Termination proof of /tmp/tmpriHz2j/24.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:

eval_3(x, y) → Cond_eval_31(>=@z(0@z, y), x, y)
Cond_eval_3(TRUE, x, y) → eval_3(x, -@z(y, 1@z))
eval_1(x, y) → Cond_eval_1(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >@z(x, y)), x, y)
Cond_eval_11(TRUE, x, y) → eval_3(x, y)
Cond_eval_1(TRUE, x, y) → eval_2(x, y)
Cond_eval_2(TRUE, x, y) → eval_2(-@z(x, 1@z), y)
eval_1(x, y) → Cond_eval_11(&&(&&(>@z(x, 0@z), >@z(y, 0@z)), >=@z(y, x)), x, y)
Cond_eval_31(TRUE, x, y) → eval_1(x, y)
Cond_eval_21(TRUE, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(>@z(y, 0@z), x, y)
eval_2(x, y) → Cond_eval_21(>=@z(0@z, x), x, y)
eval_2(x, y) → Cond_eval_2(>@z(x, 0@z), x, y)

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(TRUE, x0, x1)
eval_1(x0, x1)
Cond_eval_11(TRUE, x0, x1)
Cond_eval_1(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)
Cond_eval_31(TRUE, x0, x1)
Cond_eval_21(TRUE, x0, x1)
eval_2(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:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(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:

EVAL_3(x, y) → COND_EVAL_31(greatereq_int(pos(0), y), x, y)
EVAL_3(x, y) → GREATEREQ_INT(pos(0), y)
COND_EVAL_3(true, x, y) → EVAL_3(x, minus_int(y, pos(s(0))))
COND_EVAL_3(true, x, y) → MINUS_INT(y, pos(s(0)))
EVAL_1(x, y) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
EVAL_1(x, y) → AND(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y))
EVAL_1(x, y) → AND(greater_int(x, pos(0)), greater_int(y, pos(0)))
EVAL_1(x, y) → GREATER_INT(x, pos(0))
EVAL_1(x, y) → GREATER_INT(y, pos(0))
EVAL_1(x, y) → GREATER_INT(x, y)
COND_EVAL_11(true, x, y) → EVAL_3(x, y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
COND_EVAL_2(true, x, y) → EVAL_2(minus_int(x, pos(s(0))), y)
COND_EVAL_2(true, x, y) → MINUS_INT(x, pos(s(0)))
EVAL_1(x, y) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
EVAL_1(x, y) → AND(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x))
EVAL_1(x, y) → GREATEREQ_INT(y, x)
COND_EVAL_31(true, x, y) → EVAL_1(x, y)
COND_EVAL_21(true, x, y) → EVAL_1(x, y)
EVAL_3(x, y) → COND_EVAL_3(greater_int(y, pos(0)), x, y)
EVAL_3(x, y) → GREATER_INT(y, pos(0))
EVAL_2(x, y) → COND_EVAL_21(greatereq_int(pos(0), x), x, y)
EVAL_2(x, y) → GREATEREQ_INT(pos(0), x)
EVAL_2(x, y) → COND_EVAL_2(greater_int(x, pos(0)), x, y)
EVAL_2(x, y) → GREATER_INT(x, pos(0))
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))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(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:

EVAL_3(x, y) → COND_EVAL_31(greatereq_int(pos(0), y), x, y)
EVAL_3(x, y) → GREATEREQ_INT(pos(0), y)
COND_EVAL_3(true, x, y) → EVAL_3(x, minus_int(y, pos(s(0))))
COND_EVAL_3(true, x, y) → MINUS_INT(y, pos(s(0)))
EVAL_1(x, y) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
EVAL_1(x, y) → AND(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y))
EVAL_1(x, y) → AND(greater_int(x, pos(0)), greater_int(y, pos(0)))
EVAL_1(x, y) → GREATER_INT(x, pos(0))
EVAL_1(x, y) → GREATER_INT(y, pos(0))
EVAL_1(x, y) → GREATER_INT(x, y)
COND_EVAL_11(true, x, y) → EVAL_3(x, y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
COND_EVAL_2(true, x, y) → EVAL_2(minus_int(x, pos(s(0))), y)
COND_EVAL_2(true, x, y) → MINUS_INT(x, pos(s(0)))
EVAL_1(x, y) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
EVAL_1(x, y) → AND(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x))
EVAL_1(x, y) → GREATEREQ_INT(y, x)
COND_EVAL_31(true, x, y) → EVAL_1(x, y)
COND_EVAL_21(true, x, y) → EVAL_1(x, y)
EVAL_3(x, y) → COND_EVAL_3(greater_int(y, pos(0)), x, y)
EVAL_3(x, y) → GREATER_INT(y, pos(0))
EVAL_2(x, y) → COND_EVAL_21(greatereq_int(pos(0), x), x, y)
EVAL_2(x, y) → GREATEREQ_INT(pos(0), x)
EVAL_2(x, y) → COND_EVAL_2(greater_int(x, pos(0)), x, y)
EVAL_2(x, y) → GREATER_INT(x, pos(0))
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))
MINUS_INT(pos(x), pos(y)) → MINUS_NAT(x, y)
MINUS_INT(neg(x), neg(y)) → MINUS_NAT(y, x)
MINUS_INT(neg(x), pos(y)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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 7 SCCs with 17 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(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

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

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

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

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_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:

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

The TRS R consists of the following rules:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

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

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

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

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATER_INT(x₁, x₂)) = 2·x₁ + x₂
POL(pos(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

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

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:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ 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:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ 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

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ 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:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ 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:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ 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

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:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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

                ↳ 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

                ↳ 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:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ 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

                ↳ 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

                ↳ UsableRulesProof

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

COND_EVAL_31(true, x, y) → EVAL_1(x, y)
EVAL_1(x, y) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_21(greatereq_int(pos(0), x), x, y)
COND_EVAL_21(true, x, y) → EVAL_1(x, y)
EVAL_1(x, y) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
COND_EVAL_11(true, x, y) → EVAL_3(x, y)
EVAL_3(x, y) → COND_EVAL_31(greatereq_int(pos(0), y), x, y)
EVAL_3(x, y) → COND_EVAL_3(greater_int(y, pos(0)), x, y)
COND_EVAL_3(true, x, y) → EVAL_3(x, minus_int(y, pos(s(0))))
EVAL_2(x, y) → COND_EVAL_2(greater_int(x, pos(0)), x, y)
COND_EVAL_2(true, x, y) → EVAL_2(minus_int(x, pos(s(0))), y)

The TRS R consists of the following rules:

eval_3(x, y) → Cond_eval_31(greatereq_int(pos(0), y), x, y)
Cond_eval_3(true, x, y) → eval_3(x, minus_int(y, pos(s(0))))
eval_1(x, y) → Cond_eval_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
Cond_eval_11(true, x, y) → eval_3(x, y)
Cond_eval_1(true, x, y) → eval_2(x, y)
Cond_eval_2(true, x, y) → eval_2(minus_int(x, pos(s(0))), y)
eval_1(x, y) → Cond_eval_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
Cond_eval_31(true, x, y) → eval_1(x, y)
Cond_eval_21(true, x, y) → eval_1(x, y)
eval_3(x, y) → Cond_eval_3(greater_int(y, pos(0)), x, y)
eval_2(x, y) → Cond_eval_21(greatereq_int(pos(0), x), x, y)
eval_2(x, y) → Cond_eval_2(greater_int(x, pos(0)), 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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

COND_EVAL_31(true, x, y) → EVAL_1(x, y)
EVAL_1(x, y) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_21(greatereq_int(pos(0), x), x, y)
COND_EVAL_21(true, x, y) → EVAL_1(x, y)
EVAL_1(x, y) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
COND_EVAL_11(true, x, y) → EVAL_3(x, y)
EVAL_3(x, y) → COND_EVAL_31(greatereq_int(pos(0), y), x, y)
EVAL_3(x, y) → COND_EVAL_3(greater_int(y, pos(0)), x, y)
COND_EVAL_3(true, x, y) → EVAL_3(x, minus_int(y, pos(s(0))))
EVAL_2(x, y) → COND_EVAL_2(greater_int(x, pos(0)), x, y)
COND_EVAL_2(true, x, y) → EVAL_2(minus_int(x, pos(s(0))), y)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))

The set Q consists of the following terms:

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)
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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

eval_3(x0, x1)
Cond_eval_3(true, x0, x1)
eval_1(x0, x1)
Cond_eval_11(true, x0, x1)
Cond_eval_1(true, x0, x1)
Cond_eval_2(true, x0, x1)
Cond_eval_31(true, x0, x1)
Cond_eval_21(true, x0, x1)
eval_2(x0, x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

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

COND_EVAL_31(true, x, y) → EVAL_1(x, y)
EVAL_1(x, y) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y)
COND_EVAL_1(true, x, y) → EVAL_2(x, y)
EVAL_2(x, y) → COND_EVAL_21(greatereq_int(pos(0), x), x, y)
COND_EVAL_21(true, x, y) → EVAL_1(x, y)
EVAL_1(x, y) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y)
COND_EVAL_11(true, x, y) → EVAL_3(x, y)
EVAL_3(x, y) → COND_EVAL_31(greatereq_int(pos(0), y), x, y)
EVAL_3(x, y) → COND_EVAL_3(greater_int(y, pos(0)), x, y)
COND_EVAL_3(true, x, y) → EVAL_3(x, minus_int(y, pos(s(0))))
EVAL_2(x, y) → COND_EVAL_2(greater_int(x, pos(0)), x, y)
COND_EVAL_2(true, x, y) → EVAL_2(minus_int(x, pos(s(0))), y)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(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: COND_EVAL_3(true, x, y) → EVAL_3(x, minus_int(y, pos(s(0))))
Positions in right side of the pair:

[1,1]

Pair: COND_EVAL_2(true, x, y) → EVAL_2(minus_int(x, pos(s(0))), y)
Positions in right side of the pair:

[0,1]

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

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

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

COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
COND_EVAL_1(true, x, y, x_removed) → EVAL_2(x, y, x_removed)
EVAL_2(x, y, x_removed) → COND_EVAL_21(greatereq_int(pos(0), x), x, y, x_removed)
EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
COND_EVAL_21(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed) the following chains were created:

We consider the chain EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed), COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed) which results in the following constraint:
(1)    (COND_EVAL_31(greatereq_int(pos(0), x25), x24, x25, x26)=COND_EVAL_31(true, x27, x28, x29) ⇒ COND_EVAL_31(true, x27, x28, x29)≥EVAL_1(x27, x28, x29))

We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
(2)    (pos(0)=x486∧greatereq_int(x486, x25)=true ⇒ COND_EVAL_31(true, x24, x25, x26)≥EVAL_1(x24, x25, x26))

We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x486, x25)=true which results in the following new constraints:
(3)    (true=true∧pos(0)=pos(x487) ⇒ COND_EVAL_31(true, x24, pos(0), x26)≥EVAL_1(x24, pos(0), x26))

(4)    (true=true∧pos(0)=pos(x489) ⇒ COND_EVAL_31(true, x24, neg(x488), x26)≥EVAL_1(x24, neg(x488), x26))

(5)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_31(true, x24, pos(0), x26)≥EVAL_1(x24, pos(0), x26))

(6)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_31(true, x24, neg(x491), x26)≥EVAL_1(x24, neg(x491), x26))

(7)    (greatereq_int(pos(x497), pos(x496))=true∧pos(0)=pos(s(x497))∧(∀x498,x499:greatereq_int(pos(x497), pos(x496))=true∧pos(0)=pos(x497) ⇒ COND_EVAL_31(true, x498, pos(x496), x499)≥EVAL_1(x498, pos(x496), x499)) ⇒ COND_EVAL_31(true, x24, pos(s(x496)), x26)≥EVAL_1(x24, pos(s(x496)), x26))

(8)    (greatereq_int(neg(x501), neg(x500))=true∧pos(0)=neg(s(x501))∧(∀x502,x503:greatereq_int(neg(x501), neg(x500))=true∧pos(0)=neg(x501) ⇒ COND_EVAL_31(true, x502, neg(x500), x503)≥EVAL_1(x502, neg(x500), x503)) ⇒ COND_EVAL_31(true, x24, neg(s(x500)), x26)≥EVAL_1(x24, neg(s(x500)), x26))

We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint:
(9)    (COND_EVAL_31(true, x24, pos(0), x26)≥EVAL_1(x24, pos(0), x26))

We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint:
(10)    (COND_EVAL_31(true, x24, neg(x488), x26)≥EVAL_1(x24, neg(x488), x26))

We solved constraint (5) using rules (I), (II).We solved constraint (6) using rules (I), (II).We solved constraint (7) using rules (I), (II).We solved constraint (8) using rules (I), (II).

For Pair EVAL_1(x, y, x_removed) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed), EVAL_1(x, y, x_removed) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y, x_removed) which results in the following constraint:
(11) (EVAL_1(x39, x40, x41)=EVAL_1(x42, x43, x44) ⇒ EVAL_1(x42, x43, x44)≥COND_EVAL_1(and(and(greater_int(x42, pos(0)), greater_int(x43, pos(0))), greater_int(x42, x43)), x42, x43, x44))

We simplified constraint (11) using rules (I), (II), (III) which results in the following new constraint:
(12) (EVAL_1(x39, x40, x41)≥COND_EVAL_1(and(and(greater_int(x39, pos(0)), greater_int(x40, pos(0))), greater_int(x39, x40)), x39, x40, x41))
We consider the chain COND_EVAL_21(true, x, y, x_removed) → EVAL_1(x, y, x_removed), EVAL_1(x, y, x_removed) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y, x_removed) which results in the following constraint:
(13) (EVAL_1(x60, x61, x62)=EVAL_1(x63, x64, x65) ⇒ EVAL_1(x63, x64, x65)≥COND_EVAL_1(and(and(greater_int(x63, pos(0)), greater_int(x64, pos(0))), greater_int(x63, x64)), x63, x64, x65))

We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint:
(14) (EVAL_1(x60, x61, x62)≥COND_EVAL_1(and(and(greater_int(x60, pos(0)), greater_int(x61, pos(0))), greater_int(x60, x61)), x60, x61, x62))

For Pair EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed), EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed) which results in the following constraint:
(15) (EVAL_1(x81, x82, x83)=EVAL_1(x84, x85, x86) ⇒ EVAL_1(x84, x85, x86)≥COND_EVAL_11(and(and(greater_int(x84, pos(0)), greater_int(x85, pos(0))), greatereq_int(x85, x84)), x84, x85, x86))

We simplified constraint (15) using rules (I), (II), (III) which results in the following new constraint:
(16) (EVAL_1(x81, x82, x83)≥COND_EVAL_11(and(and(greater_int(x81, pos(0)), greater_int(x82, pos(0))), greatereq_int(x82, x81)), x81, x82, x83))
We consider the chain COND_EVAL_21(true, x, y, x_removed) → EVAL_1(x, y, x_removed), EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed) which results in the following constraint:
(17) (EVAL_1(x102, x103, x104)=EVAL_1(x105, x106, x107) ⇒ EVAL_1(x105, x106, x107)≥COND_EVAL_11(and(and(greater_int(x105, pos(0)), greater_int(x106, pos(0))), greatereq_int(x106, x105)), x105, x106, x107))

We simplified constraint (17) using rules (I), (II), (III) which results in the following new constraint:
(18) (EVAL_1(x102, x103, x104)≥COND_EVAL_11(and(and(greater_int(x102, pos(0)), greater_int(x103, pos(0))), greatereq_int(x103, x102)), x102, x103, x104))

For Pair COND_EVAL_1(true, x, y, x_removed) → EVAL_2(x, y, x_removed) the following chains were created:

We consider the chain EVAL_1(x, y, x_removed) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y, x_removed), COND_EVAL_1(true, x, y, x_removed) → EVAL_2(x, y, x_removed) which results in the following constraint:
(19)    (COND_EVAL_1(and(and(greater_int(x126, pos(0)), greater_int(x127, pos(0))), greater_int(x126, x127)), x126, x127, x128)=COND_EVAL_1(true, x129, x130, x131) ⇒ COND_EVAL_1(true, x129, x130, x131)≥EVAL_2(x129, x130, x131))

We simplified constraint (19) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(20)    (and(x506, x507)=x504∧greater_int(x126, x127)=x505∧and(x504, x505)=true ⇒ COND_EVAL_1(true, x126, x127, x128)≥EVAL_2(x126, x127, x128))

We simplified constraint (20) using rule (V) (with possible (I) afterwards) using induction on and(x504, x505)=true which results in the following new constraint:
(21)    (true=true∧and(x506, x507)=true∧greater_int(x126, x127)=true ⇒ COND_EVAL_1(true, x126, x127, x128)≥EVAL_2(x126, x127, x128))

We simplified constraint (21) using rules (I), (II) which results in the following new constraint:
(22)    (and(x506, x507)=true∧greater_int(x126, x127)=true ⇒ COND_EVAL_1(true, x126, x127, x128)≥EVAL_2(x126, x127, x128))

We simplified constraint (22) using rule (V) (with possible (I) afterwards) using induction on and(x506, x507)=true which results in the following new constraint:
(23)    (true=true∧greater_int(x126, x127)=true ⇒ COND_EVAL_1(true, x126, x127, x128)≥EVAL_2(x126, x127, x128))

We simplified constraint (23) using rules (I), (II) which results in the following new constraint:
(24)    (greater_int(x126, x127)=true ⇒ COND_EVAL_1(true, x126, x127, x128)≥EVAL_2(x126, x127, x128))

We simplified constraint (24) using rule (V) (with possible (I) afterwards) using induction on greater_int(x126, x127)=true which results in the following new constraints:
(25)    (true=true ⇒ COND_EVAL_1(true, pos(s(x510)), pos(0), x128)≥EVAL_2(pos(s(x510)), pos(0), x128))

(26)    (true=true ⇒ COND_EVAL_1(true, pos(0), neg(s(x514)), x128)≥EVAL_2(pos(0), neg(s(x514)), x128))

(27)    (true=true ⇒ COND_EVAL_1(true, neg(0), neg(s(x515)), x128)≥EVAL_2(neg(0), neg(s(x515)), x128))

(28)    (true=true ⇒ COND_EVAL_1(true, pos(s(x516)), neg(0), x128)≥EVAL_2(pos(s(x516)), neg(0), x128))

(29)    (true=true ⇒ COND_EVAL_1(true, pos(s(x519)), neg(s(x518)), x128)≥EVAL_2(pos(s(x519)), neg(s(x518)), x128))

(30)    (greater_int(pos(x523), pos(x522))=true∧(∀x524:greater_int(pos(x523), pos(x522))=true ⇒ COND_EVAL_1(true, pos(x523), pos(x522), x524)≥EVAL_2(pos(x523), pos(x522), x524)) ⇒ COND_EVAL_1(true, pos(s(x523)), pos(s(x522)), x128)≥EVAL_2(pos(s(x523)), pos(s(x522)), x128))

(31)    (greater_int(neg(x526), neg(x525))=true∧(∀x527:greater_int(neg(x526), neg(x525))=true ⇒ COND_EVAL_1(true, neg(x526), neg(x525), x527)≥EVAL_2(neg(x526), neg(x525), x527)) ⇒ COND_EVAL_1(true, neg(s(x526)), neg(s(x525)), x128)≥EVAL_2(neg(s(x526)), neg(s(x525)), x128))

We simplified constraint (25) using rules (I), (II) which results in the following new constraint:
(32)    (COND_EVAL_1(true, pos(s(x510)), pos(0), x128)≥EVAL_2(pos(s(x510)), pos(0), x128))

We simplified constraint (26) using rules (I), (II) which results in the following new constraint:
(33)    (COND_EVAL_1(true, pos(0), neg(s(x514)), x128)≥EVAL_2(pos(0), neg(s(x514)), x128))

We simplified constraint (27) using rules (I), (II) which results in the following new constraint:
(34)    (COND_EVAL_1(true, neg(0), neg(s(x515)), x128)≥EVAL_2(neg(0), neg(s(x515)), x128))

We simplified constraint (28) using rules (I), (II) which results in the following new constraint:
(35)    (COND_EVAL_1(true, pos(s(x516)), neg(0), x128)≥EVAL_2(pos(s(x516)), neg(0), x128))

We simplified constraint (29) using rules (I), (II) which results in the following new constraint:
(36)    (COND_EVAL_1(true, pos(s(x519)), neg(s(x518)), x128)≥EVAL_2(pos(s(x519)), neg(s(x518)), x128))

We simplified constraint (30) using rule (VI) where we applied the induction hypothesis (∀x524:greater_int(pos(x523), pos(x522))=true ⇒ COND_EVAL_1(true, pos(x523), pos(x522), x524)≥EVAL_2(pos(x523), pos(x522), x524)) with σ = [x524 / x128] which results in the following new constraint:
(37)    (COND_EVAL_1(true, pos(x523), pos(x522), x128)≥EVAL_2(pos(x523), pos(x522), x128) ⇒ COND_EVAL_1(true, pos(s(x523)), pos(s(x522)), x128)≥EVAL_2(pos(s(x523)), pos(s(x522)), x128))

We simplified constraint (31) using rule (VI) where we applied the induction hypothesis (∀x527:greater_int(neg(x526), neg(x525))=true ⇒ COND_EVAL_1(true, neg(x526), neg(x525), x527)≥EVAL_2(neg(x526), neg(x525), x527)) with σ = [x527 / x128] which results in the following new constraint:
(38)    (COND_EVAL_1(true, neg(x526), neg(x525), x128)≥EVAL_2(neg(x526), neg(x525), x128) ⇒ COND_EVAL_1(true, neg(s(x526)), neg(s(x525)), x128)≥EVAL_2(neg(s(x526)), neg(s(x525)), x128))

For Pair EVAL_2(x, y, x_removed) → COND_EVAL_21(greatereq_int(pos(0), x), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_1(true, x, y, x_removed) → EVAL_2(x, y, x_removed), EVAL_2(x, y, x_removed) → COND_EVAL_21(greatereq_int(pos(0), x), x, y, x_removed) which results in the following constraint:
(39) (EVAL_2(x171, x172, x173)=EVAL_2(x174, x175, x176) ⇒ EVAL_2(x174, x175, x176)≥COND_EVAL_21(greatereq_int(pos(0), x174), x174, x175, x176))

We simplified constraint (39) using rules (I), (II), (III) which results in the following new constraint:
(40) (EVAL_2(x171, x172, x173)≥COND_EVAL_21(greatereq_int(pos(0), x171), x171, x172, x173))
We consider the chain COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed), EVAL_2(x, y, x_removed) → COND_EVAL_21(greatereq_int(pos(0), x), x, y, x_removed) which results in the following constraint:
(41) (EVAL_2(minus_int(x198, x200), x199, x200)=EVAL_2(x201, x202, x203) ⇒ EVAL_2(x201, x202, x203)≥COND_EVAL_21(greatereq_int(pos(0), x201), x201, x202, x203))

We simplified constraint (41) using rules (I), (II), (III), (IV) which results in the following new constraint:
(42) (EVAL_2(x201, x199, x200)≥COND_EVAL_21(greatereq_int(pos(0), x201), x201, x199, x200))

For Pair EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_1(true, x, y, x_removed) → EVAL_2(x, y, x_removed), EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed) which results in the following constraint:
(43) (EVAL_2(x213, x214, x215)=EVAL_2(x216, x217, x218) ⇒ EVAL_2(x216, x217, x218)≥COND_EVAL_2(greater_int(x216, pos(0)), x216, x217, x218))

We simplified constraint (43) using rules (I), (II), (III) which results in the following new constraint:
(44) (EVAL_2(x213, x214, x215)≥COND_EVAL_2(greater_int(x213, pos(0)), x213, x214, x215))
We consider the chain COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed), EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed) which results in the following constraint:
(45) (EVAL_2(minus_int(x240, x242), x241, x242)=EVAL_2(x243, x244, x245) ⇒ EVAL_2(x243, x244, x245)≥COND_EVAL_2(greater_int(x243, pos(0)), x243, x244, x245))

We simplified constraint (45) using rules (I), (II), (III), (IV) which results in the following new constraint:
(46) (EVAL_2(x243, x241, x242)≥COND_EVAL_2(greater_int(x243, pos(0)), x243, x241, x242))

For Pair COND_EVAL_21(true, x, y, x_removed) → EVAL_1(x, y, x_removed) the following chains were created:

We consider the chain EVAL_2(x, y, x_removed) → COND_EVAL_21(greatereq_int(pos(0), x), x, y, x_removed), COND_EVAL_21(true, x, y, x_removed) → EVAL_1(x, y, x_removed) which results in the following constraint:
(47)    (COND_EVAL_21(greatereq_int(pos(0), x258), x258, x259, x260)=COND_EVAL_21(true, x261, x262, x263) ⇒ COND_EVAL_21(true, x261, x262, x263)≥EVAL_1(x261, x262, x263))

We simplified constraint (47) using rules (I), (II), (III), (VII) which results in the following new constraint:
(48)    (pos(0)=x528∧greatereq_int(x528, x258)=true ⇒ COND_EVAL_21(true, x258, x259, x260)≥EVAL_1(x258, x259, x260))

We simplified constraint (48) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x528, x258)=true which results in the following new constraints:
(49)    (true=true∧pos(0)=pos(x529) ⇒ COND_EVAL_21(true, pos(0), x259, x260)≥EVAL_1(pos(0), x259, x260))

(50)    (true=true∧pos(0)=pos(x531) ⇒ COND_EVAL_21(true, neg(x530), x259, x260)≥EVAL_1(neg(x530), x259, x260))

(51)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_21(true, pos(0), x259, x260)≥EVAL_1(pos(0), x259, x260))

(52)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_21(true, neg(x533), x259, x260)≥EVAL_1(neg(x533), x259, x260))

(53)    (greatereq_int(pos(x539), pos(x538))=true∧pos(0)=pos(s(x539))∧(∀x540,x541:greatereq_int(pos(x539), pos(x538))=true∧pos(0)=pos(x539) ⇒ COND_EVAL_21(true, pos(x538), x540, x541)≥EVAL_1(pos(x538), x540, x541)) ⇒ COND_EVAL_21(true, pos(s(x538)), x259, x260)≥EVAL_1(pos(s(x538)), x259, x260))

(54)    (greatereq_int(neg(x543), neg(x542))=true∧pos(0)=neg(s(x543))∧(∀x544,x545:greatereq_int(neg(x543), neg(x542))=true∧pos(0)=neg(x543) ⇒ COND_EVAL_21(true, neg(x542), x544, x545)≥EVAL_1(neg(x542), x544, x545)) ⇒ COND_EVAL_21(true, neg(s(x542)), x259, x260)≥EVAL_1(neg(s(x542)), x259, x260))

We simplified constraint (49) using rules (I), (II), (IV) which results in the following new constraint:
(55)    (COND_EVAL_21(true, pos(0), x259, x260)≥EVAL_1(pos(0), x259, x260))

We simplified constraint (50) using rules (I), (II), (IV) which results in the following new constraint:
(56)    (COND_EVAL_21(true, neg(x530), x259, x260)≥EVAL_1(neg(x530), x259, x260))

We solved constraint (51) using rules (I), (II).We solved constraint (52) using rules (I), (II).We solved constraint (53) using rules (I), (II).We solved constraint (54) using rules (I), (II).

For Pair COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed) the following chains were created:

We consider the chain EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed), COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed) which results in the following constraint:
(57)    (COND_EVAL_11(and(and(greater_int(x291, pos(0)), greater_int(x292, pos(0))), greatereq_int(x292, x291)), x291, x292, x293)=COND_EVAL_11(true, x294, x295, x296) ⇒ COND_EVAL_11(true, x294, x295, x296)≥EVAL_3(x294, x295, x296))

We simplified constraint (57) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(58)    (and(x548, x549)=x546∧greatereq_int(x292, x291)=x547∧and(x546, x547)=true ⇒ COND_EVAL_11(true, x291, x292, x293)≥EVAL_3(x291, x292, x293))

We simplified constraint (58) using rule (V) (with possible (I) afterwards) using induction on and(x546, x547)=true which results in the following new constraint:
(59)    (true=true∧and(x548, x549)=true∧greatereq_int(x292, x291)=true ⇒ COND_EVAL_11(true, x291, x292, x293)≥EVAL_3(x291, x292, x293))

We simplified constraint (59) using rules (I), (II) which results in the following new constraint:
(60)    (and(x548, x549)=true∧greatereq_int(x292, x291)=true ⇒ COND_EVAL_11(true, x291, x292, x293)≥EVAL_3(x291, x292, x293))

We simplified constraint (60) using rule (V) (with possible (I) afterwards) using induction on and(x548, x549)=true which results in the following new constraint:
(61)    (true=true∧greatereq_int(x292, x291)=true ⇒ COND_EVAL_11(true, x291, x292, x293)≥EVAL_3(x291, x292, x293))

We simplified constraint (61) using rules (I), (II) which results in the following new constraint:
(62)    (greatereq_int(x292, x291)=true ⇒ COND_EVAL_11(true, x291, x292, x293)≥EVAL_3(x291, x292, x293))

We simplified constraint (62) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x292, x291)=true which results in the following new constraints:
(63)    (true=true ⇒ COND_EVAL_11(true, pos(0), pos(x552), x293)≥EVAL_3(pos(0), pos(x552), x293))

(64)    (true=true ⇒ COND_EVAL_11(true, neg(x553), pos(x554), x293)≥EVAL_3(neg(x553), pos(x554), x293))

(65)    (true=true ⇒ COND_EVAL_11(true, pos(0), neg(0), x293)≥EVAL_3(pos(0), neg(0), x293))

(66)    (true=true ⇒ COND_EVAL_11(true, neg(x556), neg(0), x293)≥EVAL_3(neg(x556), neg(0), x293))

(67)    (greatereq_int(pos(x562), pos(x561))=true∧(∀x563:greatereq_int(pos(x562), pos(x561))=true ⇒ COND_EVAL_11(true, pos(x561), pos(x562), x563)≥EVAL_3(pos(x561), pos(x562), x563)) ⇒ COND_EVAL_11(true, pos(s(x561)), pos(s(x562)), x293)≥EVAL_3(pos(s(x561)), pos(s(x562)), x293))

(68)    (greatereq_int(neg(x565), neg(x564))=true∧(∀x566:greatereq_int(neg(x565), neg(x564))=true ⇒ COND_EVAL_11(true, neg(x564), neg(x565), x566)≥EVAL_3(neg(x564), neg(x565), x566)) ⇒ COND_EVAL_11(true, neg(s(x564)), neg(s(x565)), x293)≥EVAL_3(neg(s(x564)), neg(s(x565)), x293))

We simplified constraint (63) using rules (I), (II) which results in the following new constraint:
(69)    (COND_EVAL_11(true, pos(0), pos(x552), x293)≥EVAL_3(pos(0), pos(x552), x293))

We simplified constraint (64) using rules (I), (II) which results in the following new constraint:
(70)    (COND_EVAL_11(true, neg(x553), pos(x554), x293)≥EVAL_3(neg(x553), pos(x554), x293))

We simplified constraint (65) using rules (I), (II) which results in the following new constraint:
(71)    (COND_EVAL_11(true, pos(0), neg(0), x293)≥EVAL_3(pos(0), neg(0), x293))

We simplified constraint (66) using rules (I), (II) which results in the following new constraint:
(72)    (COND_EVAL_11(true, neg(x556), neg(0), x293)≥EVAL_3(neg(x556), neg(0), x293))

We simplified constraint (67) using rule (VI) where we applied the induction hypothesis (∀x563:greatereq_int(pos(x562), pos(x561))=true ⇒ COND_EVAL_11(true, pos(x561), pos(x562), x563)≥EVAL_3(pos(x561), pos(x562), x563)) with σ = [x563 / x293] which results in the following new constraint:
(73)    (COND_EVAL_11(true, pos(x561), pos(x562), x293)≥EVAL_3(pos(x561), pos(x562), x293) ⇒ COND_EVAL_11(true, pos(s(x561)), pos(s(x562)), x293)≥EVAL_3(pos(s(x561)), pos(s(x562)), x293))

We simplified constraint (68) using rule (VI) where we applied the induction hypothesis (∀x566:greatereq_int(neg(x565), neg(x564))=true ⇒ COND_EVAL_11(true, neg(x564), neg(x565), x566)≥EVAL_3(neg(x564), neg(x565), x566)) with σ = [x566 / x293] which results in the following new constraint:
(74)    (COND_EVAL_11(true, neg(x564), neg(x565), x293)≥EVAL_3(neg(x564), neg(x565), x293) ⇒ COND_EVAL_11(true, neg(s(x564)), neg(s(x565)), x293)≥EVAL_3(neg(s(x564)), neg(s(x565)), x293))

For Pair EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed), EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed) which results in the following constraint:
(75) (EVAL_3(x345, x346, x347)=EVAL_3(x348, x349, x350) ⇒ EVAL_3(x348, x349, x350)≥COND_EVAL_31(greatereq_int(pos(0), x349), x348, x349, x350))

We simplified constraint (75) using rules (I), (II), (III) which results in the following new constraint:
(76) (EVAL_3(x345, x346, x347)≥COND_EVAL_31(greatereq_int(pos(0), x346), x345, x346, x347))
We consider the chain COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed), EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed) which results in the following constraint:
(77) (EVAL_3(x357, minus_int(x358, x359), x359)=EVAL_3(x360, x361, x362) ⇒ EVAL_3(x360, x361, x362)≥COND_EVAL_31(greatereq_int(pos(0), x361), x360, x361, x362))

We simplified constraint (77) using rules (I), (II), (III), (IV) which results in the following new constraint:
(78) (EVAL_3(x357, x361, x359)≥COND_EVAL_31(greatereq_int(pos(0), x361), x357, x361, x359))

For Pair EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed), EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed) which results in the following constraint:
(79) (EVAL_3(x387, x388, x389)=EVAL_3(x390, x391, x392) ⇒ EVAL_3(x390, x391, x392)≥COND_EVAL_3(greater_int(x391, pos(0)), x390, x391, x392))

We simplified constraint (79) using rules (I), (II), (III) which results in the following new constraint:
(80) (EVAL_3(x387, x388, x389)≥COND_EVAL_3(greater_int(x388, pos(0)), x387, x388, x389))
We consider the chain COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed), EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed) which results in the following constraint:
(81) (EVAL_3(x399, minus_int(x400, x401), x401)=EVAL_3(x402, x403, x404) ⇒ EVAL_3(x402, x403, x404)≥COND_EVAL_3(greater_int(x403, pos(0)), x402, x403, x404))

We simplified constraint (81) using rules (I), (II), (III), (IV) which results in the following new constraint:
(82) (EVAL_3(x399, x403, x401)≥COND_EVAL_3(greater_int(x403, pos(0)), x399, x403, x401))

For Pair COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed) the following chains were created:

We consider the chain EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed), COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed) which results in the following constraint:
(83)    (COND_EVAL_3(greater_int(x436, pos(0)), x435, x436, x437)=COND_EVAL_3(true, x438, x439, x440) ⇒ COND_EVAL_3(true, x438, x439, x440)≥EVAL_3(x438, minus_int(x439, x440), x440))

We simplified constraint (83) using rules (I), (II), (III), (VII) which results in the following new constraint:
(84)    (pos(0)=x567∧greater_int(x436, x567)=true ⇒ COND_EVAL_3(true, x435, x436, x437)≥EVAL_3(x435, minus_int(x436, x437), x437))

We simplified constraint (84) using rule (V) (with possible (I) afterwards) using induction on greater_int(x436, x567)=true which results in the following new constraints:
(85)    (true=true∧pos(0)=pos(0) ⇒ COND_EVAL_3(true, x435, pos(s(x568)), x437)≥EVAL_3(x435, minus_int(pos(s(x568)), x437), x437))

(86)    (true=true∧pos(0)=neg(s(x572)) ⇒ COND_EVAL_3(true, x435, pos(0), x437)≥EVAL_3(x435, minus_int(pos(0), x437), x437))

(87)    (true=true∧pos(0)=neg(s(x573)) ⇒ COND_EVAL_3(true, x435, neg(0), x437)≥EVAL_3(x435, minus_int(neg(0), x437), x437))

(88)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_3(true, x435, pos(s(x574)), x437)≥EVAL_3(x435, minus_int(pos(s(x574)), x437), x437))

(89)    (true=true∧pos(0)=neg(s(x576)) ⇒ COND_EVAL_3(true, x435, pos(s(x577)), x437)≥EVAL_3(x435, minus_int(pos(s(x577)), x437), x437))

(90)    (greater_int(pos(x581), pos(x580))=true∧pos(0)=pos(s(x580))∧(∀x582,x583:greater_int(pos(x581), pos(x580))=true∧pos(0)=pos(x580) ⇒ COND_EVAL_3(true, x582, pos(x581), x583)≥EVAL_3(x582, minus_int(pos(x581), x583), x583)) ⇒ COND_EVAL_3(true, x435, pos(s(x581)), x437)≥EVAL_3(x435, minus_int(pos(s(x581)), x437), x437))

(91)    (greater_int(neg(x585), neg(x584))=true∧pos(0)=neg(s(x584))∧(∀x586,x587:greater_int(neg(x585), neg(x584))=true∧pos(0)=neg(x584) ⇒ COND_EVAL_3(true, x586, neg(x585), x587)≥EVAL_3(x586, minus_int(neg(x585), x587), x587)) ⇒ COND_EVAL_3(true, x435, neg(s(x585)), x437)≥EVAL_3(x435, minus_int(neg(s(x585)), x437), x437))

We simplified constraint (85) using rules (I), (II) which results in the following new constraint:
(92)    (COND_EVAL_3(true, x435, pos(s(x568)), x437)≥EVAL_3(x435, minus_int(pos(s(x568)), x437), x437))

We solved constraint (86) using rules (I), (II).We solved constraint (87) using rules (I), (II).We solved constraint (88) using rules (I), (II).We solved constraint (89) using rules (I), (II).We solved constraint (90) using rules (I), (II).We solved constraint (91) using rules (I), (II).

For Pair COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed) the following chains were created:

We consider the chain EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed), COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed) which results in the following constraint:
(93)    (COND_EVAL_2(greater_int(x462, pos(0)), x462, x463, x464)=COND_EVAL_2(true, x465, x466, x467) ⇒ COND_EVAL_2(true, x465, x466, x467)≥EVAL_2(minus_int(x465, x467), x466, x467))

We simplified constraint (93) using rules (I), (II), (III), (VII) which results in the following new constraint:
(94)    (pos(0)=x588∧greater_int(x462, x588)=true ⇒ COND_EVAL_2(true, x462, x463, x464)≥EVAL_2(minus_int(x462, x464), x463, x464))

We simplified constraint (94) using rule (V) (with possible (I) afterwards) using induction on greater_int(x462, x588)=true which results in the following new constraints:
(95)    (true=true∧pos(0)=pos(0) ⇒ COND_EVAL_2(true, pos(s(x589)), x463, x464)≥EVAL_2(minus_int(pos(s(x589)), x464), x463, x464))

(96)    (true=true∧pos(0)=neg(s(x593)) ⇒ COND_EVAL_2(true, pos(0), x463, x464)≥EVAL_2(minus_int(pos(0), x464), x463, x464))

(97)    (true=true∧pos(0)=neg(s(x594)) ⇒ COND_EVAL_2(true, neg(0), x463, x464)≥EVAL_2(minus_int(neg(0), x464), x463, x464))

(98)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_2(true, pos(s(x595)), x463, x464)≥EVAL_2(minus_int(pos(s(x595)), x464), x463, x464))

(99)    (true=true∧pos(0)=neg(s(x597)) ⇒ COND_EVAL_2(true, pos(s(x598)), x463, x464)≥EVAL_2(minus_int(pos(s(x598)), x464), x463, x464))

(100)    (greater_int(pos(x602), pos(x601))=true∧pos(0)=pos(s(x601))∧(∀x603,x604:greater_int(pos(x602), pos(x601))=true∧pos(0)=pos(x601) ⇒ COND_EVAL_2(true, pos(x602), x603, x604)≥EVAL_2(minus_int(pos(x602), x604), x603, x604)) ⇒ COND_EVAL_2(true, pos(s(x602)), x463, x464)≥EVAL_2(minus_int(pos(s(x602)), x464), x463, x464))

(101)    (greater_int(neg(x606), neg(x605))=true∧pos(0)=neg(s(x605))∧(∀x607,x608:greater_int(neg(x606), neg(x605))=true∧pos(0)=neg(x605) ⇒ COND_EVAL_2(true, neg(x606), x607, x608)≥EVAL_2(minus_int(neg(x606), x608), x607, x608)) ⇒ COND_EVAL_2(true, neg(s(x606)), x463, x464)≥EVAL_2(minus_int(neg(s(x606)), x464), x463, x464))

We simplified constraint (95) using rules (I), (II) which results in the following new constraint:
(102)    (COND_EVAL_2(true, pos(s(x589)), x463, x464)≥EVAL_2(minus_int(pos(s(x589)), x464), x463, x464))

We solved constraint (96) using rules (I), (II).We solved constraint (97) using rules (I), (II).We solved constraint (98) using rules (I), (II).We solved constraint (99) using rules (I), (II).We solved constraint (100) using rules (I), (II).We solved constraint (101) using rules (I), (II).

To summarize, we get the following constraints P_≥ for the following pairs.

COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
- (COND_EVAL_31(true, x24, pos(0), x26)≥EVAL_1(x24, pos(0), x26))
- (COND_EVAL_31(true, x24, neg(x488), x26)≥EVAL_1(x24, neg(x488), x26))
EVAL_1(x, y, x_removed) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y, x_removed)
- (EVAL_1(x39, x40, x41)≥COND_EVAL_1(and(and(greater_int(x39, pos(0)), greater_int(x40, pos(0))), greater_int(x39, x40)), x39, x40, x41))
- (EVAL_1(x60, x61, x62)≥COND_EVAL_1(and(and(greater_int(x60, pos(0)), greater_int(x61, pos(0))), greater_int(x60, x61)), x60, x61, x62))
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
- (EVAL_1(x81, x82, x83)≥COND_EVAL_11(and(and(greater_int(x81, pos(0)), greater_int(x82, pos(0))), greatereq_int(x82, x81)), x81, x82, x83))
- (EVAL_1(x102, x103, x104)≥COND_EVAL_11(and(and(greater_int(x102, pos(0)), greater_int(x103, pos(0))), greatereq_int(x103, x102)), x102, x103, x104))
COND_EVAL_1(true, x, y, x_removed) → EVAL_2(x, y, x_removed)
- (COND_EVAL_1(true, pos(s(x510)), pos(0), x128)≥EVAL_2(pos(s(x510)), pos(0), x128))
- (COND_EVAL_1(true, pos(0), neg(s(x514)), x128)≥EVAL_2(pos(0), neg(s(x514)), x128))
- (COND_EVAL_1(true, neg(0), neg(s(x515)), x128)≥EVAL_2(neg(0), neg(s(x515)), x128))
- (COND_EVAL_1(true, pos(s(x516)), neg(0), x128)≥EVAL_2(pos(s(x516)), neg(0), x128))
- (COND_EVAL_1(true, pos(s(x519)), neg(s(x518)), x128)≥EVAL_2(pos(s(x519)), neg(s(x518)), x128))
- (COND_EVAL_1(true, pos(x523), pos(x522), x128)≥EVAL_2(pos(x523), pos(x522), x128) ⇒ COND_EVAL_1(true, pos(s(x523)), pos(s(x522)), x128)≥EVAL_2(pos(s(x523)), pos(s(x522)), x128))
- (COND_EVAL_1(true, neg(x526), neg(x525), x128)≥EVAL_2(neg(x526), neg(x525), x128) ⇒ COND_EVAL_1(true, neg(s(x526)), neg(s(x525)), x128)≥EVAL_2(neg(s(x526)), neg(s(x525)), x128))
EVAL_2(x, y, x_removed) → COND_EVAL_21(greatereq_int(pos(0), x), x, y, x_removed)
- (EVAL_2(x171, x172, x173)≥COND_EVAL_21(greatereq_int(pos(0), x171), x171, x172, x173))
- (EVAL_2(x201, x199, x200)≥COND_EVAL_21(greatereq_int(pos(0), x201), x201, x199, x200))
EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
- (EVAL_2(x213, x214, x215)≥COND_EVAL_2(greater_int(x213, pos(0)), x213, x214, x215))
- (EVAL_2(x243, x241, x242)≥COND_EVAL_2(greater_int(x243, pos(0)), x243, x241, x242))
COND_EVAL_21(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
- (COND_EVAL_21(true, pos(0), x259, x260)≥EVAL_1(pos(0), x259, x260))
- (COND_EVAL_21(true, neg(x530), x259, x260)≥EVAL_1(neg(x530), x259, x260))
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
- (COND_EVAL_11(true, pos(0), pos(x552), x293)≥EVAL_3(pos(0), pos(x552), x293))
- (COND_EVAL_11(true, neg(x553), pos(x554), x293)≥EVAL_3(neg(x553), pos(x554), x293))
- (COND_EVAL_11(true, pos(0), neg(0), x293)≥EVAL_3(pos(0), neg(0), x293))
- (COND_EVAL_11(true, neg(x556), neg(0), x293)≥EVAL_3(neg(x556), neg(0), x293))
- (COND_EVAL_11(true, pos(x561), pos(x562), x293)≥EVAL_3(pos(x561), pos(x562), x293) ⇒ COND_EVAL_11(true, pos(s(x561)), pos(s(x562)), x293)≥EVAL_3(pos(s(x561)), pos(s(x562)), x293))
- (COND_EVAL_11(true, neg(x564), neg(x565), x293)≥EVAL_3(neg(x564), neg(x565), x293) ⇒ COND_EVAL_11(true, neg(s(x564)), neg(s(x565)), x293)≥EVAL_3(neg(s(x564)), neg(s(x565)), x293))
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
- (EVAL_3(x345, x346, x347)≥COND_EVAL_31(greatereq_int(pos(0), x346), x345, x346, x347))
- (EVAL_3(x357, x361, x359)≥COND_EVAL_31(greatereq_int(pos(0), x361), x357, x361, x359))
EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed)
- (EVAL_3(x387, x388, x389)≥COND_EVAL_3(greater_int(x388, pos(0)), x387, x388, x389))
- (EVAL_3(x399, x403, x401)≥COND_EVAL_3(greater_int(x403, pos(0)), x399, x403, x401))
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)
- (COND_EVAL_3(true, x435, pos(s(x568)), x437)≥EVAL_3(x435, minus_int(pos(s(x568)), x437), x437))
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)
- (COND_EVAL_2(true, pos(s(x589)), x463, x464)≥EVAL_2(minus_int(pos(s(x589)), x464), x463, x464))

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [NONINF]:

POL(0) = 1
POL(COND_EVAL_1(x₁, x₂, x₃, x₄)) = -1 - x₁ - x₃
POL(COND_EVAL_11(x₁, x₂, x₃, x₄)) = -1 - x₁ - x₂ - x₃
POL(COND_EVAL_2(x₁, x₂, x₃, x₄)) = -1 - x₁ - x₂ - x₃
POL(COND_EVAL_21(x₁, x₂, x₃, x₄)) = -1 - x₃
POL(COND_EVAL_3(x₁, x₂, x₃, x₄)) = -1 - x₁ - x₂ - x₃
POL(COND_EVAL_31(x₁, x₂, x₃, x₄)) = -1 - x₂ - x₃
POL(EVAL_1(x₁, x₂, x₃)) = -1 - x₁ - x₂
POL(EVAL_2(x₁, x₂, x₃)) = -1 - x₂
POL(EVAL_3(x₁, x₂, x₃)) = -1 - x₁ - x₂
POL(and(x₁, x₂)) = x₁
POL(c) = -3
POL(false) = 1
POL(greater_int(x₁, x₂)) = x₁
POL(greatereq_int(x₁, x₂)) = 0
POL(minus_int(x₁, x₂)) = 0
POL(minus_nat(x₁, x₂)) = x₁
POL(neg(x₁)) = 1
POL(plus_nat(x₁, x₂)) = x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 0
POL(true) = 0

The following pairs are in P_>:

COND_EVAL_21(true, x, y, x_removed) → EVAL_1(x, y, x_removed)

The following pairs are in P_bound:

COND_EVAL_1(true, x, y, x_removed) → EVAL_2(x, y, x_removed)

The following rules are usable:

true → and(true, true)
false → and(true, false)
false → greater_int(neg(s(x)), pos(0))
true → greater_int(pos(s(x)), pos(0))
false → and(false, true)
false → and(false, false)
pos(0) → minus_nat(0, 0)
neg(s(y)) → minus_nat(0, s(y))
false → greater_int(pos(0), pos(0))
false → greater_int(neg(0), pos(0))
pos(s(x)) → minus_nat(s(x), 0)
minus_nat(x, y) → minus_nat(s(x), s(y))
minus_nat(x, y) → minus_int(pos(x), pos(y))
neg(plus_nat(x, y)) → minus_int(neg(x), pos(y))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                ↳ QDP

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

COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
COND_EVAL_1(true, x, y, x_removed) → EVAL_2(x, y, x_removed)
EVAL_2(x, y, x_removed) → COND_EVAL_21(greatereq_int(pos(0), x), x, y, x_removed)
EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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 2 SCCs with 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                      ↳ QDP

                                ↳ QDP

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

EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                      ↳ QDP

                                ↳ QDP

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

EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

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)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                      ↳ QDP

                                ↳ QDP

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

EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed) at position [0] we obtained the following new rules [LPAR04]:

EVAL_2(neg(s(x0)), y1, y2) → COND_EVAL_2(false, neg(s(x0)), y1, y2)
EVAL_2(neg(0), y1, y2) → COND_EVAL_2(false, neg(0), y1, y2)
EVAL_2(pos(0), y1, y2) → COND_EVAL_2(false, pos(0), y1, y2)
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                ↳ QDP

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

COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)
EVAL_2(neg(s(x0)), y1, y2) → COND_EVAL_2(false, neg(s(x0)), y1, y2)
EVAL_2(neg(0), y1, y2) → COND_EVAL_2(false, neg(0), y1, y2)
EVAL_2(pos(0), y1, y2) → COND_EVAL_2(false, pos(0), y1, y2)
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                      ↳ QDP

                                ↳ QDP

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                      ↳ QDP

                                ↳ QDP

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                      ↳ QDP

                                ↳ QDP

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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 COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed) at position [0] we obtained the following new rules [LPAR04]:

COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
COND_EVAL_2(true, neg(x0), y1, pos(x1)) → EVAL_2(neg(plus_nat(x0, x1)), y1, pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                ↳ QDP

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
COND_EVAL_2(true, neg(x0), y1, pos(x1)) → EVAL_2(neg(plus_nat(x0, x1)), y1, pos(x1))

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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 1 SCC with 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                      ↳ QDP

                                ↳ QDP

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

COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ QReductionProof

                                      ↳ QDP

                                ↳ QDP

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

COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

The TRS R consists of the following rules:

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:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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].

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ QReductionProof

                                                                              ↳ QDP

                                                                                ↳ Instantiation

                                      ↳ QDP

                                ↳ QDP

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

COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

The TRS R consists of the following rules:

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:

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 instantiating [LPAR04] the rule COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1)) we obtained the following new rules [LPAR04]:

COND_EVAL_2(true, pos(s(z0)), z1, pos(x2)) → EVAL_2(minus_nat(s(z0), x2), z1, pos(x2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ QReductionProof

                                                                              ↳ QDP

                                                                                ↳ Instantiation

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                      ↳ QDP

                                ↳ QDP

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, pos(s(z0)), z1, pos(x2)) → EVAL_2(minus_nat(s(z0), x2), z1, pos(x2))

The TRS R consists of the following rules:

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:

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 instantiating [LPAR04] the rule EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2) we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(x0)), z1, pos(z2)) → COND_EVAL_2(true, pos(s(x0)), z1, pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ QReductionProof

                                                                              ↳ QDP

                                                                                ↳ Instantiation

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ MNOCProof

                                      ↳ QDP

                                ↳ QDP

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

COND_EVAL_2(true, pos(s(z0)), z1, pos(x2)) → EVAL_2(minus_nat(s(z0), x2), z1, pos(x2))
EVAL_2(pos(s(x0)), z1, pos(z2)) → COND_EVAL_2(true, pos(s(x0)), z1, pos(z2))

The TRS R consists of the following rules:

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:

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 modular non-overlap check [FROCOS05] to decrease Q to the empty set.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ QReductionProof

                                                                              ↳ QDP

                                                                                ↳ Instantiation

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ MNOCProof

                                                                                          ↳ QDP

                                                                                            ↳ NonTerminationProof

                                      ↳ QDP

                                ↳ QDP

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

COND_EVAL_2(true, pos(s(z0)), z1, pos(x2)) → EVAL_2(minus_nat(s(z0), x2), z1, pos(x2))
EVAL_2(pos(s(x0)), z1, pos(z2)) → COND_EVAL_2(true, pos(s(x0)), z1, pos(z2))

The TRS R consists of the following rules:

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)

Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

COND_EVAL_2(true, pos(s(z0)), z1, pos(x2)) → EVAL_2(minus_nat(s(z0), x2), z1, pos(x2))
EVAL_2(pos(s(x0)), z1, pos(z2)) → COND_EVAL_2(true, pos(s(x0)), z1, pos(z2))

The TRS R consists of the following rules:

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)

s = EVAL_2(minus_nat(s(x), 0), z1a, pos(z2)) evaluates to t =EVAL_2(minus_nat(s(x), z2), z1a, pos(z2))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [z2 / 0]
Matcher: [ ]

Rewriting sequence

EVAL_2(minus_nat(s(x), 0), z1a, pos(0)) → EVAL_2(pos(s(x)), z1a, pos(0))
with rule minus_nat(s(xa), 0) → pos(s(xa)) at position [0] and matcher [xa / x]

EVAL_2(pos(s(x)), z1a, pos(0)) → COND_EVAL_2(true, pos(s(x)), z1a, pos(0))
with rule EVAL_2(pos(s(x0)), z1aa, pos(z2)) → COND_EVAL_2(true, pos(s(x0)), z1aa, pos(z2)) at position [] and matcher [z1aa / z1a, z2 / 0, x0 / x]

COND_EVAL_2(true, pos(s(x)), z1a, pos(0)) → EVAL_2(minus_nat(s(x), 0), z1a, pos(0))
with rule COND_EVAL_2(true, pos(s(z0)), z1, pos(x2)) → EVAL_2(minus_nat(s(z0), x2), z1, pos(x2))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                ↳ QDP

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

EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                ↳ QDP

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

EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed) at position [0] we obtained the following new rules [LPAR04]:

EVAL_3(y0, neg(s(x0)), y2) → COND_EVAL_3(false, y0, neg(s(x0)), y2)
EVAL_3(y0, pos(0), y2) → COND_EVAL_3(false, y0, pos(0), y2)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
EVAL_3(y0, neg(0), y2) → COND_EVAL_3(false, y0, neg(0), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                ↳ QDP

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

EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)
EVAL_3(y0, neg(s(x0)), y2) → COND_EVAL_3(false, y0, neg(s(x0)), y2)
EVAL_3(y0, pos(0), y2) → COND_EVAL_3(false, y0, pos(0), y2)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
EVAL_3(y0, neg(0), y2) → COND_EVAL_3(false, y0, neg(0), y2)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                ↳ QDP

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

COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed) the following chains were created:

We consider the chain EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed), COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed) which results in the following constraint:
(1)    (COND_EVAL_11(and(and(greater_int(x9, pos(0)), greater_int(x10, pos(0))), greatereq_int(x10, x9)), x9, x10, x11)=COND_EVAL_11(true, x12, x13, x14) ⇒ COND_EVAL_11(true, x12, x13, x14)≥EVAL_3(x12, x13, x14))

We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(2)    (and(x134, x135)=x132∧greatereq_int(x10, x9)=x133∧and(x132, x133)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on and(x132, x133)=true which results in the following new constraint:
(3)    (true=true∧and(x134, x135)=true∧greatereq_int(x10, x9)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
(4)    (and(x134, x135)=true∧greatereq_int(x10, x9)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on and(x134, x135)=true which results in the following new constraint:
(5)    (true=true∧greatereq_int(x10, x9)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (5) using rules (I), (II) which results in the following new constraint:
(6)    (greatereq_int(x10, x9)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x10, x9)=true which results in the following new constraints:
(7)    (true=true ⇒ COND_EVAL_11(true, pos(0), pos(x138), x11)≥EVAL_3(pos(0), pos(x138), x11))

(8)    (true=true ⇒ COND_EVAL_11(true, neg(x139), pos(x140), x11)≥EVAL_3(neg(x139), pos(x140), x11))

(9)    (true=true ⇒ COND_EVAL_11(true, pos(0), neg(0), x11)≥EVAL_3(pos(0), neg(0), x11))

(10)    (true=true ⇒ COND_EVAL_11(true, neg(x142), neg(0), x11)≥EVAL_3(neg(x142), neg(0), x11))

(11)    (greatereq_int(pos(x148), pos(x147))=true∧(∀x149:greatereq_int(pos(x148), pos(x147))=true ⇒ COND_EVAL_11(true, pos(x147), pos(x148), x149)≥EVAL_3(pos(x147), pos(x148), x149)) ⇒ COND_EVAL_11(true, pos(s(x147)), pos(s(x148)), x11)≥EVAL_3(pos(s(x147)), pos(s(x148)), x11))

(12)    (greatereq_int(neg(x151), neg(x150))=true∧(∀x152:greatereq_int(neg(x151), neg(x150))=true ⇒ COND_EVAL_11(true, neg(x150), neg(x151), x152)≥EVAL_3(neg(x150), neg(x151), x152)) ⇒ COND_EVAL_11(true, neg(s(x150)), neg(s(x151)), x11)≥EVAL_3(neg(s(x150)), neg(s(x151)), x11))

We simplified constraint (7) using rules (I), (II) which results in the following new constraint:
(13)    (COND_EVAL_11(true, pos(0), pos(x138), x11)≥EVAL_3(pos(0), pos(x138), x11))

We simplified constraint (8) using rules (I), (II) which results in the following new constraint:
(14)    (COND_EVAL_11(true, neg(x139), pos(x140), x11)≥EVAL_3(neg(x139), pos(x140), x11))

We simplified constraint (9) using rules (I), (II) which results in the following new constraint:
(15)    (COND_EVAL_11(true, pos(0), neg(0), x11)≥EVAL_3(pos(0), neg(0), x11))

We simplified constraint (10) using rules (I), (II) which results in the following new constraint:
(16)    (COND_EVAL_11(true, neg(x142), neg(0), x11)≥EVAL_3(neg(x142), neg(0), x11))

We simplified constraint (11) using rule (VI) where we applied the induction hypothesis (∀x149:greatereq_int(pos(x148), pos(x147))=true ⇒ COND_EVAL_11(true, pos(x147), pos(x148), x149)≥EVAL_3(pos(x147), pos(x148), x149)) with σ = [x149 / x11] which results in the following new constraint:
(17)    (COND_EVAL_11(true, pos(x147), pos(x148), x11)≥EVAL_3(pos(x147), pos(x148), x11) ⇒ COND_EVAL_11(true, pos(s(x147)), pos(s(x148)), x11)≥EVAL_3(pos(s(x147)), pos(s(x148)), x11))

We simplified constraint (12) using rule (VI) where we applied the induction hypothesis (∀x152:greatereq_int(neg(x151), neg(x150))=true ⇒ COND_EVAL_11(true, neg(x150), neg(x151), x152)≥EVAL_3(neg(x150), neg(x151), x152)) with σ = [x152 / x11] which results in the following new constraint:
(18)    (COND_EVAL_11(true, neg(x150), neg(x151), x11)≥EVAL_3(neg(x150), neg(x151), x11) ⇒ COND_EVAL_11(true, neg(s(x150)), neg(s(x151)), x11)≥EVAL_3(neg(s(x150)), neg(s(x151)), x11))

For Pair EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed), EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed) which results in the following constraint:
(19) (EVAL_3(x21, x22, x23)=EVAL_3(x24, x25, x26) ⇒ EVAL_3(x24, x25, x26)≥COND_EVAL_31(greatereq_int(pos(0), x25), x24, x25, x26))

We simplified constraint (19) using rules (I), (II), (III) which results in the following new constraint:
(20) (EVAL_3(x21, x22, x23)≥COND_EVAL_31(greatereq_int(pos(0), x22), x21, x22, x23))
We consider the chain COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed), EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed) which results in the following constraint:
(21) (EVAL_3(x39, minus_int(x40, x41), x41)=EVAL_3(x42, x43, x44) ⇒ EVAL_3(x42, x43, x44)≥COND_EVAL_31(greatereq_int(pos(0), x43), x42, x43, x44))

We simplified constraint (21) using rules (I), (II), (III), (IV) which results in the following new constraint:
(22) (EVAL_3(x39, x43, x41)≥COND_EVAL_31(greatereq_int(pos(0), x43), x39, x43, x41))

For Pair COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed) the following chains were created:

We consider the chain EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed), COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed) which results in the following constraint:
(23)    (COND_EVAL_31(greatereq_int(pos(0), x49), x48, x49, x50)=COND_EVAL_31(true, x51, x52, x53) ⇒ COND_EVAL_31(true, x51, x52, x53)≥EVAL_1(x51, x52, x53))

We simplified constraint (23) using rules (I), (II), (III), (VII) which results in the following new constraint:
(24)    (pos(0)=x153∧greatereq_int(x153, x49)=true ⇒ COND_EVAL_31(true, x48, x49, x50)≥EVAL_1(x48, x49, x50))

We simplified constraint (24) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x153, x49)=true which results in the following new constraints:
(25)    (true=true∧pos(0)=pos(x154) ⇒ COND_EVAL_31(true, x48, pos(0), x50)≥EVAL_1(x48, pos(0), x50))

(26)    (true=true∧pos(0)=pos(x156) ⇒ COND_EVAL_31(true, x48, neg(x155), x50)≥EVAL_1(x48, neg(x155), x50))

(27)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_31(true, x48, pos(0), x50)≥EVAL_1(x48, pos(0), x50))

(28)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_31(true, x48, neg(x158), x50)≥EVAL_1(x48, neg(x158), x50))

(29)    (greatereq_int(pos(x164), pos(x163))=true∧pos(0)=pos(s(x164))∧(∀x165,x166:greatereq_int(pos(x164), pos(x163))=true∧pos(0)=pos(x164) ⇒ COND_EVAL_31(true, x165, pos(x163), x166)≥EVAL_1(x165, pos(x163), x166)) ⇒ COND_EVAL_31(true, x48, pos(s(x163)), x50)≥EVAL_1(x48, pos(s(x163)), x50))

(30)    (greatereq_int(neg(x168), neg(x167))=true∧pos(0)=neg(s(x168))∧(∀x169,x170:greatereq_int(neg(x168), neg(x167))=true∧pos(0)=neg(x168) ⇒ COND_EVAL_31(true, x169, neg(x167), x170)≥EVAL_1(x169, neg(x167), x170)) ⇒ COND_EVAL_31(true, x48, neg(s(x167)), x50)≥EVAL_1(x48, neg(s(x167)), x50))

We simplified constraint (25) using rules (I), (II), (IV) which results in the following new constraint:
(31)    (COND_EVAL_31(true, x48, pos(0), x50)≥EVAL_1(x48, pos(0), x50))

We simplified constraint (26) using rules (I), (II), (IV) which results in the following new constraint:
(32)    (COND_EVAL_31(true, x48, neg(x155), x50)≥EVAL_1(x48, neg(x155), x50))

We solved constraint (27) using rules (I), (II).We solved constraint (28) using rules (I), (II).We solved constraint (29) using rules (I), (II).We solved constraint (30) using rules (I), (II).

For Pair EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed), EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed) which results in the following constraint:
(33) (EVAL_1(x72, x73, x74)=EVAL_1(x75, x76, x77) ⇒ EVAL_1(x75, x76, x77)≥COND_EVAL_11(and(and(greater_int(x75, pos(0)), greater_int(x76, pos(0))), greatereq_int(x76, x75)), x75, x76, x77))

We simplified constraint (33) using rules (I), (II), (III) which results in the following new constraint:
(34) (EVAL_1(x72, x73, x74)≥COND_EVAL_11(and(and(greater_int(x72, pos(0)), greater_int(x73, pos(0))), greatereq_int(x73, x72)), x72, x73, x74))

For Pair EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2) the following chains were created:

We consider the chain COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed), EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2) which results in the following constraint:
(35) (EVAL_3(x87, x88, x89)=EVAL_3(x90, pos(s(x91)), x92) ⇒ EVAL_3(x90, pos(s(x91)), x92)≥COND_EVAL_3(true, x90, pos(s(x91)), x92))

We simplified constraint (35) using rules (I), (II), (III) which results in the following new constraint:
(36) (EVAL_3(x87, pos(s(x91)), x89)≥COND_EVAL_3(true, x87, pos(s(x91)), x89))
We consider the chain COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed), EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2) which results in the following constraint:
(37)    (EVAL_3(x105, minus_int(x106, x107), x107)=EVAL_3(x108, pos(s(x109)), x110) ⇒ EVAL_3(x108, pos(s(x109)), x110)≥COND_EVAL_3(true, x108, pos(s(x109)), x110))

We simplified constraint (37) using rules (I), (II), (III) which results in the following new constraint:
(38)    (minus_int(x106, x107)=pos(s(x109)) ⇒ EVAL_3(x105, pos(s(x109)), x107)≥COND_EVAL_3(true, x105, pos(s(x109)), x107))

We simplified constraint (38) using rule (V) (with possible (I) afterwards) using induction on minus_int(x106, x107)=pos(s(x109)) which results in the following new constraint:
(39)    (minus_nat(x172, x171)=pos(s(x109)) ⇒ EVAL_3(x105, pos(s(x109)), pos(x171))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(x171)))

We simplified constraint (39) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x172, x171)=pos(s(x109)) which results in the following new constraints:
(40)    (pos(0)=pos(s(x109)) ⇒ EVAL_3(x105, pos(s(x109)), pos(0))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(0)))

(41)    (pos(s(x176))=pos(s(x109)) ⇒ EVAL_3(x105, pos(s(x109)), pos(0))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(0)))

(42)    (minus_nat(x178, x177)=pos(s(x109))∧(∀x179,x180:minus_nat(x178, x177)=pos(s(x179)) ⇒ EVAL_3(x180, pos(s(x179)), pos(x177))≥COND_EVAL_3(true, x180, pos(s(x179)), pos(x177))) ⇒ EVAL_3(x105, pos(s(x109)), pos(s(x177)))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(s(x177))))

We solved constraint (40) using rules (I), (II).We simplified constraint (41) using rules (I), (II), (III) which results in the following new constraint:
(43)    (EVAL_3(x105, pos(s(x176)), pos(0))≥COND_EVAL_3(true, x105, pos(s(x176)), pos(0)))

We simplified constraint (42) using rule (VI) where we applied the induction hypothesis (∀x179,x180:minus_nat(x178, x177)=pos(s(x179)) ⇒ EVAL_3(x180, pos(s(x179)), pos(x177))≥COND_EVAL_3(true, x180, pos(s(x179)), pos(x177))) with σ = [x179 / x109, x180 / x105] which results in the following new constraint:
(44)    (EVAL_3(x105, pos(s(x109)), pos(x177))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(x177)) ⇒ EVAL_3(x105, pos(s(x109)), pos(s(x177)))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(s(x177))))

For Pair COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed) the following chains were created:

We consider the chain EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2), COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed) which results in the following constraint:
(45) (COND_EVAL_3(true, x123, pos(s(x124)), x125)=COND_EVAL_3(true, x126, x127, x128) ⇒ COND_EVAL_3(true, x126, x127, x128)≥EVAL_3(x126, minus_int(x127, x128), x128))

We simplified constraint (45) using rules (I), (II), (III) which results in the following new constraint:
(46) (COND_EVAL_3(true, x123, pos(s(x124)), x125)≥EVAL_3(x123, minus_int(pos(s(x124)), x125), x125))

To summarize, we get the following constraints P_≥ for the following pairs.

COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
- (COND_EVAL_11(true, pos(0), pos(x138), x11)≥EVAL_3(pos(0), pos(x138), x11))
- (COND_EVAL_11(true, neg(x139), pos(x140), x11)≥EVAL_3(neg(x139), pos(x140), x11))
- (COND_EVAL_11(true, pos(0), neg(0), x11)≥EVAL_3(pos(0), neg(0), x11))
- (COND_EVAL_11(true, neg(x142), neg(0), x11)≥EVAL_3(neg(x142), neg(0), x11))
- (COND_EVAL_11(true, pos(x147), pos(x148), x11)≥EVAL_3(pos(x147), pos(x148), x11) ⇒ COND_EVAL_11(true, pos(s(x147)), pos(s(x148)), x11)≥EVAL_3(pos(s(x147)), pos(s(x148)), x11))
- (COND_EVAL_11(true, neg(x150), neg(x151), x11)≥EVAL_3(neg(x150), neg(x151), x11) ⇒ COND_EVAL_11(true, neg(s(x150)), neg(s(x151)), x11)≥EVAL_3(neg(s(x150)), neg(s(x151)), x11))
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
- (EVAL_3(x21, x22, x23)≥COND_EVAL_31(greatereq_int(pos(0), x22), x21, x22, x23))
- (EVAL_3(x39, x43, x41)≥COND_EVAL_31(greatereq_int(pos(0), x43), x39, x43, x41))
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
- (COND_EVAL_31(true, x48, pos(0), x50)≥EVAL_1(x48, pos(0), x50))
- (COND_EVAL_31(true, x48, neg(x155), x50)≥EVAL_1(x48, neg(x155), x50))
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
- (EVAL_1(x72, x73, x74)≥COND_EVAL_11(and(and(greater_int(x72, pos(0)), greater_int(x73, pos(0))), greatereq_int(x73, x72)), x72, x73, x74))
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
- (EVAL_3(x87, pos(s(x91)), x89)≥COND_EVAL_3(true, x87, pos(s(x91)), x89))
- (EVAL_3(x105, pos(s(x176)), pos(0))≥COND_EVAL_3(true, x105, pos(s(x176)), pos(0)))
- (EVAL_3(x105, pos(s(x109)), pos(x177))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(x177)) ⇒ EVAL_3(x105, pos(s(x109)), pos(s(x177)))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(s(x177))))
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)
- (COND_EVAL_3(true, x123, pos(s(x124)), x125)≥EVAL_3(x123, minus_int(pos(s(x124)), x125), x125))

POL(0) = 0
POL(COND_EVAL_11(x₁, x₂, x₃, x₄)) = 1 + x₁ - x₂ + x₃
POL(COND_EVAL_3(x₁, x₂, x₃, x₄)) = x₁
POL(COND_EVAL_31(x₁, x₂, x₃, x₄)) = 1 - x₃
POL(EVAL_1(x₁, x₂, x₃)) = 1 + x₂
POL(EVAL_3(x₁, x₂, x₃)) = 1
POL(and(x₁, x₂)) = x₁
POL(c) = -1
POL(false) = 0
POL(greater_int(x₁, x₂)) = x₁
POL(greatereq_int(x₁, x₂)) = 0
POL(minus_int(x₁, x₂)) = 0
POL(minus_nat(x₁, x₂)) = 0
POL(neg(x₁)) = 0
POL(plus_nat(x₁, x₂)) = x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 1

The following pairs are in P_>:

COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)

The following pairs are in P_bound:

COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The following rules are usable:

and(true, true) → true
and(true, false) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, true) → false
and(false, false) → false
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                ↳ QDP

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

EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

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)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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 COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed) at position [1] we obtained the following new rules [LPAR04]:

COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
COND_EVAL_3(true, y0, neg(x0), pos(x1)) → EVAL_3(y0, neg(plus_nat(x0, x1)), pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
COND_EVAL_3(true, y0, neg(x0), pos(x1)) → EVAL_3(y0, neg(plus_nat(x0, x1)), pos(x1))

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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 1 SCC with 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                ↳ QDP

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

COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                ↳ QDP

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

COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)

The TRS R consists of the following rules:

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:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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].

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                ↳ QDP

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

COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)

The TRS R consists of the following rules:

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:

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 instantiating [LPAR04] the rule COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1)) we obtained the following new rules [LPAR04]:

COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))

The TRS R consists of the following rules:

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:

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 instantiating [LPAR04] the rule EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2) we obtained the following new rules [LPAR04]:

EVAL_3(z0, pos(s(x1)), pos(z2)) → COND_EVAL_3(true, z0, pos(s(x1)), pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ MNOCProof

                                ↳ QDP

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

COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))
EVAL_3(z0, pos(s(x1)), pos(z2)) → COND_EVAL_3(true, z0, pos(s(x1)), pos(z2))

The TRS R consists of the following rules:

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:

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 modular non-overlap check [FROCOS05] to decrease Q to the empty set.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ MNOCProof

                                                                                              ↳ QDP

                                                                                                ↳ NonTerminationProof

                                ↳ QDP

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

COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))
EVAL_3(z0, pos(s(x1)), pos(z2)) → COND_EVAL_3(true, z0, pos(s(x1)), pos(z2))

The TRS R consists of the following rules:

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)

COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))
EVAL_3(z0, pos(s(x1)), pos(z2)) → COND_EVAL_3(true, z0, pos(s(x1)), pos(z2))

The TRS R consists of the following rules:

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)

s = EVAL_3(z0a, minus_nat(s(x), 0), pos(z2)) evaluates to t =EVAL_3(z0a, minus_nat(s(x), z2), pos(z2))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [z2 / 0]
Matcher: [ ]

Rewriting sequence

EVAL_3(z0a, minus_nat(s(x), 0), pos(0)) → EVAL_3(z0a, pos(s(x)), pos(0))
with rule minus_nat(s(xa), 0) → pos(s(xa)) at position [1] and matcher [xa / x]

EVAL_3(z0a, pos(s(x)), pos(0)) → COND_EVAL_3(true, z0a, pos(s(x)), pos(0))
with rule EVAL_3(z0aa, pos(s(x1)), pos(z2)) → COND_EVAL_3(true, z0aa, pos(s(x1)), pos(z2)) at position [] and matcher [z2 / 0, x1 / x, z0aa / z0a]

COND_EVAL_3(true, z0a, pos(s(x)), pos(0)) → EVAL_3(z0a, minus_nat(s(x), 0), pos(0))
with rule COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

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

COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_1(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greater_int(x, y)), x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
EVAL_2(x, y, x_removed) → COND_EVAL_21(greatereq_int(pos(0), x), x, y, x_removed)
EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
COND_EVAL_21(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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 2 SCCs with 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                      ↳ QDP

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

EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                      ↳ QDP

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

EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_3(greater_int(y, pos(0)), x, y, x_removed)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

EVAL_3(y0, neg(s(x0)), y2) → COND_EVAL_3(false, y0, neg(s(x0)), y2)
EVAL_3(y0, pos(0), y2) → COND_EVAL_3(false, y0, pos(0), y2)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
EVAL_3(y0, neg(0), y2) → COND_EVAL_3(false, y0, neg(0), y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                      ↳ QDP

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

EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)
EVAL_3(y0, neg(s(x0)), y2) → COND_EVAL_3(false, y0, neg(s(x0)), y2)
EVAL_3(y0, pos(0), y2) → COND_EVAL_3(false, y0, pos(0), y2)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
EVAL_3(y0, neg(0), y2) → COND_EVAL_3(false, y0, neg(0), y2)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                      ↳ QDP

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

COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed) the following chains were created:

We consider the chain EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed), COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed) which results in the following constraint:
(1)    (COND_EVAL_11(and(and(greater_int(x9, pos(0)), greater_int(x10, pos(0))), greatereq_int(x10, x9)), x9, x10, x11)=COND_EVAL_11(true, x12, x13, x14) ⇒ COND_EVAL_11(true, x12, x13, x14)≥EVAL_3(x12, x13, x14))

We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(2)    (and(x134, x135)=x132∧greatereq_int(x10, x9)=x133∧and(x132, x133)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on and(x132, x133)=true which results in the following new constraint:
(3)    (true=true∧and(x134, x135)=true∧greatereq_int(x10, x9)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
(4)    (and(x134, x135)=true∧greatereq_int(x10, x9)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on and(x134, x135)=true which results in the following new constraint:
(5)    (true=true∧greatereq_int(x10, x9)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (5) using rules (I), (II) which results in the following new constraint:
(6)    (greatereq_int(x10, x9)=true ⇒ COND_EVAL_11(true, x9, x10, x11)≥EVAL_3(x9, x10, x11))

We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x10, x9)=true which results in the following new constraints:
(7)    (true=true ⇒ COND_EVAL_11(true, pos(0), pos(x138), x11)≥EVAL_3(pos(0), pos(x138), x11))

(8)    (true=true ⇒ COND_EVAL_11(true, neg(x139), pos(x140), x11)≥EVAL_3(neg(x139), pos(x140), x11))

(9)    (true=true ⇒ COND_EVAL_11(true, pos(0), neg(0), x11)≥EVAL_3(pos(0), neg(0), x11))

(10)    (true=true ⇒ COND_EVAL_11(true, neg(x142), neg(0), x11)≥EVAL_3(neg(x142), neg(0), x11))

(11)    (greatereq_int(pos(x148), pos(x147))=true∧(∀x149:greatereq_int(pos(x148), pos(x147))=true ⇒ COND_EVAL_11(true, pos(x147), pos(x148), x149)≥EVAL_3(pos(x147), pos(x148), x149)) ⇒ COND_EVAL_11(true, pos(s(x147)), pos(s(x148)), x11)≥EVAL_3(pos(s(x147)), pos(s(x148)), x11))

(12)    (greatereq_int(neg(x151), neg(x150))=true∧(∀x152:greatereq_int(neg(x151), neg(x150))=true ⇒ COND_EVAL_11(true, neg(x150), neg(x151), x152)≥EVAL_3(neg(x150), neg(x151), x152)) ⇒ COND_EVAL_11(true, neg(s(x150)), neg(s(x151)), x11)≥EVAL_3(neg(s(x150)), neg(s(x151)), x11))

We simplified constraint (7) using rules (I), (II) which results in the following new constraint:
(13)    (COND_EVAL_11(true, pos(0), pos(x138), x11)≥EVAL_3(pos(0), pos(x138), x11))

We simplified constraint (8) using rules (I), (II) which results in the following new constraint:
(14)    (COND_EVAL_11(true, neg(x139), pos(x140), x11)≥EVAL_3(neg(x139), pos(x140), x11))

We simplified constraint (9) using rules (I), (II) which results in the following new constraint:
(15)    (COND_EVAL_11(true, pos(0), neg(0), x11)≥EVAL_3(pos(0), neg(0), x11))

We simplified constraint (10) using rules (I), (II) which results in the following new constraint:
(16)    (COND_EVAL_11(true, neg(x142), neg(0), x11)≥EVAL_3(neg(x142), neg(0), x11))

We simplified constraint (11) using rule (VI) where we applied the induction hypothesis (∀x149:greatereq_int(pos(x148), pos(x147))=true ⇒ COND_EVAL_11(true, pos(x147), pos(x148), x149)≥EVAL_3(pos(x147), pos(x148), x149)) with σ = [x149 / x11] which results in the following new constraint:
(17)    (COND_EVAL_11(true, pos(x147), pos(x148), x11)≥EVAL_3(pos(x147), pos(x148), x11) ⇒ COND_EVAL_11(true, pos(s(x147)), pos(s(x148)), x11)≥EVAL_3(pos(s(x147)), pos(s(x148)), x11))

We simplified constraint (12) using rule (VI) where we applied the induction hypothesis (∀x152:greatereq_int(neg(x151), neg(x150))=true ⇒ COND_EVAL_11(true, neg(x150), neg(x151), x152)≥EVAL_3(neg(x150), neg(x151), x152)) with σ = [x152 / x11] which results in the following new constraint:
(18)    (COND_EVAL_11(true, neg(x150), neg(x151), x11)≥EVAL_3(neg(x150), neg(x151), x11) ⇒ COND_EVAL_11(true, neg(s(x150)), neg(s(x151)), x11)≥EVAL_3(neg(s(x150)), neg(s(x151)), x11))

For Pair EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed), EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed) which results in the following constraint:
(19) (EVAL_3(x21, x22, x23)=EVAL_3(x24, x25, x26) ⇒ EVAL_3(x24, x25, x26)≥COND_EVAL_31(greatereq_int(pos(0), x25), x24, x25, x26))

We simplified constraint (19) using rules (I), (II), (III) which results in the following new constraint:
(20) (EVAL_3(x21, x22, x23)≥COND_EVAL_31(greatereq_int(pos(0), x22), x21, x22, x23))
We consider the chain COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed), EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed) which results in the following constraint:
(21) (EVAL_3(x39, minus_int(x40, x41), x41)=EVAL_3(x42, x43, x44) ⇒ EVAL_3(x42, x43, x44)≥COND_EVAL_31(greatereq_int(pos(0), x43), x42, x43, x44))

We simplified constraint (21) using rules (I), (II), (III), (IV) which results in the following new constraint:
(22) (EVAL_3(x39, x43, x41)≥COND_EVAL_31(greatereq_int(pos(0), x43), x39, x43, x41))

For Pair COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed) the following chains were created:

We consider the chain EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed), COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed) which results in the following constraint:
(23)    (COND_EVAL_31(greatereq_int(pos(0), x49), x48, x49, x50)=COND_EVAL_31(true, x51, x52, x53) ⇒ COND_EVAL_31(true, x51, x52, x53)≥EVAL_1(x51, x52, x53))

We simplified constraint (23) using rules (I), (II), (III), (VII) which results in the following new constraint:
(24)    (pos(0)=x153∧greatereq_int(x153, x49)=true ⇒ COND_EVAL_31(true, x48, x49, x50)≥EVAL_1(x48, x49, x50))

We simplified constraint (24) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x153, x49)=true which results in the following new constraints:
(25)    (true=true∧pos(0)=pos(x154) ⇒ COND_EVAL_31(true, x48, pos(0), x50)≥EVAL_1(x48, pos(0), x50))

(26)    (true=true∧pos(0)=pos(x156) ⇒ COND_EVAL_31(true, x48, neg(x155), x50)≥EVAL_1(x48, neg(x155), x50))

(27)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_31(true, x48, pos(0), x50)≥EVAL_1(x48, pos(0), x50))

(28)    (true=true∧pos(0)=neg(0) ⇒ COND_EVAL_31(true, x48, neg(x158), x50)≥EVAL_1(x48, neg(x158), x50))

(29)    (greatereq_int(pos(x164), pos(x163))=true∧pos(0)=pos(s(x164))∧(∀x165,x166:greatereq_int(pos(x164), pos(x163))=true∧pos(0)=pos(x164) ⇒ COND_EVAL_31(true, x165, pos(x163), x166)≥EVAL_1(x165, pos(x163), x166)) ⇒ COND_EVAL_31(true, x48, pos(s(x163)), x50)≥EVAL_1(x48, pos(s(x163)), x50))

(30)    (greatereq_int(neg(x168), neg(x167))=true∧pos(0)=neg(s(x168))∧(∀x169,x170:greatereq_int(neg(x168), neg(x167))=true∧pos(0)=neg(x168) ⇒ COND_EVAL_31(true, x169, neg(x167), x170)≥EVAL_1(x169, neg(x167), x170)) ⇒ COND_EVAL_31(true, x48, neg(s(x167)), x50)≥EVAL_1(x48, neg(s(x167)), x50))

We simplified constraint (25) using rules (I), (II), (IV) which results in the following new constraint:
(31)    (COND_EVAL_31(true, x48, pos(0), x50)≥EVAL_1(x48, pos(0), x50))

We simplified constraint (26) using rules (I), (II), (IV) which results in the following new constraint:
(32)    (COND_EVAL_31(true, x48, neg(x155), x50)≥EVAL_1(x48, neg(x155), x50))

We solved constraint (27) using rules (I), (II).We solved constraint (28) using rules (I), (II).We solved constraint (29) using rules (I), (II).We solved constraint (30) using rules (I), (II).

For Pair EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed) the following chains were created:

We consider the chain COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed), EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed) which results in the following constraint:
(33) (EVAL_1(x72, x73, x74)=EVAL_1(x75, x76, x77) ⇒ EVAL_1(x75, x76, x77)≥COND_EVAL_11(and(and(greater_int(x75, pos(0)), greater_int(x76, pos(0))), greatereq_int(x76, x75)), x75, x76, x77))

We simplified constraint (33) using rules (I), (II), (III) which results in the following new constraint:
(34) (EVAL_1(x72, x73, x74)≥COND_EVAL_11(and(and(greater_int(x72, pos(0)), greater_int(x73, pos(0))), greatereq_int(x73, x72)), x72, x73, x74))

For Pair EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2) the following chains were created:

We consider the chain COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed), EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2) which results in the following constraint:
(35) (EVAL_3(x87, x88, x89)=EVAL_3(x90, pos(s(x91)), x92) ⇒ EVAL_3(x90, pos(s(x91)), x92)≥COND_EVAL_3(true, x90, pos(s(x91)), x92))

We simplified constraint (35) using rules (I), (II), (III) which results in the following new constraint:
(36) (EVAL_3(x87, pos(s(x91)), x89)≥COND_EVAL_3(true, x87, pos(s(x91)), x89))
We consider the chain COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed), EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2) which results in the following constraint:
(37)    (EVAL_3(x105, minus_int(x106, x107), x107)=EVAL_3(x108, pos(s(x109)), x110) ⇒ EVAL_3(x108, pos(s(x109)), x110)≥COND_EVAL_3(true, x108, pos(s(x109)), x110))

We simplified constraint (37) using rules (I), (II), (III) which results in the following new constraint:
(38)    (minus_int(x106, x107)=pos(s(x109)) ⇒ EVAL_3(x105, pos(s(x109)), x107)≥COND_EVAL_3(true, x105, pos(s(x109)), x107))

We simplified constraint (38) using rule (V) (with possible (I) afterwards) using induction on minus_int(x106, x107)=pos(s(x109)) which results in the following new constraint:
(39)    (minus_nat(x172, x171)=pos(s(x109)) ⇒ EVAL_3(x105, pos(s(x109)), pos(x171))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(x171)))

We simplified constraint (39) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x172, x171)=pos(s(x109)) which results in the following new constraints:
(40)    (pos(0)=pos(s(x109)) ⇒ EVAL_3(x105, pos(s(x109)), pos(0))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(0)))

(41)    (pos(s(x176))=pos(s(x109)) ⇒ EVAL_3(x105, pos(s(x109)), pos(0))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(0)))

(42)    (minus_nat(x178, x177)=pos(s(x109))∧(∀x179,x180:minus_nat(x178, x177)=pos(s(x179)) ⇒ EVAL_3(x180, pos(s(x179)), pos(x177))≥COND_EVAL_3(true, x180, pos(s(x179)), pos(x177))) ⇒ EVAL_3(x105, pos(s(x109)), pos(s(x177)))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(s(x177))))

We solved constraint (40) using rules (I), (II).We simplified constraint (41) using rules (I), (II), (III) which results in the following new constraint:
(43)    (EVAL_3(x105, pos(s(x176)), pos(0))≥COND_EVAL_3(true, x105, pos(s(x176)), pos(0)))

We simplified constraint (42) using rule (VI) where we applied the induction hypothesis (∀x179,x180:minus_nat(x178, x177)=pos(s(x179)) ⇒ EVAL_3(x180, pos(s(x179)), pos(x177))≥COND_EVAL_3(true, x180, pos(s(x179)), pos(x177))) with σ = [x179 / x109, x180 / x105] which results in the following new constraint:
(44)    (EVAL_3(x105, pos(s(x109)), pos(x177))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(x177)) ⇒ EVAL_3(x105, pos(s(x109)), pos(s(x177)))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(s(x177))))

For Pair COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed) the following chains were created:

We consider the chain EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2), COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed) which results in the following constraint:
(45) (COND_EVAL_3(true, x123, pos(s(x124)), x125)=COND_EVAL_3(true, x126, x127, x128) ⇒ COND_EVAL_3(true, x126, x127, x128)≥EVAL_3(x126, minus_int(x127, x128), x128))

We simplified constraint (45) using rules (I), (II), (III) which results in the following new constraint:
(46) (COND_EVAL_3(true, x123, pos(s(x124)), x125)≥EVAL_3(x123, minus_int(pos(s(x124)), x125), x125))

To summarize, we get the following constraints P_≥ for the following pairs.

COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
- (COND_EVAL_11(true, pos(0), pos(x138), x11)≥EVAL_3(pos(0), pos(x138), x11))
- (COND_EVAL_11(true, neg(x139), pos(x140), x11)≥EVAL_3(neg(x139), pos(x140), x11))
- (COND_EVAL_11(true, pos(0), neg(0), x11)≥EVAL_3(pos(0), neg(0), x11))
- (COND_EVAL_11(true, neg(x142), neg(0), x11)≥EVAL_3(neg(x142), neg(0), x11))
- (COND_EVAL_11(true, pos(x147), pos(x148), x11)≥EVAL_3(pos(x147), pos(x148), x11) ⇒ COND_EVAL_11(true, pos(s(x147)), pos(s(x148)), x11)≥EVAL_3(pos(s(x147)), pos(s(x148)), x11))
- (COND_EVAL_11(true, neg(x150), neg(x151), x11)≥EVAL_3(neg(x150), neg(x151), x11) ⇒ COND_EVAL_11(true, neg(s(x150)), neg(s(x151)), x11)≥EVAL_3(neg(s(x150)), neg(s(x151)), x11))
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
- (EVAL_3(x21, x22, x23)≥COND_EVAL_31(greatereq_int(pos(0), x22), x21, x22, x23))
- (EVAL_3(x39, x43, x41)≥COND_EVAL_31(greatereq_int(pos(0), x43), x39, x43, x41))
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
- (COND_EVAL_31(true, x48, pos(0), x50)≥EVAL_1(x48, pos(0), x50))
- (COND_EVAL_31(true, x48, neg(x155), x50)≥EVAL_1(x48, neg(x155), x50))
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
- (EVAL_1(x72, x73, x74)≥COND_EVAL_11(and(and(greater_int(x72, pos(0)), greater_int(x73, pos(0))), greatereq_int(x73, x72)), x72, x73, x74))
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
- (EVAL_3(x87, pos(s(x91)), x89)≥COND_EVAL_3(true, x87, pos(s(x91)), x89))
- (EVAL_3(x105, pos(s(x176)), pos(0))≥COND_EVAL_3(true, x105, pos(s(x176)), pos(0)))
- (EVAL_3(x105, pos(s(x109)), pos(x177))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(x177)) ⇒ EVAL_3(x105, pos(s(x109)), pos(s(x177)))≥COND_EVAL_3(true, x105, pos(s(x109)), pos(s(x177))))
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)
- (COND_EVAL_3(true, x123, pos(s(x124)), x125)≥EVAL_3(x123, minus_int(pos(s(x124)), x125), x125))

POL(0) = 0
POL(COND_EVAL_11(x₁, x₂, x₃, x₄)) = 1 + x₁ - x₂ + x₃
POL(COND_EVAL_3(x₁, x₂, x₃, x₄)) = x₁
POL(COND_EVAL_31(x₁, x₂, x₃, x₄)) = 1 - x₃
POL(EVAL_1(x₁, x₂, x₃)) = 1 + x₂
POL(EVAL_3(x₁, x₂, x₃)) = 1
POL(and(x₁, x₂)) = x₁
POL(c) = -1
POL(false) = 0
POL(greater_int(x₁, x₂)) = x₁
POL(greatereq_int(x₁, x₂)) = 0
POL(minus_int(x₁, x₂)) = 0
POL(minus_nat(x₁, x₂)) = 0
POL(neg(x₁)) = 0
POL(plus_nat(x₁, x₂)) = x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 1

The following pairs are in P_>:

COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)

The following pairs are in P_bound:

COND_EVAL_11(true, x, y, x_removed) → EVAL_3(x, y, x_removed)
EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The following rules are usable:

and(true, true) → true
and(true, false) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, true) → false
and(false, false) → false
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                      ↳ QDP

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

EVAL_3(x, y, x_removed) → COND_EVAL_31(greatereq_int(pos(0), y), x, y, x_removed)
COND_EVAL_31(true, x, y, x_removed) → EVAL_1(x, y, x_removed)
EVAL_1(x, y, x_removed) → COND_EVAL_11(and(and(greater_int(x, pos(0)), greater_int(y, pos(0))), greatereq_int(y, x)), x, y, x_removed)
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                      ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                      ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

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)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                      ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, x, y, x_removed) → EVAL_3(x, minus_int(y, x_removed), x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
COND_EVAL_3(true, y0, neg(x0), pos(x1)) → EVAL_3(y0, neg(plus_nat(x0, x1)), pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                      ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
COND_EVAL_3(true, y0, neg(x0), pos(x1)) → EVAL_3(y0, neg(plus_nat(x0, x1)), pos(x1))

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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 1 SCC with 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                      ↳ QDP

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

COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                      ↳ QDP

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

COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)

The TRS R consists of the following rules:

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:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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].

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                      ↳ QDP

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

COND_EVAL_3(true, y0, pos(x0), pos(x1)) → EVAL_3(y0, minus_nat(x0, x1), pos(x1))
EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)

The TRS R consists of the following rules:

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:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                      ↳ QDP

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

EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2)
COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))

The TRS R consists of the following rules:

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:

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 instantiating [LPAR04] the rule EVAL_3(y0, pos(s(x0)), y2) → COND_EVAL_3(true, y0, pos(s(x0)), y2) we obtained the following new rules [LPAR04]:

EVAL_3(z0, pos(s(x1)), pos(z2)) → COND_EVAL_3(true, z0, pos(s(x1)), pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ MNOCProof

                                      ↳ QDP

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

COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))
EVAL_3(z0, pos(s(x1)), pos(z2)) → COND_EVAL_3(true, z0, pos(s(x1)), pos(z2))

The TRS R consists of the following rules:

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:

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 modular non-overlap check [FROCOS05] to decrease Q to the empty set.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ NonInfProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                                          ↳ QDP

                                                            ↳ UsableRulesProof

                                                              ↳ QDP

                                                                ↳ QReductionProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ UsableRulesProof

                                                                              ↳ QDP

                                                                                ↳ QReductionProof

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ MNOCProof

                                                                                              ↳ QDP

                                                                                                ↳ NonTerminationProof

                                      ↳ QDP

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

COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))
EVAL_3(z0, pos(s(x1)), pos(z2)) → COND_EVAL_3(true, z0, pos(s(x1)), pos(z2))

The TRS R consists of the following rules:

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)

COND_EVAL_3(true, z0, pos(s(z1)), pos(x2)) → EVAL_3(z0, minus_nat(s(z1), x2), pos(x2))
EVAL_3(z0, pos(s(x1)), pos(z2)) → COND_EVAL_3(true, z0, pos(s(x1)), pos(z2))

The TRS R consists of the following rules:

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)

s = EVAL_3(z0a, minus_nat(s(x), 0), pos(z2)) evaluates to t =EVAL_3(z0a, minus_nat(s(x), z2), pos(z2))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [z2 / 0]
Matcher: [ ]

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

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

EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)
greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(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(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
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))

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

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

EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

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)))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

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)))
and(false, false)
and(false, true)
and(true, false)
and(true, true)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

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

EVAL_2(x, y, x_removed) → COND_EVAL_2(greater_int(x, pos(0)), x, y, x_removed)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

EVAL_2(neg(s(x0)), y1, y2) → COND_EVAL_2(false, neg(s(x0)), y1, y2)
EVAL_2(neg(0), y1, y2) → COND_EVAL_2(false, neg(0), y1, y2)
EVAL_2(pos(0), y1, y2) → COND_EVAL_2(false, pos(0), y1, y2)
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

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

COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)
EVAL_2(neg(s(x0)), y1, y2) → COND_EVAL_2(false, neg(s(x0)), y1, y2)
EVAL_2(neg(0), y1, y2) → COND_EVAL_2(false, neg(0), y1, y2)
EVAL_2(pos(0), y1, y2) → COND_EVAL_2(false, pos(0), y1, y2)
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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 3 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

greater_int(pos(0), pos(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_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].

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, x, y, x_removed) → EVAL_2(minus_int(x, x_removed), y, x_removed)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
COND_EVAL_2(true, neg(x0), y1, pos(x1)) → EVAL_2(neg(plus_nat(x0, x1)), y1, pos(x1))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
COND_EVAL_2(true, neg(x0), y1, pos(x1)) → EVAL_2(neg(plus_nat(x0, x1)), y1, pos(x1))

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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 1 SCC with 1 less node.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

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

COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(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)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ QReductionProof

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

COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

The TRS R consists of the following rules:

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:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), 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].

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ QReductionProof

                                                                              ↳ QDP

                                                                                ↳ Instantiation

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

COND_EVAL_2(true, pos(x0), y1, pos(x1)) → EVAL_2(minus_nat(x0, x1), y1, pos(x1))
EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)

The TRS R consists of the following rules:

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:

minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))

COND_EVAL_2(true, pos(s(z0)), z1, pos(x2)) → EVAL_2(minus_nat(s(z0), x2), z1, pos(x2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ QReductionProof

                                                                              ↳ QDP

                                                                                ↳ Instantiation

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

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

EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2)
COND_EVAL_2(true, pos(s(z0)), z1, pos(x2)) → EVAL_2(minus_nat(s(z0), x2), z1, pos(x2))

The TRS R consists of the following rules:

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:

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 instantiating [LPAR04] the rule EVAL_2(pos(s(x0)), y1, y2) → COND_EVAL_2(true, pos(s(x0)), y1, y2) we obtained the following new rules [LPAR04]:

EVAL_2(pos(s(x0)), z1, pos(z2)) → COND_EVAL_2(true, pos(s(x0)), z1, pos(z2))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ RemovalProof

                          ↳ QDP

                            ↳ NonInfProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ AND

                                      ↳ QDP

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QReductionProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ UsableRulesProof

                                                          ↳ QDP

                                                            ↳ QReductionProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ UsableRulesProof

                                                                          ↳ QDP

                                                                            ↳ QReductionProof

                                                                              ↳ QDP

                                                                                ↳ Instantiation

                                                                                  ↳ QDP

                                                                                    ↳ Instantiation

                                                                                      ↳ QDP

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

COND_EVAL_2(true, pos(s(z0)), z1, pos(x2)) → EVAL_2(minus_nat(s(z0), x2), z1, pos(x2))
EVAL_2(pos(s(x0)), z1, pos(z2)) → COND_EVAL_2(true, pos(s(x0)), z1, pos(z2))

The TRS R consists of the following rules:

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:

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.