The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳4 CdtProblem
↳5 CdtUnreachableProof (⇔)
↳6 CdtProblem
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳8 CdtProblem
↳9 SIsEmptyProof (⇔)
↳10 BOUNDS(O(1), O(1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

h(z, e(x)) → h(c(z), d(z, x))
d(z, g(0, 0)) → e(0)
d(z, g(x, y)) → g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) → g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) → e(g(x, y))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, e(z1)) → h(c(z0), d(z0, z1))
d(z0, g(0, 0)) → e(0)
d(z0, g(z1, z2)) → g(e(z1), d(z0, z2))
d(c(z0), g(g(z1, z2), 0)) → g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

H(z0, e(z1)) → c1(H(c(z0), d(z0, z1)), D(z0, z1))
D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2))
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2))
G(e(z0), e(z1)) → c5(G(z0, z1))
S tuples:

H(z0, e(z1)) → c1(H(c(z0), d(z0, z1)), D(z0, z1))
D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2))
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2))
G(e(z0), e(z1)) → c5(G(z0, z1))
K tuples:none
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

H, D, G

Compound Symbols:

c1, c3, c4, c5

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

Use narrowing to replace H(z0, e(z1)) → c1(H(c(z0), d(z0, z1)), D(z0, z1)) by

H(z0, e(g(0, 0))) → c1(H(c(z0), e(0)), D(z0, g(0, 0)))
H(z0, e(g(z1, z2))) → c1(H(c(z0), g(e(z1), d(z0, z2))), D(z0, g(z1, z2)))
H(c(z0), e(g(g(z1, z2), 0))) → c1(H(c(c(z0)), g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))), D(c(z0), g(g(z1, z2), 0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, e(z1)) → h(c(z0), d(z0, z1))
d(z0, g(0, 0)) → e(0)
d(z0, g(z1, z2)) → g(e(z1), d(z0, z2))
d(c(z0), g(g(z1, z2), 0)) → g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2))
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2))
G(e(z0), e(z1)) → c5(G(z0, z1))
H(z0, e(g(0, 0))) → c1(H(c(z0), e(0)), D(z0, g(0, 0)))
H(z0, e(g(z1, z2))) → c1(H(c(z0), g(e(z1), d(z0, z2))), D(z0, g(z1, z2)))
H(c(z0), e(g(g(z1, z2), 0))) → c1(H(c(c(z0)), g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))), D(c(z0), g(g(z1, z2), 0)))
S tuples:

D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2))
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2))
G(e(z0), e(z1)) → c5(G(z0, z1))
H(z0, e(g(0, 0))) → c1(H(c(z0), e(0)), D(z0, g(0, 0)))
H(z0, e(g(z1, z2))) → c1(H(c(z0), g(e(z1), d(z0, z2))), D(z0, g(z1, z2)))
H(c(z0), e(g(g(z1, z2), 0))) → c1(H(c(c(z0)), g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))), D(c(z0), g(g(z1, z2), 0)))
K tuples:none
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

D, G, H

Compound Symbols:

c3, c4, c5, c1

(5) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

D(z0, g(z1, z2)) → c3(G(e(z1), d(z0, z2)), D(z0, z2))
D(c(z0), g(g(z1, z2), 0)) → c4(G(d(c(z0), g(z1, z2)), d(z0, g(z1, z2))), D(c(z0), g(z1, z2)), G(z1, z2), D(z0, g(z1, z2)), G(z1, z2))
H(z0, e(g(0, 0))) → c1(H(c(z0), e(0)), D(z0, g(0, 0)))
H(z0, e(g(z1, z2))) → c1(H(c(z0), g(e(z1), d(z0, z2))), D(z0, g(z1, z2)))
H(c(z0), e(g(g(z1, z2), 0))) → c1(H(c(c(z0)), g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))), D(c(z0), g(g(z1, z2), 0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, e(z1)) → h(c(z0), d(z0, z1))
d(z0, g(0, 0)) → e(0)
d(z0, g(z1, z2)) → g(e(z1), d(z0, z2))
d(c(z0), g(g(z1, z2), 0)) → g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
S tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
K tuples:none
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

G

Compound Symbols:

c5

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

G(e(z0), e(z1)) → c5(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(G(x1, x2)) = [2]x1   
POL(c5(x1)) = x1   
POL(e(x1)) = [1] + x1   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, e(z1)) → h(c(z0), d(z0, z1))
d(z0, g(0, 0)) → e(0)
d(z0, g(z1, z2)) → g(e(z1), d(z0, z2))
d(c(z0), g(g(z1, z2), 0)) → g(d(c(z0), g(z1, z2)), d(z0, g(z1, z2)))
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
S tuples:none
K tuples:

G(e(z0), e(z1)) → c5(G(z0, z1))
Defined Rule Symbols:

h, d, g

Defined Pair Symbols:

G

Compound Symbols:

c5

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))