(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
fac(0, x) → x
fac(s(x), y) → fac(p(s(x)), times(s(x), y))
factorial(x) → fac(x, s(0))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
fac(0, x) → x
fac(s(x), y) → fac(p(s(x)), times(s(x), y))
factorial(x) → fac(x, s(0))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(0))
p(s(s(x0)))
fac(0, x0)
fac(s(x0), x1)
factorial(x0)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(s(x), y) → PLUS(p(s(x)), y)
PLUS(s(x), y) → P(s(x))
TIMES(s(x), y) → PLUS(y, times(p(s(x)), y))
TIMES(s(x), y) → TIMES(p(s(x)), y)
TIMES(s(x), y) → P(s(x))
P(s(s(x))) → P(s(x))
FAC(s(x), y) → FAC(p(s(x)), times(s(x), y))
FAC(s(x), y) → P(s(x))
FAC(s(x), y) → TIMES(s(x), y)
FACTORIAL(x) → FAC(x, s(0))
The TRS R consists of the following rules:
plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
fac(0, x) → x
fac(s(x), y) → fac(p(s(x)), times(s(x), y))
factorial(x) → fac(x, s(0))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(0))
p(s(s(x0)))
fac(0, x0)
fac(s(x0), x1)
factorial(x0)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 6 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
P(s(s(x))) → P(s(x))
The TRS R consists of the following rules:
plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
fac(0, x) → x
fac(s(x), y) → fac(p(s(x)), times(s(x), y))
factorial(x) → fac(x, s(0))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(0))
p(s(s(x0)))
fac(0, x0)
fac(s(x0), x1)
factorial(x0)
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(s(x), y) → PLUS(p(s(x)), y)
The TRS R consists of the following rules:
plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
fac(0, x) → x
fac(s(x), y) → fac(p(s(x)), times(s(x), y))
factorial(x) → fac(x, s(0))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(0))
p(s(s(x0)))
fac(0, x0)
fac(s(x0), x1)
factorial(x0)
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TIMES(s(x), y) → TIMES(p(s(x)), y)
The TRS R consists of the following rules:
plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
fac(0, x) → x
fac(s(x), y) → fac(p(s(x)), times(s(x), y))
factorial(x) → fac(x, s(0))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(0))
p(s(s(x0)))
fac(0, x0)
fac(s(x0), x1)
factorial(x0)
We have to consider all minimal (P,Q,R)-chains.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
FAC(s(x), y) → FAC(p(s(x)), times(s(x), y))
The TRS R consists of the following rules:
plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
fac(0, x) → x
fac(s(x), y) → fac(p(s(x)), times(s(x), y))
factorial(x) → fac(x, s(0))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(0))
p(s(s(x0)))
fac(0, x0)
fac(s(x0), x1)
factorial(x0)
We have to consider all minimal (P,Q,R)-chains.