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:

:(:(:(:(C, x), y), z), u) → :(:(x, z), :(:(:(x, y), z), u))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

:(:(:(:(C, z0), z1), z2), z3) → :(:(z0, z2), :(:(:(z0, z1), z2), z3))

Tuples:

:'(:(:(:(C, z0), z1), z2), z3) → c(:'(:(z0, z2), :(:(:(z0, z1), z2), z3)), :'(z0, z2), :'(:(:(z0, z1), z2), z3), :'(:(z0, z1), z2), :'(z0, z1))

S tuples:

:'(:(:(:(C, z0), z1), z2), z3) → c(:'(:(z0, z2), :(:(:(z0, z1), z2), z3)), :'(z0, z2), :'(:(:(z0, z1), z2), z3), :'(:(z0, z1), z2), :'(z0, z1))

K tuples:none
Defined Rule Symbols:

:

Defined Pair Symbols:

:'

Compound Symbols:

c

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

:'(:(:(:(C, z0), z1), z2), z3) → c(:'(:(z0, z2), :(:(:(z0, z1), z2), z3)), :'(z0, z2), :'(:(:(z0, z1), z2), z3), :'(:(z0, z1), z2), :'(z0, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

:(:(:(:(C, z0), z1), z2), z3) → :(:(z0, z2), :(:(:(z0, z1), z2), z3))

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

:

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))