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

(0) Obligation:

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

gcd(x, 0) → x
gcd(0, y) → y
gcd(s(x), s(y)) → if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y)))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

gcd(z0, 0) → z0
gcd(0, z0) → z0
gcd(s(z0), s(z1)) → if(<(z0, z1), gcd(s(z0), -(z1, z0)), gcd(-(z0, z1), s(z1)))

Tuples:

GCD(s(z0), s(z1)) → c2(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))

S tuples:

GCD(s(z0), s(z1)) → c2(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))

K tuples:none
Defined Rule Symbols:

gcd

Defined Pair Symbols:

GCD

Compound Symbols:

c2

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

GCD(s(z0), s(z1)) → c2(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

gcd(z0, 0) → z0
gcd(0, z0) → z0
gcd(s(z0), s(z1)) → if(<(z0, z1), gcd(s(z0), -(z1, z0)), gcd(-(z0, z1), s(z1)))

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

gcd

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))