(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
Rewrite Strategy: INNERMOST
(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:
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
reverse(nil) → nil
reverse(add(n, x)) → app(reverse(x), add(n, nil))
shuffle(nil) → nil
shuffle(add(n, x)) → add(n, shuffle(reverse(x)))
S is empty.
Rewrite Strategy: INNERMOST
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
add/0
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
app(nil, y) → y
app(add(x), y) → add(app(x, y))
reverse(nil) → nil
reverse(add(x)) → app(reverse(x), add(nil))
shuffle(nil) → nil
shuffle(add(x)) → add(shuffle(reverse(x)))
S is empty.
Rewrite Strategy: INNERMOST
(5) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
app(add(x), y) →+ add(app(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / add(x)].
The result substitution is [ ].
(6) BOUNDS(n^1, INF)