### (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(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

The duplicating contexts are:
f(0, 1, [])

The defined contexts are:
f([], x1, x2)
f(x0, [], x2)

[] just represents basic- or constructor-terms in the following defined contexts:
f([], x1, x2)
f(x0, [], x2)

As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, 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(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
H(0) → c1
H(g(z0, z1)) → c2
S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
H(0) → c1
H(g(z0, z1)) → c2
K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F, H

Compound Symbols:

c, c1, c2

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

Removed 2 trailing nodes:

H(0) → c1
H(g(z0, z1)) → c2

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 2 trailing tuple parts

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))
S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))
K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

### (9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(0, 1, z0) → f(h(z0), h(z0), z0)

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))
S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(0, 1, z0) → c(F(h(z0), h(z0), z0)) by

F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, 0) → c(F(h(0), 0, 0))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1))) by

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
S tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(0, 1, 0) → c(F(0, h(0), 0)) by

F(0, 1, 0) → c(F(0, 0, 0))

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, 0) → c(F(0, 0, 0))
S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, 0) → c(F(0, 0, 0))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, 0) → c(F(0, 0, 0))

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

### (23) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

h(0) → 0

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1))) by

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(g(z0, z1)) → z1
Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(g(z0, z1)) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty