### (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(x, x) → f(g(x), x)
g(x) → s(x)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, 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(x, x) → f(g(x), x)
g(x) → s(x)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)
Tuples:

F(z0, z0) → c(F(g(z0), z0), G(z0))
G(z0) → c1
S tuples:

F(z0, z0) → c(F(g(z0), z0), G(z0))
G(z0) → c1
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1

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

Removed 1 trailing nodes:

G(z0) → c1

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)
Tuples:

F(z0, z0) → c(F(g(z0), z0), G(z0))
S tuples:

F(z0, z0) → c(F(g(z0), z0), G(z0))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing tuple parts

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)
Tuples:

F(z0, z0) → c(F(g(z0), z0))
S tuples:

F(z0, z0) → c(F(g(z0), z0))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F

Compound Symbols:

c

### (9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(z0, z0) → f(g(z0), z0)

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0) → s(z0)
Tuples:

F(z0, z0) → c(F(g(z0), z0))
S tuples:

F(z0, z0) → c(F(g(z0), z0))
K tuples:none
Defined Rule Symbols:

g

Defined Pair Symbols:

F

Compound Symbols:

c

### (11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(z0, z0) → c(F(g(z0), z0)) by

F(z0, z0) → c(F(s(z0), z0))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0) → s(z0)
Tuples:

F(z0, z0) → c(F(s(z0), z0))
S tuples:

F(z0, z0) → c(F(s(z0), z0))
K tuples:none
Defined Rule Symbols:

g

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(z0, z0) → c(F(s(z0), z0))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0) → s(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

g

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty