### (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(s(a), s(b), x) → f(x, x, x)
g(f(s(x), s(y), z)) → g(f(x, y, z))
cons(x, y) → x
cons(x, y) → y

Rewrite Strategy: FULL

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
g(f(s(x), s(y), z)) → g(f(x, y, z))

### (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:

cons(x, y) → x
cons(x, y) → y
f(s(a), s(b), x) → f(x, x, 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:

cons(x, y) → x
cons(x, y) → y
f(s(a), s(b), x) → f(x, x, x)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cons(z0, z1) → z0
cons(z0, z1) → z1
f(s(a), s(b), z0) → f(z0, z0, z0)
Tuples:

CONS(z0, z1) → c
CONS(z0, z1) → c1
F(s(a), s(b), z0) → c2(F(z0, z0, z0))
S tuples:

CONS(z0, z1) → c
CONS(z0, z1) → c1
F(s(a), s(b), z0) → c2(F(z0, z0, z0))
K tuples:none
Defined Rule Symbols:

cons, f

Defined Pair Symbols:

CONS, F

Compound Symbols:

c, c1, c2

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

Removed 3 trailing nodes:

F(s(a), s(b), z0) → c2(F(z0, z0, z0))
CONS(z0, z1) → c
CONS(z0, z1) → c1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

cons(z0, z1) → z0
cons(z0, z1) → z1
f(s(a), s(b), z0) → f(z0, z0, z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

cons, f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty