### (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:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))

f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

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

### (2) 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:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))

f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT
### (4) Obligation:

Complexity Dependency Tuples Problem

Rules:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))

f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

Tuples:

F(g(i(a, b, b'), c), d) → c1(F(.(b, c), d'), F(.(b', c), d'))

F(g(h(a, b), c), d) → c2(F(.(b, g(h(a, b), c)), d), F(c, d'))

S tuples:

F(g(i(a, b, b'), c), d) → c1(F(.(b, c), d'), F(.(b', c), d'))

F(g(h(a, b), c), d) → c2(F(.(b, g(h(a, b), c)), d), F(c, d'))

K tuples:none

Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2

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

Removed 2 trailing nodes:

F(g(h(a, b), c), d) → c2(F(.(b, g(h(a, b), c)), d), F(c, d'))

F(g(i(a, b, b'), c), d) → c1(F(.(b, c), d'), F(.(b', c), d'))

### (6) Obligation:

Complexity Dependency Tuples Problem

Rules:

f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))

f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))

Tuples:none

S tuples:none

K tuples:none

Defined Rule Symbols:

f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

### (8) BOUNDS(1, 1)