(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
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 Cpx (relative) TRS 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(z0, 0) → c
GCD(0, z0) → c1
GCD(s(z0), s(z1)) → c2(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))
S tuples:
GCD(z0, 0) → c
GCD(0, z0) → c1
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:
c, c1, c2
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
GCD(z0, 0) → c
GCD(s(z0), s(z1)) → c2(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))
GCD(0, z0) → c1
(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 (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)