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

(0) Obligation:

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

rw(Val(n), c) → Op(Val(n), rewrite(c))
rewrite(Op(x, y)) → rw(x, y)
rw(Op(x, y), c) → rw[Let](Op(x, y), c, rewrite(x))
rewrite(Val(n)) → Val(n)
second(Op(x, y)) → y
isOp(Val(n)) → False
isOp(Op(x, y)) → True
first(Val(n)) → Val(n)
first(Op(x, y)) → x
assrewrite(exp) → rewrite(exp)

The (relative) TRS S consists of the following rules:

rw[Let](Op(x, y), c, a1) → rw[Let][Let](Op(x, y), c, a1, rewrite(y))
rw[Let][Let](ab, c, a1, b1) → rw[Let][Let][Let](c, a1, b1, rewrite(c))
rw[Let][Let][Let](c, a1, b1, c1) → rw(a1, Op(b1, c1))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
rewrite(Op(Val(n59683_0), y)) →⁺ Op(Val(n59683_0), rewrite(y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [y / Op(Val(n59683_0), y)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)