(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))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
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))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
p(s(0))
p(s(s(x0)))
tower(x0, x1)
towerIter(0, x0, x1)
towerIter(s(x0), x1, x2)
(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))
EXP(x, s(y)) → TIMES(x, exp(x, y))
EXP(x, s(y)) → EXP(x, y)
P(s(s(x))) → P(s(x))
TOWER(x, y) → TOWERITER(x, y, s(0))
TOWERITER(s(x), y, z) → TOWERITER(p(s(x)), y, exp(y, z))
TOWERITER(s(x), y, z) → P(s(x))
TOWERITER(s(x), y, z) → EXP(y, z)
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))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
p(s(0))
p(s(s(x0)))
tower(x0, x1)
towerIter(0, x0, x1)
towerIter(s(x0), x1, x2)
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 7 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))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
p(s(0))
p(s(s(x0)))
tower(x0, x1)
towerIter(0, x0, x1)
towerIter(s(x0), x1, x2)
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))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
p(s(0))
p(s(s(x0)))
tower(x0, x1)
towerIter(0, x0, x1)
towerIter(s(x0), x1, x2)
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))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
p(s(0))
p(s(s(x0)))
tower(x0, x1)
towerIter(0, x0, x1)
towerIter(s(x0), x1, x2)
We have to consider all minimal (P,Q,R)-chains.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
EXP(x, s(y)) → EXP(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))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
p(s(0))
p(s(s(x0)))
tower(x0, x1)
towerIter(0, x0, x1)
towerIter(s(x0), x1, x2)
We have to consider all minimal (P,Q,R)-chains.
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOWERITER(s(x), y, z) → TOWERITER(p(s(x)), y, exp(y, z))
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))
exp(x, 0) → s(0)
exp(x, s(y)) → times(x, exp(x, y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
tower(x, y) → towerIter(x, y, s(0))
towerIter(0, y, z) → z
towerIter(s(x), y, z) → towerIter(p(s(x)), y, exp(y, z))
The set Q consists of the following terms:
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
exp(x0, 0)
exp(x0, s(x1))
p(s(0))
p(s(s(x0)))
tower(x0, x1)
towerIter(0, x0, x1)
towerIter(s(x0), x1, x2)
We have to consider all minimal (P,Q,R)-chains.