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:

f_0(x) → a
f_1(x) → g_1(x, x)
g_1(s(x), y) → b(f_0(y), g_1(x, y))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))
f_6(x) → g_6(x, x)
g_6(s(x), y) → b(f_5(y), g_6(x, y))
f_7(x) → g_7(x, x)
g_7(s(x), y) → b(f_6(y), g_7(x, y))
f_8(x) → g_8(x, x)
g_8(s(x), y) → b(f_7(y), g_8(x, y))
f_9(x) → g_9(x, x)
g_9(s(x), y) → b(f_8(y), g_9(x, y))
f_10(x) → g_10(x, x)
g_10(s(x), y) → b(f_9(y), g_10(x, y))

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:

f_0(x) → a
f_1(x) → g_1(x, x)
g_1(s(x), y) → b(f_0(y), g_1(x, y))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))
f_6(x) → g_6(x, x)
g_6(s(x), y) → b(f_5(y), g_6(x, y))
f_7(x) → g_7(x, x)
g_7(s(x), y) → b(f_6(y), g_7(x, y))
f_8(x) → g_8(x, x)
g_8(s(x), y) → b(f_7(y), g_8(x, y))
f_9(x) → g_9(x, x)
g_9(s(x), y) → b(f_8(y), g_9(x, y))
f_10(x) → g_10(x, x)
g_10(s(x), y) → b(f_9(y), g_10(x, y))

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
f_0/0
g_1/1

(4) Obligation:

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

f_0 → a
f_1(x) → g_1(x)
g_1(s(x)) → b(f_0, g_1(x))
f_2(x) → g_2(x, x)
g_2(s(x), y) → b(f_1(y), g_2(x, y))
f_3(x) → g_3(x, x)
g_3(s(x), y) → b(f_2(y), g_3(x, y))
f_4(x) → g_4(x, x)
g_4(s(x), y) → b(f_3(y), g_4(x, y))
f_5(x) → g_5(x, x)
g_5(s(x), y) → b(f_4(y), g_5(x, y))
f_6(x) → g_6(x, x)
g_6(s(x), y) → b(f_5(y), g_6(x, y))
f_7(x) → g_7(x, x)
g_7(s(x), y) → b(f_6(y), g_7(x, y))
f_8(x) → g_8(x, x)
g_8(s(x), y) → b(f_7(y), g_8(x, y))
f_9(x) → g_9(x, x)
g_9(s(x), y) → b(f_8(y), g_9(x, y))
f_10(x) → g_10(x, x)
g_10(s(x), y) → b(f_9(y), g_10(x, y))

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
g_1(s(x)) →⁺ b(f_0, g_1(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / s(x)].
The result substitution is [ ].

(6) BOUNDS(n^1, INF)