(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
minus(x, 0) → x
minus(s(x), s(y)) → s(minus(x, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(x)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MIN(s(x), s(y)) → MIN(x, y)
MAX(s(x), s(y)) → MAX(x, y)
MINUS(s(x), s(y)) → MINUS(x, y)
GCD(s(x), s(y)) → GCD(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
GCD(s(x), s(y)) → MINUS(max(x, y), min(x, transform(y)))
GCD(s(x), s(y)) → MAX(x, y)
GCD(s(x), s(y)) → MIN(x, transform(y))
GCD(s(x), s(y)) → TRANSFORM(y)
GCD(s(x), s(y)) → MIN(x, y)
TRANSFORM(cons(x, y)) → CONS(cons(x, x), x)
TRANSFORM(cons(x, y)) → CONS(x, x)
TRANSFORM(s(x)) → TRANSFORM(x)
CONS(x, cons(y, s(z))) → CONS(y, x)
CONS(cons(x, z), s(y)) → TRANSFORM(x)
The TRS R consists of the following rules:
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
minus(x, 0) → x
minus(s(x), s(y)) → s(minus(x, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 5 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
CONS(x, cons(y, s(z))) → CONS(y, x)
CONS(cons(x, z), s(y)) → TRANSFORM(x)
TRANSFORM(cons(x, y)) → CONS(cons(x, x), x)
TRANSFORM(cons(x, y)) → CONS(x, x)
TRANSFORM(s(x)) → TRANSFORM(x)
The TRS R consists of the following rules:
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
minus(x, 0) → x
minus(s(x), s(y)) → s(minus(x, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MINUS(s(x), s(y)) → MINUS(x, y)
The TRS R consists of the following rules:
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
minus(x, 0) → x
minus(s(x), s(y)) → s(minus(x, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MAX(s(x), s(y)) → MAX(x, y)
The TRS R consists of the following rules:
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
minus(x, 0) → x
minus(s(x), s(y)) → s(minus(x, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MIN(s(x), s(y)) → MIN(x, y)
The TRS R consists of the following rules:
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
minus(x, 0) → x
minus(s(x), s(y)) → s(minus(x, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
GCD(s(x), s(y)) → GCD(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
The TRS R consists of the following rules:
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
minus(x, 0) → x
minus(s(x), s(y)) → s(minus(x, y))
gcd(s(x), s(y)) → gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y)))
transform(x) → s(s(x))
transform(cons(x, y)) → cons(cons(x, x), x)
transform(cons(x, y)) → y
transform(s(x)) → s(s(transform(x)))
cons(x, y) → y
cons(x, cons(y, s(z))) → cons(y, x)
cons(cons(x, z), s(y)) → transform(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.