/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) FlatCCProof [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RootLabelingProof [EQUIVALENT, 0 ms]
(4) RelTRS
(5) FlatCCProof [EQUIVALENT, 1 ms]
(6) RelTRS
(7) RootLabelingProof [EQUIVALENT, 0 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 117 ms]
(10) RelTRS
(11) RelTRSRRRProof [EQUIVALENT, 65 ms]
(12) RelTRS
(13) RelTRSRRRProof [EQUIVALENT, 19.3 s]
(14) RelTRS
(15) RelTRSRRRProof [EQUIVALENT, 12 ms]
(16) RelTRS
(17) RelTRSRRRProof [EQUIVALENT, 11 ms]
(18) RelTRS
(19) RelTRSSemanticLabellingPOLOProof [EQUIVALENT, 709 ms]
(20) RelTRS
(21) RelTRSSemanticLabellingPOLOProof [EQUIVALENT, 719 ms]
(22) RelTRS
(23) RelTRSRRRProof [EQUIVALENT, 1918 ms]
(24) RelTRS
(25) RelTRSRRRProof [EQUIVALENT, 9 ms]
(26) RelTRS
(27) RelTRSSemanticLabellingPOLOProof [EQUIVALENT, 621 ms]
(28) RelTRS
(29) RelTRSRRRProof [EQUIVALENT, 8 ms]
(30) RelTRS
(31) RelTRSRRRProof [EQUIVALENT, 5239 ms]
(32) RelTRS
(33) RelTRSRRRProof [EQUIVALENT, 2019 ms]
(34) RelTRS
(35) RelTRSRRRProof [EQUIVALENT, 1530 ms]
(36) RelTRS
(37) RelTRSRRRProof [EQUIVALENT, 938 ms]
(38) RelTRS
(39) RIsEmptyProof [EQUIVALENT, 0 ms]
(40) YES


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(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a(b(a(b(a(x1))))) -> x1

The relative TRS consists of the following S rules:

   a(b(x1)) -> b(b(a(a(x1))))


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(1) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]

----------------------------------------

(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a(a(b(a(b(a(x1)))))) -> a(x1)
   b(a(b(a(b(a(x1)))))) -> b(x1)

The relative TRS consists of the following S rules:

   a(a(b(x1))) -> a(b(b(a(a(x1)))))
   b(a(b(x1))) -> b(b(b(a(a(x1)))))


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(3) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled


----------------------------------------

(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> a_{a_1}(x1)
   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> a_{b_1}(x1)
   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> b_{a_1}(x1)
   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(x1)

The relative TRS consists of the following S rules:

   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))


----------------------------------------

(5) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]

----------------------------------------

(6)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> a_{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}(x1)
   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) -> a_{a_1}(a_{b_1}(x1))
   a_{b_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) -> a_{b_1}(a_{b_1}(x1))
   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) -> b_{a_1}(a_{b_1}(x1))
   b_{b_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) -> b_{b_1}(a_{b_1}(x1))
   a_{a_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) -> a_{a_1}(b_{b_1}(x1))
   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(x1))
   b_{a_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) -> b_{a_1}(b_{b_1}(x1))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(x1))

The relative TRS consists of the following S rules:

   a_{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_{a_1}(a_{a_1}(x1))))))
   a_{b_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
   b_{b_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
   a_{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_{a_1}(a_{b_1}(x1))))))
   a_{b_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
   b_{b_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
   a_{a_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
   b_{a_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
   a_{a_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
   b_{a_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))


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(7) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled


----------------------------------------

(8)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1)))))) -> a_{a_1}_{a_{a_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1)))))) -> a_{a_1}_{a_{b_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1)))))) -> a_{a_1}_{b_{a_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1)))))) -> a_{a_1}_{b_{b_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1)))))) -> b_{a_1}_{a_{a_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1)))))) -> b_{a_1}_{a_{b_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1)))))) -> b_{a_1}_{b_{a_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1)))))) -> b_{a_1}_{b_{b_1}_1}(x1)
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


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(9) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(a_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{b_{a_1}_1}(x_1)) = 1 + x_1
   POL(a_{a_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{a_1}_1}(x_1)) = 1 + x_1
   POL(a_{b_1}_{a_{b_1}_1}(x_1)) = 1 + x_1
   POL(a_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{b_{a_1}_1}(x_1)) = 1 + x_1
   POL(b_{a_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{a_{a_1}_1}(x_1)) = 1 + x_1
   POL(b_{b_1}_{a_{b_1}_1}(x_1)) = 1 + x_1
   POL(b_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{b_1}_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
Rules from S:

   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))




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(10)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1)))))) -> a_{a_1}_{a_{a_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1)))))) -> a_{a_1}_{a_{b_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1)))))) -> a_{a_1}_{b_{a_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1)))))) -> a_{a_1}_{b_{b_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1)))))) -> b_{a_1}_{a_{a_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1)))))) -> b_{a_1}_{a_{b_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1)))))) -> b_{a_1}_{b_{a_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1)))))) -> b_{a_1}_{b_{b_1}_1}(x1)
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


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(11) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(a_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{a_1}_1}(x_1)) = 1 + x_1
   POL(a_{b_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{a_{a_1}_1}(x_1)) = 1 + x_1
   POL(b_{b_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{b_1}_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
Rules from S:

   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))




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(12)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1)))))) -> a_{a_1}_{a_{a_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1)))))) -> a_{a_1}_{a_{b_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1)))))) -> a_{a_1}_{b_{a_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1)))))) -> a_{a_1}_{b_{b_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1)))))) -> b_{a_1}_{a_{a_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1)))))) -> b_{a_1}_{a_{b_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1)))))) -> b_{a_1}_{b_{a_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1)))))) -> b_{a_1}_{b_{b_1}_1}(x1)
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


----------------------------------------

(13) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :

   <<<
 POL(a_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 1, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 1, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 1, 0], [0, 1, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [1], [0]] 	 +  	[[1, 0, 0], [0, 1, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 1, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [1, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1)))))) -> a_{a_1}_{a_{a_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1)))))) -> a_{a_1}_{a_{b_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1)))))) -> a_{a_1}_{b_{a_1}_1}(x1)
   a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1)))))) -> a_{a_1}_{b_{b_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1)))))) -> b_{a_1}_{a_{a_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1)))))) -> b_{a_1}_{a_{b_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1)))))) -> b_{a_1}_{b_{a_1}_1}(x1)
   b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1)))))) -> b_{a_1}_{b_{b_1}_1}(x1)
Rules from S:

   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))




----------------------------------------

(14)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


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(15) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(a_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{b_{a_1}_1}(x_1)) = 1 + x_1
   POL(a_{a_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{b_{a_1}_1}(x_1)) = 1 + x_1
   POL(b_{a_1}_{b_{b_1}_1}(x_1)) = 1 + x_1
   POL(b_{b_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{b_1}_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{b_1}_1}(x1))))))




----------------------------------------

(16)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


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(17) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(a_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{b_{a_1}_1}(x_1)) = 1 + x_1
   POL(b_{b_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{b_1}_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{b_{a_1}_1}(x1))))))




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(18)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


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(19) RelTRSSemanticLabellingPOLOProof (EQUIVALENT)
We use Semantic Labelling over tuples of bools combined with a polynomial order [SEMLAB]



We use semantic labelling over boolean tuples of size 1.



We used the following model:

a_{a_1}_{a_{a_1}_1}_1: component 1: OR[x_1^1]

a_{a_1}_{a_{b_1}_1}_1: component 1: OR[]

a_{b_1}_{b_{a_1}_1}_1: component 1: OR[]

b_{a_1}_{a_{a_1}_1}_1: component 1: OR[x_1^1]

a_{b_1}_{b_{b_1}_1}_1: component 1: OR[]

b_{b_1}_{b_{a_1}_1}_1: component 1: OR[]

b_{a_1}_{a_{b_1}_1}_1: component 1: OR[]

b_{b_1}_{a_{a_1}_1}_1: component 1: OR[]

a_{b_1}_{a_{a_1}_1}_1: component 1: AND[]

b_{b_1}_{a_{b_1}_1}_1: component 1: OR[]

a_{b_1}_{a_{b_1}_1}_1: component 1: OR[]

b_{b_1}_{b_{b_1}_1}_1: component 1: OR[]



Our labelling function was:

a_{a_1}_{a_{a_1}_1}_1:component 1: OR[]

a_{a_1}_{a_{b_1}_1}_1:component 1: OR[]

a_{b_1}_{b_{a_1}_1}_1:component 1: XOR[]

b_{a_1}_{a_{a_1}_1}_1:component 1: OR[]

a_{b_1}_{b_{b_1}_1}_1:component 1: OR[]

b_{b_1}_{b_{a_1}_1}_1:component 1: OR[x_1^1]

b_{a_1}_{a_{b_1}_1}_1:component 1: OR[x_1^1]

b_{b_1}_{a_{a_1}_1}_1:component 1: OR[]

a_{b_1}_{a_{a_1}_1}_1:component 1: XOR[]

b_{b_1}_{a_{b_1}_1}_1:component 1: OR[]

a_{b_1}_{a_{b_1}_1}_1:component 1: OR[]

b_{b_1}_{b_{b_1}_1}_1:component 1: OR[]





Our labelled system was:

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[t]b_{a_1}_{a_{a_1}_1}^[f](^[t]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[t](^[t]b_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[f](^[t]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[t]b_{a_1}_{a_{a_1}_1}^[f](^[t]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[t](^[t]b_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[f](^[t]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[t](^[t]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[t](^[t]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[t](^[t]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[t](^[t]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[t](^[t]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[t](^[t]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1))





Our polynomial interpretation was:

P(a_{a_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{a_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{a_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{a_1}_{a_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{b_1}_1}^[true])(x_1) = 1 + 1*x_1

P(b_{b_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{b_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{a_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{b_1}_1}^[true])(x_1) = 0 + 1*x_1


The following rules were deleted from R:

   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))

The following rules were deleted from S:
none




----------------------------------------

(20)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


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(21) RelTRSSemanticLabellingPOLOProof (EQUIVALENT)
We use Semantic Labelling over tuples of bools combined with a polynomial order [SEMLAB]



We use semantic labelling over boolean tuples of size 1.



We used the following model:

a_{a_1}_{a_{a_1}_1}_1: component 1: OR[x_1^1]

a_{a_1}_{a_{b_1}_1}_1: component 1: OR[]

a_{b_1}_{b_{a_1}_1}_1: component 1: OR[x_1^1]

b_{a_1}_{a_{a_1}_1}_1: component 1: OR[x_1^1]

a_{b_1}_{b_{b_1}_1}_1: component 1: OR[]

b_{b_1}_{b_{a_1}_1}_1: component 1: OR[x_1^1]

b_{a_1}_{a_{b_1}_1}_1: component 1: OR[]

b_{b_1}_{a_{a_1}_1}_1: component 1: OR[]

a_{b_1}_{a_{a_1}_1}_1: component 1: OR[]

b_{b_1}_{a_{b_1}_1}_1: component 1: OR[]

a_{b_1}_{a_{b_1}_1}_1: component 1: AND[]

b_{b_1}_{b_{b_1}_1}_1: component 1: OR[]



Our labelling function was:

a_{a_1}_{a_{a_1}_1}_1:component 1: XOR[x_1^1]

a_{a_1}_{a_{b_1}_1}_1:component 1: OR[]

a_{b_1}_{b_{a_1}_1}_1:component 1: OR[]

b_{a_1}_{a_{a_1}_1}_1:component 1: XOR[]

a_{b_1}_{b_{b_1}_1}_1:component 1: XOR[x_1^1]

b_{b_1}_{b_{a_1}_1}_1:component 1: OR[]

b_{a_1}_{a_{b_1}_1}_1:component 1: XOR[x_1^1]

b_{b_1}_{a_{a_1}_1}_1:component 1: OR[]

a_{b_1}_{a_{a_1}_1}_1:component 1: XOR[]

b_{b_1}_{a_{b_1}_1}_1:component 1: XOR[]

a_{b_1}_{a_{b_1}_1}_1:component 1: OR[]

b_{b_1}_{b_{b_1}_1}_1:component 1: OR[]





Our labelled system was:

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]b_{a_1}_{a_{a_1}_1}^[f](^[t]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]b_{b_1}_{b_{a_1}_1}^[f](^[t]b_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[t](^[t]a_{a_1}_{a_{a_1}_1}^[t](^[t]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]b_{a_1}_{a_{a_1}_1}^[f](^[t]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]b_{b_1}_{b_{a_1}_1}^[f](^[t]b_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[t](^[t]a_{a_1}_{a_{a_1}_1}^[t](^[t]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]b_{b_1}_{b_{a_1}_1}^[f](^[t]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[t]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]b_{b_1}_{b_{a_1}_1}^[f](^[t]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[t]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]b_{b_1}_{b_{a_1}_1}^[f](^[t]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[t]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]b_{b_1}_{b_{a_1}_1}^[f](^[t]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[t]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{a_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]b_{b_1}_{b_{a_1}_1}^[f](^[t]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[t]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{b_{a_1}_1}^[f](^[t]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[t]b_{b_1}_{b_{a_1}_1}^[f](^[t]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[t](^[t]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[t]x1))





Our polynomial interpretation was:

P(a_{a_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{a_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{a_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{a_1}_{a_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{b_1}_1}^[true])(x_1) = 1 + 1*x_1

P(b_{b_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{b_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{a_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{b_1}_1}^[true])(x_1) = 0 + 1*x_1


The following rules were deleted from R:

   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))

The following rules were deleted from S:
none




----------------------------------------

(22)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


----------------------------------------

(23) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(a_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [2, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{a_{a_1}_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [2, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1))
Rules from S:
none




----------------------------------------

(24)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


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(25) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(a_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{a_{a_1}_1}(x_1)) = 1 + x_1
   POL(b_{b_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{b_1}_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))




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(26)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


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(27) RelTRSSemanticLabellingPOLOProof (EQUIVALENT)
We use Semantic Labelling over tuples of bools combined with a polynomial order [SEMLAB]



We use semantic labelling over boolean tuples of size 1.



We used the following model:

a_{a_1}_{a_{a_1}_1}_1: component 1: OR[x_1^1]

a_{a_1}_{a_{b_1}_1}_1: component 1: OR[]

a_{b_1}_{b_{a_1}_1}_1: component 1: OR[]

b_{a_1}_{a_{a_1}_1}_1: component 1: OR[]

a_{b_1}_{b_{b_1}_1}_1: component 1: OR[]

b_{b_1}_{b_{a_1}_1}_1: component 1: OR[]

b_{a_1}_{a_{b_1}_1}_1: component 1: OR[]

b_{b_1}_{a_{b_1}_1}_1: component 1: OR[]

a_{b_1}_{a_{b_1}_1}_1: component 1: AND[]

b_{b_1}_{b_{b_1}_1}_1: component 1: OR[]

a_{b_1}_{a_{a_1}_1}_1: component 1: OR[]



Our labelling function was:

a_{a_1}_{a_{a_1}_1}_1:component 1: XOR[]

a_{a_1}_{a_{b_1}_1}_1:component 1: XOR[]

a_{b_1}_{b_{a_1}_1}_1:component 1: OR[]

b_{a_1}_{a_{a_1}_1}_1:component 1: OR[]

a_{b_1}_{b_{b_1}_1}_1:component 1: XOR[]

b_{b_1}_{b_{a_1}_1}_1:component 1: OR[]

b_{a_1}_{a_{b_1}_1}_1:component 1: XOR[x_1^1]

b_{b_1}_{a_{b_1}_1}_1:component 1: OR[]

a_{b_1}_{a_{b_1}_1}_1:component 1: XOR[]

b_{b_1}_{b_{b_1}_1}_1:component 1: XOR[]

a_{b_1}_{a_{a_1}_1}_1:component 1: OR[]





Our labelled system was:

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[t]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[f](^[t]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[t]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[f](^[t]a_{a_1}_{a_{a_1}_1}^[f](^[t]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1)))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]a_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{a_{a_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]b_{a_1}_{a_{a_1}_1}^[f](^[f]a_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[t](^[t]a_{b_1}_{a_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{a_{b_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]x1))

^[f]b_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{a_1}_1}^[f](^[f]b_{a_1}_{a_{b_1}_1}^[f](^[f]a_{b_1}_{b_{b_1}_1}^[f](^[f]x1))))))) -> ^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]b_{b_1}_{b_{b_1}_1}^[f](^[f]x1))





Our polynomial interpretation was:

P(a_{a_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{a_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{a_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{a_1}_{a_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{b_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{a_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{a_1}_{a_{b_1}_1}^[true])(x_1) = 1 + 1*x_1

P(b_{b_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{a_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{b_1}_1}^[false])(x_1) = 0 + 1*x_1

P(b_{b_1}_{b_{b_1}_1}^[true])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{a_1}_1}^[false])(x_1) = 0 + 1*x_1

P(a_{b_1}_{a_{a_1}_1}^[true])(x_1) = 0 + 1*x_1


The following rules were deleted from R:

   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1))

The following rules were deleted from S:
none




----------------------------------------

(28)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


----------------------------------------

(29) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(a_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(a_{b_1}_{b_{b_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{a_1}_1}(x_1)) = x_1
   POL(b_{a_1}_{a_{b_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{a_{b_1}_1}(x_1)) = 1 + x_1
   POL(b_{b_1}_{b_{a_1}_1}(x_1)) = x_1
   POL(b_{b_1}_{b_{b_1}_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))




----------------------------------------

(30)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


----------------------------------------

(31) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(a_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{a_{b_1}_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
Rules from S:
none




----------------------------------------

(32)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


----------------------------------------

(33) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :

   <<<
 POL(a_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 1, 1], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 1, 0], [0, 0, 1], [0, 0, 1]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [1], [1]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0], [1]] 	 +  	[[1, 0, 0], [1, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{a_1}_1}(x1))
Rules from S:
none




----------------------------------------

(34)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


----------------------------------------

(35) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(a_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [2, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [2, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{a_{b_1}_1}(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{a_{b_1}_1}(x1))
Rules from S:
none




----------------------------------------

(36)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))

The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


----------------------------------------

(37) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :

   <<<
 POL(a_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [1], [0]] 	 +  	[[1, 0, 0], [0, 1, 0], [1, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [1], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [1], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 1, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}_{a_{a_1}_1}(x_1)) =  	[[0], [0], [1]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 1, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{a_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}_{b_{b_1}_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1))
Rules from S:

   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))




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(38)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   a_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{a_1}_1}(x1))))))
   b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(x1)))) -> b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{b_1}_1}(b_{b_1}_{b_{a_1}_1}(b_{a_1}_{a_{a_1}_1}(a_{a_1}_{a_{b_1}_1}(a_{b_1}_{b_{b_1}_1}(x1))))))


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(39) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(40)
YES