_{A_1}.1(x1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A_{A_1}.1(x_1)) = x_1
POL(a_{a_1}.0(x_1)) = 0
POL(a_{a_1}.1(x_1)) = 1 + x_1
POL(a_{b_1}.0(x_1)) = x_1
POL(a_{b_1}.1(x_1)) = 1 + x_1
POL(b_{a_1}.0(x_1)) = x_1
POL(b_{a_1}.1(x_1)) = 1 + x_1
POL(b_{b_1}.0(x_1)) = 1 + x_1
POL(b_{b_1}.1(x_1)) = 1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
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(26)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(a_{b_1}.0(b_{a_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{a_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(x1)))))))))))))))
b_{a_1}.1(a_{b_1}.1(b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(x1))))))))))) -> b_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{b_1}.0(b_{a_1}.0(a_{a_1}.1(a_{b_1}.1(b_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(b_{a_1}.1(a_{b_1}.1(x1)))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(27) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(28)
YES