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

(0) Obligation:

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

1(2(1(x1))) → 2(0(2(x1)))
0(2(1(x1))) → 1(0(2(x1)))
L(2(1(x1))) → L(1(0(2(x1))))
1(2(0(x1))) → 2(0(1(x1)))
1(2(R(x1))) → 2(0(1(R(x1))))
0(2(0(x1))) → 1(0(1(x1)))
L(2(0(x1))) → L(1(0(1(x1))))
0(2(R(x1))) → 1(0(1(R(x1))))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

1(2(1(z0))) → 2(0(2(z0)))
1(2(0(z0))) → 2(0(1(z0)))
1(2(R(z0))) → 2(0(1(R(z0))))
0(2(1(z0))) → 1(0(2(z0)))
0(2(0(z0))) → 1(0(1(z0)))
0(2(R(z0))) → 1(0(1(R(z0))))
L(2(1(z0))) → L(1(0(2(z0))))
L(2(0(z0))) → L(1(0(1(z0))))

Tuples:

1'(2(1(z0))) → c(0'(2(z0)))
1'(2(0(z0))) → c1(0'(1(z0)), 1'(z0))
1'(2(R(z0))) → c2(0'(1(R(z0))), 1'(R(z0)))
0'(2(1(z0))) → c3(1'(0(2(z0))), 0'(2(z0)))
0'(2(0(z0))) → c4(1'(0(1(z0))), 0'(1(z0)), 1'(z0))
0'(2(R(z0))) → c5(1'(0(1(R(z0)))), 0'(1(R(z0))), 1'(R(z0)))
L'(2(1(z0))) → c6(L'(1(0(2(z0)))), 1'(0(2(z0))), 0'(2(z0)))
L'(2(0(z0))) → c7(L'(1(0(1(z0)))), 1'(0(1(z0))), 0'(1(z0)), 1'(z0))

S tuples:

1'(2(1(z0))) → c(0'(2(z0)))
1'(2(0(z0))) → c1(0'(1(z0)), 1'(z0))
1'(2(R(z0))) → c2(0'(1(R(z0))), 1'(R(z0)))
0'(2(1(z0))) → c3(1'(0(2(z0))), 0'(2(z0)))
0'(2(0(z0))) → c4(1'(0(1(z0))), 0'(1(z0)), 1'(z0))
0'(2(R(z0))) → c5(1'(0(1(R(z0)))), 0'(1(R(z0))), 1'(R(z0)))
L'(2(1(z0))) → c6(L'(1(0(2(z0)))), 1'(0(2(z0))), 0'(2(z0)))
L'(2(0(z0))) → c7(L'(1(0(1(z0)))), 1'(0(1(z0))), 0'(1(z0)), 1'(z0))

K tuples:none
Defined Rule Symbols:

1, 0, L

Defined Pair Symbols:

1', 0', L'

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

1'(2(1(z0))) → c(0'(2(z0)))
1'(2(0(z0))) → c1(0'(1(z0)), 1'(z0))
0'(2(1(z0))) → c3(1'(0(2(z0))), 0'(2(z0)))
0'(2(0(z0))) → c4(1'(0(1(z0))), 0'(1(z0)), 1'(z0))
L'(2(1(z0))) → c6(L'(1(0(2(z0)))), 1'(0(2(z0))), 0'(2(z0)))
L'(2(0(z0))) → c7(L'(1(0(1(z0)))), 1'(0(1(z0))), 0'(1(z0)), 1'(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

1(2(1(z0))) → 2(0(2(z0)))
1(2(0(z0))) → 2(0(1(z0)))
1(2(R(z0))) → 2(0(1(R(z0))))
0(2(1(z0))) → 1(0(2(z0)))
0(2(0(z0))) → 1(0(1(z0)))
0(2(R(z0))) → 1(0(1(R(z0))))
L(2(1(z0))) → L(1(0(2(z0))))
L(2(0(z0))) → L(1(0(1(z0))))

Tuples:

1'(2(R(z0))) → c2(0'(1(R(z0))), 1'(R(z0)))
0'(2(R(z0))) → c5(1'(0(1(R(z0)))), 0'(1(R(z0))), 1'(R(z0)))

S tuples:

1'(2(R(z0))) → c2(0'(1(R(z0))), 1'(R(z0)))
0'(2(R(z0))) → c5(1'(0(1(R(z0)))), 0'(1(R(z0))), 1'(R(z0)))

K tuples:none
Defined Rule Symbols:

1, 0, L

Defined Pair Symbols:

1', 0'

Compound Symbols:

c2, c5

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 2 dangling nodes:

1'(2(R(z0))) → c2(0'(1(R(z0))), 1'(R(z0)))
0'(2(R(z0))) → c5(1'(0(1(R(z0)))), 0'(1(R(z0))), 1'(R(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

1(2(1(z0))) → 2(0(2(z0)))
1(2(0(z0))) → 2(0(1(z0)))
1(2(R(z0))) → 2(0(1(R(z0))))
0(2(1(z0))) → 1(0(2(z0)))
0(2(0(z0))) → 1(0(1(z0)))
0(2(R(z0))) → 1(0(1(R(z0))))
L(2(1(z0))) → L(1(0(2(z0))))
L(2(0(z0))) → L(1(0(1(z0))))

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

1, 0, L

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))