Termination proof of /tmp/tmp6tyJyZ/sqrt.itrs

Termination of the given ITRSProblem could not be shown:

↳ ITRS

  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(||(>=@z(*@z(y, y), x), <@z(y, 0@z)), x, y)
sqrt(x) → sqrtAcc(x, 0@z)
condAcc(FALSE, x, y) → sqrtAcc(x, +@z(y, 1@z))
condAcc(TRUE, x, y) → y

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(FALSE, x0, x1)
condAcc(TRUE, x0, x1)

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

SQRTACC(x, y) → CONDACC(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
SQRTACC(x, y) → OR(greatereq_int(mult_int(y, y), x), less_int(y, pos(0)))
SQRTACC(x, y) → GREATEREQ_INT(mult_int(y, y), x)
SQRTACC(x, y) → MULT_INT(y, y)
SQRTACC(x, y) → LESS_INT(y, pos(0))
SQRT(x) → SQRTACC(x, pos(0))
CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
CONDACC(false, x, y) → PLUS_INT(pos(s(0)), y)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
MULT_INT(pos(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(pos(x), neg(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), neg(y)) → MULT_NAT(x, y)
MULT_NAT(s(x), s(y)) → PLUS_NAT(mult_nat(x, s(y)), s(y))
MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))
LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(x, y) → CONDACC(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
SQRTACC(x, y) → OR(greatereq_int(mult_int(y, y), x), less_int(y, pos(0)))
SQRTACC(x, y) → GREATEREQ_INT(mult_int(y, y), x)
SQRTACC(x, y) → MULT_INT(y, y)
SQRTACC(x, y) → LESS_INT(y, pos(0))
SQRT(x) → SQRTACC(x, pos(0))
CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
CONDACC(false, x, y) → PLUS_INT(pos(s(0)), y)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
MULT_INT(pos(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(pos(x), neg(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), pos(y)) → MULT_NAT(x, y)
MULT_INT(neg(x), neg(y)) → MULT_NAT(x, y)
MULT_NAT(s(x), s(y)) → PLUS_NAT(mult_nat(x, s(y)), s(y))
MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))
LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 8 SCCs with 15 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

R is empty.
The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))

R is empty.
The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

LESS_INT(neg(s(x)), neg(s(y))) → LESS_INT(neg(x), neg(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESS_INT(x₁, x₂)) = 2·x₁ + x₂
POL(neg(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))

R is empty.
The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LESS_INT(pos(s(x)), pos(s(y))) → LESS_INT(pos(x), pos(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(LESS_INT(x₁, x₂)) = 2·x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

R is empty.
The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)

From the DPs we obtained the following set of size-change graphs:

PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 >= 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

R is empty.
The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))

From the DPs we obtained the following set of size-change graphs:

MULT_NAT(s(x), s(y)) → MULT_NAT(x, s(y))
The graph contains the following edges 1 > 1, 2 >= 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

R is empty.
The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATEREQ_INT(x₁, x₂)) = 2·x₁ + x₂
POL(neg(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))

R is empty.
The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATEREQ_INT(x₁, x₂)) = 2·x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(x, y) → CONDACC(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)

The TRS R consists of the following rules:

sqrtAcc(x, y) → condAcc(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)
sqrt(x) → sqrtAcc(x, pos(0))
condAcc(false, x, y) → sqrtAcc(x, plus_int(pos(s(0)), y))
condAcc(true, x, y) → y
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
less_int(pos(0), pos(0)) → false
less_int(pos(0), neg(0)) → false
less_int(neg(0), pos(0)) → false
less_int(neg(0), neg(0)) → false
less_int(pos(0), pos(s(y))) → true
less_int(neg(0), pos(s(y))) → true
less_int(pos(0), neg(s(y))) → false
less_int(neg(0), neg(s(y))) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
less_int(pos(s(x)), neg(0)) → false
less_int(neg(s(x)), neg(0)) → true
less_int(pos(s(x)), neg(s(y))) → false
less_int(neg(s(x)), pos(s(y))) → true
less_int(pos(s(x)), pos(s(y))) → less_int(pos(x), pos(y))
less_int(neg(s(x)), neg(s(y))) → less_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(x, y) → CONDACC(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)
or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

sqrtAcc(x0, x1)
sqrt(x0)
condAcc(false, x0, x1)
condAcc(true, x0, x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(x, y) → CONDACC(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
Positions in right side of the pair:

[1,0]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                        ↳ RemovalProof

                        ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y, x_removed) → SQRTACC(x, plus_int(x_removed, y), x_removed)
SQRTACC(x, y, x_removed) → CONDACC(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y, x_removed)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

[1,0]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                          ↳ QDP

                        ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y, x_removed) → SQRTACC(x, plus_int(x_removed, y), x_removed)
SQRTACC(x, y, x_removed) → CONDACC(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y, x_removed)

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule SQRTACC(x, y) → CONDACC(or(greatereq_int(mult_int(y, y), x), less_int(y, pos(0))), x, y) at position [0] we obtained the following new rules [LPAR04]:

SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(mult_int(neg(0), neg(0)), y0), false), y0, neg(0))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(mult_int(neg(s(x0)), neg(s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(mult_int(pos(0), pos(0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(mult_int(pos(s(x0)), pos(s(x0))), y0), false), y0, pos(s(x0)))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(mult_int(neg(0), neg(0)), y0), false), y0, neg(0))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(mult_int(neg(s(x0)), neg(s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(mult_int(pos(0), pos(0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(mult_int(pos(s(x0)), pos(s(x0))), y0), false), y0, pos(s(x0)))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(mult_int(neg(0), neg(0)), y0), false), y0, neg(0)) at position [0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(mult_int(neg(s(x0)), neg(s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(mult_int(pos(0), pos(0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(mult_int(pos(s(x0)), pos(s(x0))), y0), false), y0, pos(s(x0)))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, neg(0))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(mult_int(neg(s(x0)), neg(s(x0))), y0), true), y0, neg(s(x0))) at position [0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), true), y0, neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(mult_int(pos(0), pos(0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(mult_int(pos(s(x0)), pos(s(x0))), y0), false), y0, pos(s(x0)))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), true), y0, neg(s(x0)))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
less_int(pos(0), pos(0)) → false
less_int(neg(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(false, true) → true
or(true, false) → true
or(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(mult_int(pos(0), pos(0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(mult_int(pos(s(x0)), pos(s(x0))), y0), false), y0, pos(s(x0)))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), true), y0, neg(s(x0)))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(mult_int(pos(0), pos(0)), y0), false), y0, pos(0)) at position [0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(mult_int(pos(s(x0)), pos(s(x0))), y0), false), y0, pos(s(x0)))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, pos(0))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(mult_int(pos(s(x0)), pos(s(x0))), y0), false), y0, pos(s(x0))) at position [0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), false), y0, pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), false), y0, pos(s(x0)))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), false), y0, pos(s(x0)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), false), y0, pos(s(x0)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, neg(0)) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, neg(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), false), y0, pos(s(x0)))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, neg(0))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), true), y0, neg(s(x0))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), true), y0, neg(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), false), y0, pos(s(x0)))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), true), y0, neg(s(x0)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(mult_nat(0, 0)), y0), false), y0, pos(0)) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, pos(0))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), false), y0, pos(s(x0)))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, pos(0))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(mult_nat(s(x0), s(x0))), y0), false), y0, pos(s(x0))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule CONDACC(false, x, y) → SQRTACC(x, plus_int(pos(s(0)), y)) at position [1] we obtained the following new rules [LPAR04]:

CONDACC(false, y0, neg(x1)) → SQRTACC(y0, minus_nat(s(0), x1))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(plus_nat(s(0), x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
SQRTACC(y0, neg(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, neg(0))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), true), y0, neg(s(x0)))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, y0, neg(x1)) → SQRTACC(y0, minus_nat(s(0), x1))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(plus_nat(s(0), x1)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(plus_nat(s(0), x1)))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(plus_nat(s(0), x1)))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
or(false, false) → false
or(true, false) → true
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(plus_nat(s(0), x1)))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
or(false, false) → false
or(true, false) → true
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(plus_nat(s(0), x1))) at position [1,0] we obtained the following new rules [LPAR04]:

CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(plus_nat(0, x1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, pos(0)) → CONDACC(or(greatereq_int(pos(0), y0), false), y0, pos(0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(plus_nat(0, x1))))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
or(false, false) → false
or(true, false) → true
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(plus_nat(0, x1))))
SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
or(false, false) → false
or(true, false) → true
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(plus_nat(0, x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:

CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0))
SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(x1)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
or(false, false) → false
or(true, false) → true
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule SQRTACC(y0, pos(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(pos(x0), pos(0))), y0, pos(x0)) we obtained the following new rules [LPAR04]:

SQRTACC(z0, pos(s(z1))) → CONDACC(or(greatereq_int(pos(mult_nat(s(z1), s(z1))), z0), less_int(pos(s(z1)), pos(0))), z0, pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(x1)))
SQRTACC(z0, pos(s(z1))) → CONDACC(or(greatereq_int(pos(mult_nat(s(z1), s(z1))), z0), less_int(pos(s(z1)), pos(0))), z0, pos(s(z1)))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
or(false, false) → false
or(true, false) → true
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(x1)))
SQRTACC(z0, pos(s(z1))) → CONDACC(or(greatereq_int(pos(mult_nat(s(z1), s(z1))), z0), less_int(pos(s(z1)), pos(0))), z0, pos(s(z1)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(z0, pos(s(z1))) → CONDACC(or(greatereq_int(pos(mult_nat(s(z1), s(z1))), z0), less_int(pos(s(z1)), pos(0))), z0, pos(s(z1))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(z0, pos(s(z1))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(z1, s(z1)), s(z1))), z0), less_int(pos(s(z1)), pos(0))), z0, pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(x1)))
SQRTACC(z0, pos(s(z1))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(z1, s(z1)), s(z1))), z0), less_int(pos(s(z1)), pos(0))), z0, pos(s(z1)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(z0, pos(s(z1))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(z1, s(z1)), s(z1))), z0), less_int(pos(s(z1)), pos(0))), z0, pos(s(z1))) at position [0,1] we obtained the following new rules [LPAR04]:

SQRTACC(z0, pos(s(z1))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(z1, s(z1)), s(z1))), z0), false), z0, pos(s(z1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                                                    ↳ QDP

                                                                                                                      ↳ UsableRulesProof

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(x1)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                                                    ↳ QDP

                                                                                                                      ↳ UsableRulesProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QReductionProof

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(x1)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                                                    ↳ QDP

                                                                                                                      ↳ UsableRulesProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QReductionProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Instantiation

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(x1)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule CONDACC(false, y0, pos(x1)) → SQRTACC(y0, pos(s(x1))) we obtained the following new rules [LPAR04]:

CONDACC(false, z0, pos(s(z1))) → SQRTACC(z0, pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                                                    ↳ QDP

                                                                                                                      ↳ UsableRulesProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QReductionProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Instantiation

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0)))
CONDACC(false, z0, pos(s(z1))) → SQRTACC(z0, pos(s(s(z1))))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule SQRTACC(y0, pos(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), false), y0, pos(s(x0))) we obtained the following new rules [LPAR04]:

SQRTACC(z0, pos(s(s(z1)))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(s(z1), s(s(z1))), s(s(z1)))), z0), false), z0, pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                                                    ↳ QDP

                                                                                                                      ↳ UsableRulesProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QReductionProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Instantiation

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Rewriting

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, z0, pos(s(z1))) → SQRTACC(z0, pos(s(s(z1))))
SQRTACC(z0, pos(s(s(z1)))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(s(z1), s(s(z1))), s(s(z1)))), z0), false), z0, pos(s(s(z1))))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(z0, pos(s(s(z1)))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(s(z1), s(s(z1))), s(s(z1)))), z0), false), z0, pos(s(s(z1)))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(z0, pos(s(s(z1)))) → CONDACC(or(greatereq_int(pos(plus_nat(plus_nat(mult_nat(z1, s(s(z1))), s(s(z1))), s(s(z1)))), z0), false), z0, pos(s(s(z1))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                                                    ↳ QDP

                                                                                                                      ↳ UsableRulesProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QReductionProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Instantiation

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Rewriting

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, z0, pos(s(z1))) → SQRTACC(z0, pos(s(s(z1))))
SQRTACC(z0, pos(s(s(z1)))) → CONDACC(or(greatereq_int(pos(plus_nat(plus_nat(mult_nat(z1, s(s(z1))), s(s(z1))), s(s(z1)))), z0), false), z0, pos(s(s(z1))))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule CONDACC(false, z0, pos(s(z1))) → SQRTACC(z0, pos(s(s(z1)))) we obtained the following new rules [LPAR04]:

CONDACC(false, z0, pos(s(s(z1)))) → SQRTACC(z0, pos(s(s(s(z1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                                                    ↳ QDP

                                                                                                                      ↳ UsableRulesProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QReductionProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Instantiation

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Rewriting

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Instantiation

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(z0, pos(s(s(z1)))) → CONDACC(or(greatereq_int(pos(plus_nat(plus_nat(mult_nat(z1, s(s(z1))), s(s(z1))), s(s(z1)))), z0), false), z0, pos(s(s(z1))))
CONDACC(false, z0, pos(s(s(z1)))) → SQRTACC(z0, pos(s(s(s(z1)))))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule SQRTACC(z0, pos(s(s(z1)))) → CONDACC(or(greatereq_int(pos(plus_nat(plus_nat(mult_nat(z1, s(s(z1))), s(s(z1))), s(s(z1)))), z0), false), z0, pos(s(s(z1)))) we obtained the following new rules [LPAR04]:

SQRTACC(z0, pos(s(s(s(z1))))) → CONDACC(or(greatereq_int(pos(plus_nat(plus_nat(mult_nat(s(z1), s(s(s(z1)))), s(s(s(z1)))), s(s(s(z1))))), z0), false), z0, pos(s(s(s(z1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                                                    ↳ QDP

                                                                                                                      ↳ UsableRulesProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QReductionProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Instantiation

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Rewriting

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Instantiation

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ Rewriting

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, z0, pos(s(s(z1)))) → SQRTACC(z0, pos(s(s(s(z1)))))
SQRTACC(z0, pos(s(s(s(z1))))) → CONDACC(or(greatereq_int(pos(plus_nat(plus_nat(mult_nat(s(z1), s(s(s(z1)))), s(s(s(z1)))), s(s(s(z1))))), z0), false), z0, pos(s(s(s(z1)))))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule SQRTACC(z0, pos(s(s(s(z1))))) → CONDACC(or(greatereq_int(pos(plus_nat(plus_nat(mult_nat(s(z1), s(s(s(z1)))), s(s(s(z1)))), s(s(s(z1))))), z0), false), z0, pos(s(s(s(z1))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]:

SQRTACC(z0, pos(s(s(s(z1))))) → CONDACC(or(greatereq_int(pos(plus_nat(plus_nat(plus_nat(mult_nat(z1, s(s(s(z1)))), s(s(s(z1)))), s(s(s(z1)))), s(s(s(z1))))), z0), false), z0, pos(s(s(s(z1)))))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ Rewriting

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Rewriting

                                                                                                    ↳ QDP

                                                                                                      ↳ Instantiation

                                                                                                        ↳ QDP

                                                                                                          ↳ UsableRulesProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Rewriting

                                                                                                                ↳ QDP

                                                                                                                  ↳ Rewriting

                                                                                                                    ↳ QDP

                                                                                                                      ↳ UsableRulesProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QReductionProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Instantiation

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Instantiation

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Rewriting

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ Instantiation

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Instantiation

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ Rewriting

                                                                                                                                                    ↳ QDP

                                                                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONDACC(false, z0, pos(s(s(z1)))) → SQRTACC(z0, pos(s(s(s(z1)))))
SQRTACC(z0, pos(s(s(s(z1))))) → CONDACC(or(greatereq_int(pos(plus_nat(plus_nat(plus_nat(mult_nat(z1, s(s(s(z1)))), s(s(s(z1)))), s(s(s(z1)))), s(s(s(z1))))), z0), false), z0, pos(s(s(s(z1)))))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

We have to consider all minimal (P,Q,R)-chains.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
CONDACC(false, y0, neg(x1)) → SQRTACC(y0, minus_nat(s(0), x1))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), true), y0, neg(s(x0)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
or(false, false) → false
or(true, false) → true
mult_nat(0, y) → 0
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
less_int(pos(0), pos(0)) → false
less_int(pos(s(x)), pos(0)) → false
or(false, true) → true
or(true, true) → true
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
CONDACC(false, y0, neg(x1)) → SQRTACC(y0, minus_nat(s(0), x1))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), true), y0, neg(s(x0)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(true, false) → true
or(false, true) → true
or(true, true) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
CONDACC(false, y0, neg(x1)) → SQRTACC(y0, minus_nat(s(0), x1))
SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), true), y0, neg(s(x0)))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(true, false) → true
or(false, true) → true
or(true, true) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SQRTACC(y0, neg(s(x0))) → CONDACC(or(greatereq_int(pos(plus_nat(mult_nat(x0, s(x0)), s(x0))), y0), true), y0, neg(s(x0)))

The remaining pairs can at least be oriented weakly.

SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
CONDACC(false, y0, neg(x1)) → SQRTACC(y0, minus_nat(s(0), x1))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(CONDACC(x₁, x₂, x₃)) = x₁
POL(SQRTACC(x₁, x₂)) = x₂
POL(false) = 1
POL(greatereq_int(x₁, x₂)) = 0
POL(less_int(x₁, x₂)) = x₁
POL(minus_nat(x₁, x₂)) = 1
POL(mult_nat(x₁, x₂)) = 0
POL(neg(x₁)) = x₁
POL(or(x₁, x₂)) = x₂
POL(plus_nat(x₁, x₂)) = 0
POL(pos(x₁)) = 0
POL(s(x₁)) = 1
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

or(true, false) → true
or(false, false) → false
less_int(neg(s(x)), pos(0)) → true
less_int(neg(0), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
or(false, true) → true
or(true, true) → true

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                                                                                            ↳ QDP

                                                                                              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))
CONDACC(false, y0, neg(x1)) → SQRTACC(y0, minus_nat(s(0), x1))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(true, false) → true
or(false, true) → true
or(true, true) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONDACC(false, y0, neg(x1)) → SQRTACC(y0, minus_nat(s(0), x1))

The remaining pairs can at least be oriented weakly.

SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(CONDACC(x₁, x₂, x₃)) = x₁
POL(SQRTACC(x₁, x₂)) = x₂
POL(false) = 1
POL(greatereq_int(x₁, x₂)) = 0
POL(less_int(x₁, x₂)) = x₁
POL(minus_nat(x₁, x₂)) = 0
POL(mult_nat(x₁, x₂)) = 1 + x₁
POL(neg(x₁)) = x₁
POL(or(x₁, x₂)) = x₂
POL(plus_nat(x₁, x₂)) = x₂
POL(pos(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

or(true, false) → true
or(false, false) → false
less_int(neg(s(x)), pos(0)) → true
less_int(neg(0), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
or(false, true) → true
or(true, true) → true

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                        ↳ RemovalProof

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Rewriting

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ UsableRulesProof

                                      ↳ QDP

                                        ↳ Rewriting

                                          ↳ QDP

                                            ↳ Rewriting

                                              ↳ QDP

                                                ↳ UsableRulesProof

                                                  ↳ QDP

                                                    ↳ QReductionProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ Rewriting

                                                                  ↳ QDP

                                                                    ↳ Rewriting

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ AND

                                                                                ↳ QDP

                                                                                ↳ QDP

                                                                                  ↳ UsableRulesProof

                                                                                    ↳ QDP

                                                                                      ↳ QReductionProof

                                                                                        ↳ QDP

                                                                                          ↳ QDPOrderProof

                                                                                            ↳ QDP

                                                                                              ↳ QDPOrderProof

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

SQRTACC(y0, neg(x0)) → CONDACC(or(greatereq_int(pos(mult_nat(x0, x0)), y0), less_int(neg(x0), pos(0))), y0, neg(x0))

The TRS R consists of the following rules:

mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
mult_nat(0, y) → 0
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
less_int(neg(0), pos(0)) → false
less_int(neg(s(x)), pos(0)) → true
or(false, false) → false
or(true, false) → true
or(false, true) → true
or(true, true) → true
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))

The set Q consists of the following terms:

or(false, false)
or(false, true)
or(true, false)
or(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
less_int(pos(0), pos(0))
less_int(pos(0), neg(0))
less_int(neg(0), pos(0))
less_int(neg(0), neg(0))
less_int(pos(0), pos(s(x0)))
less_int(neg(0), pos(s(x0)))
less_int(pos(0), neg(s(x0)))
less_int(neg(0), neg(s(x0)))
less_int(pos(s(x0)), pos(0))
less_int(neg(s(x0)), pos(0))
less_int(pos(s(x0)), neg(0))
less_int(neg(s(x0)), neg(0))
less_int(pos(s(x0)), neg(s(x1)))
less_int(neg(s(x0)), pos(s(x1)))
less_int(pos(s(x0)), pos(s(x1)))
less_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.