double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
↳ QTRS
↳ DependencyPairsProof
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
PERMUTE3(x, y, a) -> ISZERO1(x)
PLUS2(x, s1(s1(y))) -> PLUS2(s1(x), y)
PLUS2(s1(x), y) -> PLUS2(x, s1(y))
PERMUTE3(false, x, b) -> PERMUTE3(ack2(x, x), p1(x), c)
ACK2(s1(x), 0) -> ACK2(x, s1(0))
ACK2(0, x) -> PLUS2(x, s1(0))
PERMUTE3(x, y, a) -> PERMUTE3(isZero1(x), x, b)
ACK2(s1(x), s1(y)) -> ACK2(s1(x), y)
DOUBLE1(x) -> PERMUTE3(x, x, a)
ACK2(s1(x), s1(y)) -> ACK2(x, ack2(s1(x), y))
PERMUTE3(false, x, b) -> ACK2(x, x)
PERMUTE3(y, x, c) -> PERMUTE3(x, y, a)
PERMUTE3(false, x, b) -> P1(x)
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
PERMUTE3(x, y, a) -> ISZERO1(x)
PLUS2(x, s1(s1(y))) -> PLUS2(s1(x), y)
PLUS2(s1(x), y) -> PLUS2(x, s1(y))
PERMUTE3(false, x, b) -> PERMUTE3(ack2(x, x), p1(x), c)
ACK2(s1(x), 0) -> ACK2(x, s1(0))
ACK2(0, x) -> PLUS2(x, s1(0))
PERMUTE3(x, y, a) -> PERMUTE3(isZero1(x), x, b)
ACK2(s1(x), s1(y)) -> ACK2(s1(x), y)
DOUBLE1(x) -> PERMUTE3(x, x, a)
ACK2(s1(x), s1(y)) -> ACK2(x, ack2(s1(x), y))
PERMUTE3(false, x, b) -> ACK2(x, x)
PERMUTE3(y, x, c) -> PERMUTE3(x, y, a)
PERMUTE3(false, x, b) -> P1(x)
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
PLUS2(x, s1(s1(y))) -> PLUS2(s1(x), y)
PLUS2(s1(x), y) -> PLUS2(x, s1(y))
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PLUS2(s1(x), y) -> PLUS2(x, s1(y))
Used ordering: Polynomial interpretation [21]:
PLUS2(x, s1(s1(y))) -> PLUS2(s1(x), y)
POL(PLUS2(x1, x2)) = 2·x1 + x2
POL(s1(x1)) = 2 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
PLUS2(x, s1(s1(y))) -> PLUS2(s1(x), y)
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
PLUS2(x, s1(s1(y))) -> PLUS2(s1(x), y)
POL(PLUS2(x1, x2)) = x2
POL(s1(x1)) = 2 + 3·x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
ACK2(s1(x), s1(y)) -> ACK2(s1(x), y)
ACK2(s1(x), s1(y)) -> ACK2(x, ack2(s1(x), y))
ACK2(s1(x), 0) -> ACK2(x, s1(0))
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACK2(s1(x), s1(y)) -> ACK2(x, ack2(s1(x), y))
ACK2(s1(x), 0) -> ACK2(x, s1(0))
Used ordering: Polynomial interpretation [21]:
ACK2(s1(x), s1(y)) -> ACK2(s1(x), y)
POL(0) = 0
POL(ACK2(x1, x2)) = 3·x1
POL(ack2(x1, x2)) = 0
POL(plus2(x1, x2)) = 0
POL(s1(x1)) = 1 + x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
ACK2(s1(x), s1(y)) -> ACK2(s1(x), y)
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ACK2(s1(x), s1(y)) -> ACK2(s1(x), y)
POL(ACK2(x1, x2)) = x1·x2
POL(s1(x1)) = 2 + 3·x1 + 3·x12
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
PERMUTE3(y, x, c) -> PERMUTE3(x, y, a)
PERMUTE3(false, x, b) -> PERMUTE3(ack2(x, x), p1(x), c)
PERMUTE3(x, y, a) -> PERMUTE3(isZero1(x), x, b)
double1(x) -> permute3(x, x, a)
permute3(x, y, a) -> permute3(isZero1(x), x, b)
permute3(false, x, b) -> permute3(ack2(x, x), p1(x), c)
permute3(true, x, b) -> 0
permute3(y, x, c) -> s1(s1(permute3(x, y, a)))
p1(0) -> 0
p1(s1(x)) -> x
ack2(0, x) -> plus2(x, s1(0))
ack2(s1(x), 0) -> ack2(x, s1(0))
ack2(s1(x), s1(y)) -> ack2(x, ack2(s1(x), y))
plus2(0, y) -> y
plus2(s1(x), y) -> plus2(x, s1(y))
plus2(x, s1(s1(y))) -> s1(plus2(s1(x), y))
plus2(x, s1(0)) -> s1(x)
plus2(x, 0) -> x
isZero1(0) -> true
isZero1(s1(x)) -> false