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

(0) Obligation:

The Runtime Complexity (full) 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: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

(2) 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

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

Converted Cpx (relative) TRS to CDT

(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:

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

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

Removed 3 trailing nodes:

GCD(s(z0), s(z1)) → c2(GCD(s(z0), -(z1, z0)), GCD(-(z0, z1), s(z1)))
GCD(0, z0) → c1
GCD(z0, 0) → c

(6) 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

(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(8) BOUNDS(1, 1)