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 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳4 CdtProblem
↳5 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳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:

f(0, n) → g(0, n)
f(m, 0) → g(m, 0)
f(s(m), s(n)) → h(m, n, f(m, p(m, n)), f(s(m), n))
g(n, m) → bot
p(m, n) → bot
h(k, l, m, n) → bot

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, z0) → g(0, z0)
f(z0, 0) → g(z0, 0)
f(s(z0), s(z1)) → h(z0, z1, f(z0, p(z0, z1)), f(s(z0), z1))
g(z0, z1) → bot
p(z0, z1) → bot
h(z0, z1, z2, z3) → bot
Tuples:

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

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

f, g, p, h

Defined Pair Symbols:

F

Compound Symbols:

c, c1, c2

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 3 dangling nodes:

F(0, z0) → c(G(0, z0))
F(z0, 0) → c1(G(z0, 0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, z0) → g(0, z0)
f(z0, 0) → g(z0, 0)
f(s(z0), s(z1)) → h(z0, z1, f(z0, p(z0, z1)), f(s(z0), z1))
g(z0, z1) → bot
p(z0, z1) → bot
h(z0, z1, z2, z3) → bot
Tuples:

F(s(z0), s(z1)) → c2(H(z0, z1, f(z0, p(z0, z1)), f(s(z0), z1)), F(z0, p(z0, z1)), P(z0, z1), F(s(z0), z1))
S tuples:

F(s(z0), s(z1)) → c2(H(z0, z1, f(z0, p(z0, z1)), f(s(z0), z1)), F(z0, p(z0, z1)), P(z0, z1), F(s(z0), z1))
K tuples:none
Defined Rule Symbols:

f, g, p, h

Defined Pair Symbols:

F

Compound Symbols:

c2

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, z0) → g(0, z0)
f(z0, 0) → g(z0, 0)
f(s(z0), s(z1)) → h(z0, z1, f(z0, p(z0, z1)), f(s(z0), z1))
g(z0, z1) → bot
p(z0, z1) → bot
h(z0, z1, z2, z3) → bot
Tuples:

F(s(z0), s(z1)) → c2(F(z0, p(z0, z1)), F(s(z0), z1))
S tuples:

F(s(z0), s(z1)) → c2(F(z0, p(z0, z1)), F(s(z0), z1))
K tuples:none
Defined Rule Symbols:

f, g, p, h

Defined Pair Symbols:

F

Compound Symbols:

c2

(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.

F(s(z0), s(z1)) → c2(F(z0, p(z0, z1)), F(s(z0), z1))
We considered the (Usable) Rules:

p(z0, z1) → bot
And the Tuples:

F(s(z0), s(z1)) → c2(F(z0, p(z0, z1)), F(s(z0), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = [1] + [2]x2   
POL(bot) = 0   
POL(c2(x1, x2)) = x1 + x2   
POL(p(x1, x2)) = 0   
POL(s(x1)) = [2] + x1   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, z0) → g(0, z0)
f(z0, 0) → g(z0, 0)
f(s(z0), s(z1)) → h(z0, z1, f(z0, p(z0, z1)), f(s(z0), z1))
g(z0, z1) → bot
p(z0, z1) → bot
h(z0, z1, z2, z3) → bot
Tuples:

F(s(z0), s(z1)) → c2(F(z0, p(z0, z1)), F(s(z0), z1))
S tuples:none
K tuples:

F(s(z0), s(z1)) → c2(F(z0, p(z0, z1)), F(s(z0), z1))
Defined Rule Symbols:

f, g, p, h

Defined Pair Symbols:

F

Compound Symbols:

c2

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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