### (0) Obligation:

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

The TRS R consists of the following rules:

i(x, x) → i(a, b)
g(x, x) → g(a, b)
h(s(f(x))) → h(f(x))
f(s(x)) → s(s(f(h(s(x)))))
f(g(s(x), y)) → f(g(x, s(y)))
h(g(x, s(y))) → h(g(s(x), y))
h(i(x, y)) → i(i(c, h(h(y))), x)
g(a, g(x, g(b, g(a, g(x, y))))) → g(a, g(a, g(a, g(x, g(b, g(b, y))))))

Rewrite Strategy: FULL

### (1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of f: h
The following defined symbols can occur below the 0th argument of h: h

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
h(s(f(x))) → h(f(x))
f(g(s(x), y)) → f(g(x, s(y)))
h(g(x, s(y))) → h(g(s(x), y))
h(i(x, y)) → i(i(c, h(h(y))), x)
g(a, g(x, g(b, g(a, g(x, y))))) → g(a, g(a, g(a, g(x, g(b, g(b, y))))))

### (2) Obligation:

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

The TRS R consists of the following rules:

g(x, x) → g(a, b)
i(x, x) → i(a, b)
f(s(x)) → s(s(f(h(s(x)))))

Rewrite Strategy: FULL

### (3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

### (4) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).

The TRS R consists of the following rules:

g(x, x) → g(a, b)
i(x, x) → i(a, b)
f(s(x)) → s(s(f(h(s(x)))))

Rewrite Strategy: INNERMOST

### (5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0, z0) → g(a, b)
i(z0, z0) → i(a, b)
f(s(z0)) → s(s(f(h(s(z0)))))
Tuples:

G(z0, z0) → c(G(a, b))
I(z0, z0) → c1(I(a, b))
F(s(z0)) → c2(F(h(s(z0))))
S tuples:

G(z0, z0) → c(G(a, b))
I(z0, z0) → c1(I(a, b))
F(s(z0)) → c2(F(h(s(z0))))
K tuples:none
Defined Rule Symbols:

g, i, f

Defined Pair Symbols:

G, I, F

Compound Symbols:

c, c1, c2

### (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

F(s(z0)) → c2(F(h(s(z0))))
G(z0, z0) → c(G(a, b))
I(z0, z0) → c1(I(a, b))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0, z0) → g(a, b)
i(z0, z0) → i(a, b)
f(s(z0)) → s(s(f(h(s(z0)))))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

g, i, f

Defined Pair Symbols:none

Compound Symbols:none

### (9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty