The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

0(0(1(x1))) → 1(2(0(2(0(x1)))))
0(0(1(x1))) → 2(0(2(1(0(x1)))))
0(0(1(x1))) → 2(1(0(2(0(x1)))))
0(0(1(x1))) → 1(1(2(0(2(0(x1))))))
0(0(1(x1))) → 1(2(0(2(0(2(x1))))))
0(0(1(x1))) → 1(2(0(2(0(3(x1))))))
0(0(1(x1))) → 1(2(0(2(0(4(x1))))))
0(0(1(x1))) → 1(2(0(2(2(0(x1))))))
0(0(1(x1))) → 1(2(0(2(3(0(x1))))))
0(0(1(x1))) → 1(2(0(2(5(0(x1))))))
0(0(1(x1))) → 1(2(0(3(2(0(x1))))))
0(0(1(x1))) → 1(2(1(0(2(0(x1))))))
0(0(1(x1))) → 1(2(2(0(2(0(x1))))))
0(0(1(x1))) → 1(2(2(0(3(0(x1))))))
0(0(1(x1))) → 1(2(2(0(4(0(x1))))))
0(0(1(x1))) → 1(2(3(0(2(0(x1))))))
0(0(1(x1))) → 1(2(5(0(2(0(x1))))))
0(0(1(x1))) → 1(3(2(0(2(0(x1))))))
0(0(1(x1))) → 2(0(1(2(0(2(x1))))))
0(0(1(x1))) → 2(0(1(2(2(0(x1))))))
0(0(1(x1))) → 2(0(1(3(0(2(x1))))))
0(0(1(x1))) → 2(0(2(0(2(1(x1))))))
0(0(1(x1))) → 2(0(2(1(0(2(x1))))))
0(0(1(x1))) → 2(0(2(1(0(4(x1))))))
0(0(1(x1))) → 2(0(2(1(2(0(x1))))))
0(0(1(x1))) → 2(0(2(3(1(0(x1))))))
0(0(1(x1))) → 2(0(2(5(1(0(x1))))))
0(0(1(x1))) → 2(1(0(2(0(2(x1))))))
0(0(1(x1))) → 2(1(0(2(2(0(x1))))))
0(0(1(x1))) → 2(1(0(2(5(0(x1))))))
0(0(1(x1))) → 2(1(2(0(2(0(x1))))))
0(0(1(x1))) → 2(1(2(0(4(0(x1))))))
0(0(1(x1))) → 2(1(4(0(2(0(x1))))))
0(0(1(x1))) → 2(1(4(0(4(0(x1))))))
0(0(1(x1))) → 2(2(0(2(1(0(x1))))))
0(0(1(x1))) → 2(2(0(4(1(0(x1))))))
0(0(1(x1))) → 2(2(1(0(2(0(x1))))))
0(0(1(x1))) → 3(1(2(0(2(0(x1))))))
0(0(0(1(x1)))) → 0(0(1(2(0(2(x1))))))
0(0(0(1(x1)))) → 0(2(1(0(2(0(x1))))))
0(0(0(1(x1)))) → 2(0(2(1(0(0(x1))))))
0(0(1(0(x1)))) → 0(0(2(0(2(1(x1))))))
0(0(1(1(x1)))) → 2(0(1(2(1(0(x1))))))
0(0(1(1(x1)))) → 2(0(2(0(1(1(x1))))))
0(0(1(1(x1)))) → 2(1(0(2(0(1(x1))))))
0(0(3(1(x1)))) → 2(0(1(3(0(x1)))))
0(0(3(1(x1)))) → 2(0(1(3(0(2(x1))))))
0(0(3(1(x1)))) → 2(0(2(3(1(0(x1))))))
0(0(3(1(x1)))) → 2(1(0(2(0(3(x1))))))
0(0(3(1(x1)))) → 3(2(0(2(1(0(x1))))))
0(0(5(1(x1)))) → 0(2(5(1(0(x1)))))
0(0(5(1(x1)))) → 0(5(0(2(1(x1)))))
0(0(5(1(x1)))) → 2(5(0(0(1(x1)))))
0(0(5(1(x1)))) → 2(5(1(0(0(x1)))))
0(0(5(1(x1)))) → 0(1(2(5(2(0(x1))))))
0(0(5(1(x1)))) → 0(2(1(2(5(0(x1))))))
0(0(5(1(x1)))) → 0(2(1(4(5(0(x1))))))
0(0(5(1(x1)))) → 1(2(0(0(2(5(x1))))))
0(0(5(1(x1)))) → 2(0(2(1(0(5(x1))))))
0(0(5(1(x1)))) → 2(1(0(4(5(0(x1))))))
0(0(5(1(x1)))) → 2(5(0(2(1(0(x1))))))
0(1(0(1(x1)))) → 0(0(1(2(1(x1)))))
0(1(0(1(x1)))) → 1(2(1(0(0(x1)))))
0(1(0(1(x1)))) → 2(1(0(0(1(x1)))))
0(1(0(1(x1)))) → 0(0(1(1(2(2(x1))))))
0(1(0(1(x1)))) → 0(1(1(3(2(0(x1))))))
0(1(0(1(x1)))) → 1(1(3(2(0(0(x1))))))
0(1(0(1(x1)))) → 1(2(0(2(0(1(x1))))))
0(1(0(1(x1)))) → 1(2(2(0(0(1(x1))))))
0(1(0(1(x1)))) → 1(2(5(1(0(0(x1))))))
0(1(0(1(x1)))) → 2(1(2(1(0(0(x1))))))
0(1(0(1(x1)))) → 3(1(2(1(0(0(x1))))))
0(3(0(1(x1)))) → 1(2(3(0(2(0(x1))))))
0(3(0(1(x1)))) → 3(1(2(0(2(0(x1))))))
0(5(0(1(x1)))) → 2(5(0(0(1(x1)))))
0(5(0(1(x1)))) → 0(0(2(1(2(5(x1))))))
0(5(0(1(x1)))) → 0(0(2(5(2(1(x1))))))
0(5(0(1(x1)))) → 0(1(2(0(2(5(x1))))))
0(5(0(1(x1)))) → 1(2(0(2(5(0(x1))))))
0(5(0(1(x1)))) → 2(0(0(1(2(5(x1))))))
0(5(0(1(x1)))) → 2(0(1(5(2(0(x1))))))
0(5(0(1(x1)))) → 2(0(2(1(5(0(x1))))))
0(5(0(1(x1)))) → 2(5(0(0(1(2(x1))))))
0(5(0(1(x1)))) → 2(5(0(2(0(1(x1))))))
0(5(1(1(x1)))) → 1(2(5(1(0(x1)))))
0(5(1(1(x1)))) → 2(5(0(1(1(x1)))))
0(5(1(1(x1)))) → 1(2(1(0(4(5(x1))))))
0(5(1(1(x1)))) → 1(2(1(4(5(0(x1))))))
0(5(1(1(x1)))) → 2(0(1(2(5(1(x1))))))
0(5(1(1(x1)))) → 2(5(0(2(1(1(x1))))))
0(5(1(1(x1)))) → 2(5(1(2(0(1(x1))))))
0(5(1(1(x1)))) → 2(5(1(2(1(0(x1))))))
0(5(1(1(x1)))) → 2(5(2(1(0(1(x1))))))
0(5(5(1(x1)))) → 2(5(5(1(0(x1)))))
0(5(5(1(x1)))) → 0(1(2(5(2(5(x1))))))
0(5(5(1(x1)))) → 1(2(0(2(5(5(x1))))))
0(5(5(1(x1)))) → 2(0(2(5(1(5(x1))))))
0(5(5(1(x1)))) → 2(2(5(0(5(1(x1))))))
0(5(5(1(x1)))) → 2(2(5(5(1(0(x1))))))
0(5(5(1(x1)))) → 2(5(0(2(5(1(x1))))))
0(5(5(1(x1)))) → 2(5(2(5(0(1(x1))))))
5(0(0(1(x1)))) → 1(2(5(2(0(0(x1))))))
5(0(0(1(x1)))) → 1(4(5(0(2(0(x1))))))
5(0(0(1(x1)))) → 2(0(2(5(1(0(x1))))))
5(0(0(1(x1)))) → 2(5(1(0(2(0(x1))))))
0(0(0(3(1(x1))))) → 2(0(3(0(0(1(x1))))))
0(0(0(5(1(x1))))) → 2(5(1(0(0(0(x1))))))
0(0(1(4(0(x1))))) → 0(2(1(0(4(0(x1))))))
0(0(1(5(0(x1))))) → 0(2(5(1(0(0(x1))))))
0(0(1(5(0(x1))))) → 0(5(0(2(1(0(x1))))))
0(0(5(1(1(x1))))) → 0(0(2(5(1(1(x1))))))
0(0(5(1(1(x1))))) → 2(1(0(5(0(1(x1))))))
0(0(5(3(1(x1))))) → 3(0(0(2(5(1(x1))))))
0(0(5(3(1(x1))))) → 3(2(5(1(0(0(x1))))))
0(1(0(1(1(x1))))) → 0(0(1(1(1(3(x1))))))
0(1(0(3(1(x1))))) → 2(0(1(3(1(0(x1))))))
0(1(1(0(1(x1))))) → 0(0(1(1(1(3(x1))))))
0(1(4(1(1(x1))))) → 1(2(1(4(0(1(x1))))))
0(1(5(0(1(x1))))) → 1(2(5(1(0(0(x1))))))
0(3(0(5(1(x1))))) → 2(5(0(0(3(1(x1))))))
0(3(1(0(1(x1))))) → 2(3(1(0(0(1(x1))))))
0(3(5(0(1(x1))))) → 0(0(3(2(5(1(x1))))))
0(3(5(1(1(x1))))) → 1(2(5(1(0(3(x1))))))
0(4(4(1(1(x1))))) → 1(2(1(4(0(4(x1))))))
0(4(5(5(1(x1))))) → 5(0(2(1(4(5(x1))))))
0(5(0(0(1(x1))))) → 2(5(0(0(0(1(x1))))))
0(5(0(1(0(x1))))) → 0(0(4(1(0(5(x1))))))
0(5(1(0(1(x1))))) → 2(1(0(1(5(0(x1))))))
0(5(1(3(1(x1))))) → 2(5(0(1(3(1(x1))))))
0(5(1(3(1(x1))))) → 2(5(1(0(1(3(x1))))))
0(5(2(5(1(x1))))) → 2(0(2(5(5(1(x1))))))
0(5(2(5(1(x1))))) → 2(5(5(0(2(1(x1))))))
0(5(3(0(1(x1))))) → 0(0(3(2(5(1(x1))))))
0(5(3(0(1(x1))))) → 2(0(3(0(5(1(x1))))))
0(5(3(1(1(x1))))) → 1(1(3(0(2(5(x1))))))
0(5(3(1(1(x1))))) → 1(2(5(1(3(0(x1))))))
0(5(4(1(1(x1))))) → 2(5(0(4(1(1(x1))))))
5(0(1(0(1(x1))))) → 5(1(2(1(0(0(x1))))))
5(5(4(1(1(x1))))) → 1(4(5(2(5(1(x1))))))
5(5(4(1(1(x1))))) → 2(5(1(4(5(1(x1))))))

Rewrite Strategy: INNERMOST

(1) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.
The certificate found is represented by the following graph.

Start state: 8209
Accept states: [8210, 8211]
Transitions:
8209→8210[0_1|0]
8209→8211[5_1|0]
8209→8209[1_1|0, 2_1|0, 3_1|0, 4_1|0]
8209→8212[1_1|1]
8209→8217[4_1|1]
8212→8213[0_1|1]
8213→8214[4_1|1]
8214→8215[1_1|1]
8215→8216[2_1|1]
8216→8210[1_1|1]
8216→8213[1_1|1]
8217→8218[0_1|1]
8218→8219[4_1|1]
8219→8220[1_1|1]
8220→8221[2_1|1]
8221→8210[1_1|1]
8221→8218[1_1|1]

(2) BOUNDS(O(1), O(n^1))