(0) Obligation:

Runtime Complexity TRS:
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

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) 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

(3) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(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))
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

(5) 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)))

(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, 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:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

(7) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 4 dangling nodes:

F(0, 1, 0) → c(F(h(0), 0, 0))

(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, 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:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

(9) 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)))

(10) 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, 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:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

(11) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 3 dangling nodes:

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

(12) 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, 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:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

(13) 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))

(14) 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, 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:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

(15) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 2 dangling nodes:

F(0, 1, 0) → c(F(0, 0, 0))

(16) 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, 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:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

(17) 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)))

(18) 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, 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:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

(19) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

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

(20) 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:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:none

Compound Symbols:none

(21) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(22) BOUNDS(O(1), O(1))