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

(0) Obligation:

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

or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(x, nil) → false
mem(x, set(y)) → =(x, y)
mem(x, union(y, z)) → or(mem(x, y), mem(x, z))

Rewrite Strategy: FULL

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

Transformed TRS to relative TRS where S is empty.

(2) Obligation:

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

or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(x, nil) → false
mem(x, set(y)) → =(x, y)
mem(x, union(y, z)) → or(mem(x, y), mem(x, z))

S is empty.
Rewrite Strategy: FULL

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
mem/0
set/0
=/0
=/1

(4) Obligation:

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

or(true, y) → true
or(x, true) → true
or(false, false) → false
mem(nil) → false
mem(set) → =
mem(union(y, z)) → or(mem(y), mem(z))

S is empty.
Rewrite Strategy: FULL

(5) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
mem(union(y, z)) →⁺ or(mem(y), mem(z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [y / union(y, z)].
The result substitution is [ ].

(6) BOUNDS(n^1, INF)