(0) Obligation:

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

f(s(a), s(b), x) → f(x, x, x)
g(f(s(x), s(y), z)) → g(f(x, y, z))
cons(x, y) → x
cons(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(s(a), s(b), z0) → f(z0, z0, z0)
g(f(s(z0), s(z1), z2)) → g(f(z0, z1, z2))
cons(z0, z1) → z0
cons(z0, z1) → z1
Tuples:

F(s(a), s(b), z0) → c(F(z0, z0, z0))
G(f(s(z0), s(z1), z2)) → c1(G(f(z0, z1, z2)), F(z0, z1, z2))
S tuples:

F(s(a), s(b), z0) → c(F(z0, z0, z0))
G(f(s(z0), s(z1), z2)) → c1(G(f(z0, z1, z2)), F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

f, g, cons

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

G(f(s(z0), s(z1), z2)) → c1(G(f(z0, z1, z2)), F(z0, z1, z2))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(s(a), s(b), z0) → f(z0, z0, z0)
g(f(s(z0), s(z1), z2)) → g(f(z0, z1, z2))
cons(z0, z1) → z0
cons(z0, z1) → z1
Tuples:

F(s(a), s(b), z0) → c(F(z0, z0, z0))
S tuples:

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

f, g, cons

Defined Pair Symbols:

F

Compound Symbols:

c

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

F(s(a), s(b), z0) → c(F(z0, z0, z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(s(a), s(b), z0) → f(z0, z0, z0)
g(f(s(z0), s(z1), z2)) → g(f(z0, z1, z2))
cons(z0, z1) → z0
cons(z0, z1) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, g, cons

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))