(0) Obligation:

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

f(g(x), g(y)) → f(p(f(g(x), s(y))), g(s(p(x))))
p(0) → g(0)
g(s(p(x))) → p(x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(g(z0), g(z1)) → f(p(f(g(z0), s(z1))), g(s(p(z0))))
p(0) → g(0)
g(s(p(z0))) → p(z0)
Tuples:

F(g(z0), g(z1)) → c(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
P(0) → c1(G(0))
G(s(p(z0))) → c2(P(z0))
S tuples:

F(g(z0), g(z1)) → c(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
P(0) → c1(G(0))
G(s(p(z0))) → c2(P(z0))
K tuples:none
Defined Rule Symbols:

f, p, g

Defined Pair Symbols:

F, P, G

Compound Symbols:

c, c1, c2

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(g(z0), g(z1)) → c(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
G(s(p(z0))) → c2(P(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(g(z0), g(z1)) → f(p(f(g(z0), s(z1))), g(s(p(z0))))
p(0) → g(0)
g(s(p(z0))) → p(z0)
Tuples:

P(0) → c1(G(0))
S tuples:

P(0) → c1(G(0))
K tuples:none
Defined Rule Symbols:

f, p, g

Defined Pair Symbols:

P

Compound Symbols:

c1

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

P(0) → c1(G(0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(g(z0), g(z1)) → f(p(f(g(z0), s(z1))), g(s(p(z0))))
p(0) → g(0)
g(s(p(z0))) → p(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, p, g

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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