(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
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:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
The set Q consists of the following terms:
log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))
(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:
LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)
LOG(x, s(s(y))) → LE(x, s(s(y)))
COND(false, x, y) → DOUBLE(log(x, square(s(s(y)))))
COND(false, x, y) → LOG(x, square(s(s(y))))
COND(false, x, y) → SQUARE(s(s(y)))
LE(s(u), s(v)) → LE(u, v)
DOUBLE(s(x)) → DOUBLE(x)
SQUARE(s(x)) → PLUS(square(x), double(x))
SQUARE(s(x)) → SQUARE(x)
SQUARE(s(x)) → DOUBLE(x)
PLUS(n, s(m)) → PLUS(n, m)
The TRS R consists of the following rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
The set Q consists of the following terms:
log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 5 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(n, s(m)) → PLUS(n, m)
The TRS R consists of the following rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
The set Q consists of the following terms:
log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
DOUBLE(s(x)) → DOUBLE(x)
The TRS R consists of the following rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
The set Q consists of the following terms:
log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SQUARE(s(x)) → SQUARE(x)
The TRS R consists of the following rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
The set Q consists of the following terms:
log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LE(s(u), s(v)) → LE(u, v)
The TRS R consists of the following rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
The set Q consists of the following terms:
log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
COND(false, x, y) → LOG(x, square(s(s(y))))
LOG(x, s(s(y))) → COND(le(x, s(s(y))), x, y)
The TRS R consists of the following rules:
log(x, s(s(y))) → cond(le(x, s(s(y))), x, y)
cond(true, x, y) → s(0)
cond(false, x, y) → double(log(x, square(s(s(y)))))
le(0, v) → true
le(s(u), 0) → false
le(s(u), s(v)) → le(u, v)
double(0) → 0
double(s(x)) → s(s(double(x)))
square(0) → 0
square(s(x)) → s(plus(square(x), double(x)))
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))
The set Q consists of the following terms:
log(x0, s(s(x1)))
cond(true, x0, x1)
cond(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
double(0)
double(s(x0))
square(0)
square(s(x0))
plus(x0, 0)
plus(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.