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

(0) Obligation:

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

h(X, Z) → f(X, s(X), Z)
f(X, Y, g(X, Y)) → h(0, g(X, Y))
g(0, Y) → 0
g(X, s(Y)) → 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, z1) → f(z0, s(z0), z1)
f(z0, z1, g(z0, z1)) → h(0, g(z0, z1))
g(0, z0) → 0
g(z0, s(z1)) → g(z0, z1)

Tuples:

H(z0, z1) → c(F(z0, s(z0), z1))
F(z0, z1, g(z0, z1)) → c1(H(0, g(z0, z1)), G(z0, z1))
G(z0, s(z1)) → c3(G(z0, z1))

S tuples:

H(z0, z1) → c(F(z0, s(z0), z1))
F(z0, z1, g(z0, z1)) → c1(H(0, g(z0, z1)), G(z0, z1))
G(z0, s(z1)) → c3(G(z0, z1))

K tuples:none
Defined Rule Symbols:

h, f, g

Defined Pair Symbols:

H, F, G

Compound Symbols:

c, c1, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(z0, z1, g(z0, z1)) → c1(H(0, g(z0, z1)), G(z0, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, z1) → f(z0, s(z0), z1)
f(z0, z1, g(z0, z1)) → h(0, g(z0, z1))
g(0, z0) → 0
g(z0, s(z1)) → g(z0, z1)

Tuples:

H(z0, z1) → c(F(z0, s(z0), z1))
G(z0, s(z1)) → c3(G(z0, z1))

S tuples:

H(z0, z1) → c(F(z0, s(z0), z1))
G(z0, s(z1)) → c3(G(z0, z1))

K tuples:none
Defined Rule Symbols:

h, f, g

Defined Pair Symbols:

H, G

Compound Symbols:

c, c3

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 2 dangling nodes:

H(z0, z1) → c(F(z0, s(z0), z1))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, z1) → f(z0, s(z0), z1)
f(z0, z1, g(z0, z1)) → h(0, g(z0, z1))
g(0, z0) → 0
g(z0, s(z1)) → g(z0, z1)

Tuples:

G(z0, s(z1)) → c3(G(z0, z1))

S tuples:

G(z0, s(z1)) → c3(G(z0, z1))

K tuples:none
Defined Rule Symbols:

h, f, g

Defined Pair Symbols:

G

Compound Symbols:

c3

(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(z0, s(z1)) → c3(G(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

G(z0, s(z1)) → c3(G(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(G(x₁, x₂)) = [5]x₂   
POL(c3(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(z0, z1) → f(z0, s(z0), z1)
f(z0, z1, g(z0, z1)) → h(0, g(z0, z1))
g(0, z0) → 0
g(z0, s(z1)) → g(z0, z1)

Tuples:

G(z0, s(z1)) → c3(G(z0, z1))

S tuples:none
K tuples:

G(z0, s(z1)) → c3(G(z0, z1))

Defined Rule Symbols:

h, f, g

Defined Pair Symbols:

G

Compound Symbols:

c3

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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