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:
Cond_eval2(TRUE, x, y) → eval(-@z(x, 1@z), y)
eval(x, y) → Cond_eval1(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
Cond_eval1(TRUE, x, y) → eval(x, y)
Cond_eval(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval2(&&(>@z(+@z(x, y), 0@z), >@z(x, 0@z)), x, y)
eval(x, y) → Cond_eval(&&(&&(>@z(+@z(x, y), 0@z), >=@z(0@z, x)), >@z(y, 0@z)), x, y)
The set Q consists of the following terms:
Cond_eval2(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Represented integers and predefined function symbols by Terms
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
COND_EVAL2(true, x, y) → MINUS_INT(x, pos(s(0)))
EVAL(x, y) → COND_EVAL1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
EVAL(x, y) → AND(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y))
EVAL(x, y) → AND(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x))
EVAL(x, y) → GREATER_INT(plus_int(x, y), pos(0))
EVAL(x, y) → PLUS_INT(x, y)
EVAL(x, y) → GREATEREQ_INT(pos(0), x)
EVAL(x, y) → GREATEREQ_INT(pos(0), y)
COND_EVAL1(true, x, y) → EVAL(x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
COND_EVAL(true, x, y) → MINUS_INT(y, pos(s(0)))
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
EVAL(x, y) → AND(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0)))
EVAL(x, y) → GREATER_INT(x, pos(0))
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
EVAL(x, y) → AND(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0)))
EVAL(x, y) → GREATER_INT(y, pos(0))
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))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
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 TRS R consists of the following rules:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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:
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
COND_EVAL2(true, x, y) → MINUS_INT(x, pos(s(0)))
EVAL(x, y) → COND_EVAL1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
EVAL(x, y) → AND(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y))
EVAL(x, y) → AND(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x))
EVAL(x, y) → GREATER_INT(plus_int(x, y), pos(0))
EVAL(x, y) → PLUS_INT(x, y)
EVAL(x, y) → GREATEREQ_INT(pos(0), x)
EVAL(x, y) → GREATEREQ_INT(pos(0), y)
COND_EVAL1(true, x, y) → EVAL(x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
COND_EVAL(true, x, y) → MINUS_INT(y, pos(s(0)))
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
EVAL(x, y) → AND(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0)))
EVAL(x, y) → GREATER_INT(x, pos(0))
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
EVAL(x, y) → AND(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0)))
EVAL(x, y) → GREATER_INT(y, pos(0))
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))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
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 TRS R consists of the following rules:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 20 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:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
The TRS R consists of the following rules:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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:
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:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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].
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
↳ 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:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_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:
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
↳ 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:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
The TRS R consists of the following rules:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ 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:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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].
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
↳ 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:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(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:
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
↳ 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:
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
The TRS R consists of the following rules:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ 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:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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].
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ 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
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
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:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ 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:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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].
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ 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.
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(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
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ 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:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ 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:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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].
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ 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
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ 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:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ 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:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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].
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_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:
- 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
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(x, y) → COND_EVAL1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
COND_EVAL1(true, x, y) → EVAL(x, y)
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
The TRS R consists of the following rules:
Cond_eval2(true, x, y) → eval(minus_int(x, pos(s(0))), y)
eval(x, y) → Cond_eval1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
Cond_eval1(true, x, y) → eval(x, y)
Cond_eval(true, x, y) → eval(x, minus_int(y, pos(s(0))))
eval(x, y) → Cond_eval2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
eval(x, y) → Cond_eval(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, 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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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))
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(x, y) → COND_EVAL1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
COND_EVAL1(true, x, y) → EVAL(x, y)
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, 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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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].
Cond_eval2(true, x0, x1)
eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, x1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
EVAL(x, y) → COND_EVAL1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y)
COND_EVAL1(true, x, y) → EVAL(x, y)
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
Positions in right side of the pair: Pair: COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
Positions in right side of the pair: 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
↳ RemovalProof
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
EVAL(x, y, x_removed) → COND_EVAL1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, x_removed)
COND_EVAL1(true, x, y, x_removed) → EVAL(x, y, x_removed)
EVAL(x, y, x_removed) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y, x_removed)
EVAL(x, y, x_removed) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, x_removed)
COND_EVAL2(true, x, y, x_removed) → EVAL(minus_int(x, x_removed), y, x_removed)
COND_EVAL(true, x, y, x_removed) → EVAL(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
Positions in right side of the pair: Pair: COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
Positions in right side of the pair: 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
↳ RemovalProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
EVAL(x, y, x_removed) → COND_EVAL1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y, x_removed)
COND_EVAL1(true, x, y, x_removed) → EVAL(x, y, x_removed)
EVAL(x, y, x_removed) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y, x_removed)
EVAL(x, y, x_removed) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y, x_removed)
COND_EVAL2(true, x, y, x_removed) → EVAL(minus_int(x, x_removed), y, x_removed)
COND_EVAL(true, x, y, x_removed) → EVAL(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(x, y) → COND_EVAL1(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greatereq_int(pos(0), y)), x, y) at position [0] we obtained the following new rules [LPAR04]:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), neg(x1))), pos(x0), neg(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y) → EVAL(x, y)
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), neg(x1))), pos(x0), neg(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), neg(x1))), neg(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1)) at position [0,0,1] we obtained the following new rules [LPAR04]:
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y) → EVAL(x, y)
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), neg(x1))), pos(x0), neg(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), 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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), neg(x1))), pos(x0), neg(x1)) at position [0,1] we obtained the following new rules [LPAR04]:
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y) → EVAL(x, y)
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), neg(x0))), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1)) at position [0,0,1] we obtained the following new rules [LPAR04]:
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y) → EVAL(x, y)
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greatereq_int(pos(0), neg(x1))), neg(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greatereq_int(pos(0), neg(x1))), neg(x0), neg(x1)) at position [0,1] we obtained the following new rules [LPAR04]:
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y) → EVAL(x, y)
EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(x, y) → COND_EVAL2(and(greater_int(plus_int(x, y), pos(0)), greater_int(x, pos(0))), x, y) at position [0] we obtained the following new rules [LPAR04]:
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y) → EVAL(x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(x, y) → COND_EVAL(and(and(greater_int(plus_int(x, y), pos(0)), greatereq_int(pos(0), x)), greater_int(y, pos(0))), x, y) at position [0] we obtained the following new rules [LPAR04]:
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), neg(x0))), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), greatereq_int(pos(0), neg(x0))), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y) → EVAL(x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), neg(x0))), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), greatereq_int(pos(0), neg(x0))), greater_int(pos(x1), pos(0))), neg(x0), 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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), neg(x0))), greater_int(neg(x1), pos(0))), neg(x0), neg(x1)) at position [0,0,1] we obtained the following new rules [LPAR04]:
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y) → EVAL(x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), greatereq_int(pos(0), neg(x0))), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), greatereq_int(pos(0), neg(x0))), greater_int(pos(x1), pos(0))), neg(x0), pos(x1)) at position [0,0,1] we obtained the following new rules [LPAR04]:
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y) → EVAL(x, y)
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), 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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL1(true, x, y) → EVAL(x, y) we obtained the following new rules [LPAR04]:
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y)
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL2(true, x, y) → EVAL(minus_int(x, pos(s(0))), y) we obtained the following new rules [LPAR04]:
COND_EVAL2(true, pos(0), z0) → EVAL(minus_int(pos(0), pos(s(0))), z0)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(minus_int(neg(0), pos(s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(minus_int(pos(0), pos(s(0))), z0)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(minus_int(neg(0), pos(s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, pos(0), z0) → EVAL(minus_int(pos(0), pos(s(0))), z0) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(minus_int(neg(0), pos(s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), neg(z1)) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(minus_int(neg(0), pos(s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), pos(z1)) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(minus_int(neg(0), pos(s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(minus_int(neg(z0), pos(s(0))), neg(z1)) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, neg(0), z0) → EVAL(minus_int(neg(0), pos(s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, neg(0), z0) → EVAL(minus_int(neg(0), pos(s(0))), z0) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, neg(0), z0) → EVAL(neg(plus_nat(0, s(0))), z0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(plus_nat(0, s(0))), z0)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_int(pos(s(z0)), pos(s(0))), z1) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(s(z0), s(0)), z1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(plus_nat(0, s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(s(z0), s(0)), z1)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_int(pos(z0), pos(s(0))), pos(z1)) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(plus_nat(0, s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(s(z0), s(0)), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, neg(s(z0)), z1) → EVAL(minus_int(neg(s(z0)), pos(s(0))), z1) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(plus_nat(s(z0), s(0))), z1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(plus_nat(0, s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(s(z0), s(0)), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(plus_nat(s(z0), s(0))), z1)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, pos(0), z0) → EVAL(minus_nat(0, s(0)), z0) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(plus_nat(0, s(0))), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(s(z0), s(0)), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(plus_nat(s(z0), s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, neg(0), z0) → EVAL(neg(plus_nat(0, s(0))), z0) at position [0,0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(s(z0), s(0)), z1)
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(plus_nat(s(z0), s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(s(z0), s(0)), z1) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(plus_nat(s(z0), s(0))), z1)
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(plus_nat(s(z0), s(0))), z1) at position [0,0] we obtained the following new rules [LPAR04]:
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0))))
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL(true, x, y) → EVAL(x, minus_int(y, pos(s(0)))) we obtained the following new rules [LPAR04]:
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_int(pos(0), pos(s(0))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_int(pos(0), pos(s(0))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0))))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, minus_int(neg(s(z1)), pos(s(0)))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_int(pos(0), pos(s(0))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0))))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), minus_int(neg(z1), pos(s(0)))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_int(pos(0), pos(s(0))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_int(pos(0), pos(s(0)))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_nat(0, s(0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_nat(0, s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_int(pos(s(z1)), pos(s(0)))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(s(z1), s(0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_nat(0, s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(s(z1), s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_int(pos(z1), pos(s(0)))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_nat(0, s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(s(z1), s(0)))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), minus_int(neg(z1), pos(s(0)))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_nat(0, s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(s(z1), s(0)))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_int(pos(z1), pos(s(0)))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_nat(0, s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(s(z1), s(0)))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, z0, neg(0)) → EVAL(z0, minus_int(neg(0), pos(s(0)))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(plus_nat(0, s(0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_nat(0, s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(s(z1), s(0)))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(plus_nat(0, s(0))))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(plus_nat(s(z1), s(0)))) at position [1,0] we obtained the following new rules [LPAR04]:
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_nat(0, s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(s(z1), s(0)))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(plus_nat(0, s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, z0, pos(0)) → EVAL(z0, minus_nat(0, s(0))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, z0, pos(0)) → EVAL(z0, neg(s(0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(s(z1), s(0)))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(plus_nat(0, s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, neg(s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(s(z1), s(0))) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(plus_nat(0, s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, neg(s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(plus_nat(0, s(0)))) at position [1,0] we obtained the following new rules [LPAR04]:
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, neg(s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
COND_EVAL2(true, pos(0), z0) → EVAL(neg(s(0)), z0)
The remaining pairs can at least be oriented weakly.
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, neg(s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 1
POL(COND_EVAL(x1, x2, x3)) = x2
POL(COND_EVAL1(x1, x2, x3)) = x2
POL(COND_EVAL2(x1, x2, x3)) = x2
POL(EVAL(x1, x2)) = x1
POL(and(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(greatereq_int(x1, x2)) = 0
POL(minus_int(x1, x2)) = 1 + x1
POL(minus_nat(x1, x2)) = x1
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 1 + x1 + x2
POL(plus_nat(x1, x2)) = x1
POL(pos(x1)) = x1
POL(s(x1)) = x1
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(0)) → EVAL(z0, neg(s(0)))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
COND_EVAL(true, z0, pos(0)) → EVAL(z0, neg(s(0)))
The remaining pairs can at least be oriented weakly.
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 1
POL(COND_EVAL(x1, x2, x3)) = x3
POL(COND_EVAL1(x1, x2, x3)) = x3
POL(COND_EVAL2(x1, x2, x3)) = x3
POL(EVAL(x1, x2)) = x2
POL(and(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(greatereq_int(x1, x2)) = 0
POL(minus_int(x1, x2)) = x1
POL(minus_nat(x1, x2)) = x1
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = x1
POL(s(x1)) = x1
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
EVAL(neg(0), y1) → COND_EVAL2(and(greater_int(plus_int(neg(0), y1), pos(0)), false), neg(0), y1)
The remaining pairs can at least be oriented weakly.
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 1
POL(COND_EVAL(x1, x2, x3)) = x2
POL(COND_EVAL1(x1, x2, x3)) = x2
POL(COND_EVAL2(x1, x2, x3)) = 0
POL(EVAL(x1, x2)) = x1
POL(and(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(greatereq_int(x1, x2)) = 0
POL(minus_int(x1, x2)) = 0
POL(minus_nat(x1, x2)) = 0
POL(neg(x1)) = x1
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = x2
POL(pos(x1)) = 0
POL(s(x1)) = 0
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
COND_EVAL2(true, neg(0), z0) → EVAL(neg(s(0)), z0)
The remaining pairs can at least be oriented weakly.
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 1
POL(COND_EVAL(x1, x2, x3)) = x2
POL(COND_EVAL1(x1, x2, x3)) = x2
POL(COND_EVAL2(x1, x2, x3)) = x2
POL(EVAL(x1, x2)) = x1
POL(and(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(greatereq_int(x1, x2)) = 0
POL(minus_int(x1, x2)) = 0
POL(minus_nat(x1, x2)) = 0
POL(neg(x1)) = x1
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = x1 + x2
POL(pos(x1)) = 0
POL(s(x1)) = 0
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
COND_EVAL(true, z0, pos(s(z1))) → EVAL(z0, minus_nat(z1, 0))
The remaining pairs can at least be oriented weakly.
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(COND_EVAL(x1, x2, x3)) = x3
POL(COND_EVAL1(x1, x2, x3)) = x3
POL(COND_EVAL2(x1, x2, x3)) = x3
POL(EVAL(x1, x2)) = x2
POL(and(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(greatereq_int(x1, x2)) = 0
POL(minus_int(x1, x2)) = x1
POL(minus_nat(x1, x2)) = x1
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = x1
POL(s(x1)) = 1 + x1
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
COND_EVAL(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(plus_nat(z1, s(0))))
COND_EVAL(true, z0, neg(s(z1))) → EVAL(z0, neg(s(plus_nat(z1, s(0)))))
COND_EVAL(true, z0, neg(0)) → EVAL(z0, neg(s(0)))
The remaining pairs can at least be oriented weakly.
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(COND_EVAL(x1, x2, x3)) = x1
POL(COND_EVAL1(x1, x2, x3)) = x3
POL(COND_EVAL2(x1, x2, x3)) = x3
POL(EVAL(x1, x2)) = x2
POL(and(x1, x2)) = x2
POL(false) = 0
POL(greater_int(x1, x2)) = x1
POL(greatereq_int(x1, x2)) = 0
POL(minus_int(x1, x2)) = 1
POL(minus_nat(x1, x2)) = 1
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = x1
POL(s(x1)) = 1
POL(true) = 1
The following usable rules [FROCOS05] were oriented:
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(0)) → false
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
and(false, false) → false
and(true, true) → true
and(false, true) → false
and(true, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
COND_EVAL2(true, pos(s(z0)), z1) → EVAL(minus_nat(z0, 0), z1)
The remaining pairs can at least be oriented weakly.
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(COND_EVAL(x1, x2, x3)) = x2
POL(COND_EVAL1(x1, x2, x3)) = x2
POL(COND_EVAL2(x1, x2, x3)) = x2
POL(EVAL(x1, x2)) = x1
POL(and(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(greatereq_int(x1, x2)) = 0
POL(minus_int(x1, x2)) = 0
POL(minus_nat(x1, x2)) = x1
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = x1
POL(s(x1)) = 1 + x1
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
EVAL(pos(0), y1) → COND_EVAL2(and(greater_int(plus_int(pos(0), y1), pos(0)), false), pos(0), y1)
COND_EVAL2(true, neg(z0), pos(z1)) → EVAL(neg(plus_nat(z0, s(0))), pos(z1))
COND_EVAL2(true, neg(z0), neg(z1)) → EVAL(neg(plus_nat(z0, s(0))), neg(z1))
COND_EVAL2(true, neg(s(z0)), z1) → EVAL(neg(s(plus_nat(z0, s(0)))), z1)
The remaining pairs can at least be oriented weakly.
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(COND_EVAL(x1, x2, x3)) = x2
POL(COND_EVAL1(x1, x2, x3)) = x2
POL(COND_EVAL2(x1, x2, x3)) = x1
POL(EVAL(x1, x2)) = x1
POL(and(x1, x2)) = x2
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)) = 1
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = 1
POL(s(x1)) = 0
POL(true) = 1
The following usable rules [FROCOS05] were oriented:
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(0)) → false
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
and(false, false) → false
and(true, true) → true
and(false, true) → false
and(true, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL2(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), neg(x1))
EVAL(neg(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(neg(s(x0)), y1), pos(0)), false), neg(s(x0)), y1)
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL2(and(greater_int(minus_nat(x1, x0), pos(0)), greater_int(neg(x0), pos(0))), neg(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
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
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
EVAL(y0, pos(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(0))
The remaining pairs can at least be oriented weakly.
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 1
POL(COND_EVAL(x1, x2, x3)) = x1
POL(COND_EVAL1(x1, x2, x3)) = x3
POL(COND_EVAL2(x1, x2, x3)) = x3
POL(EVAL(x1, x2)) = x2
POL(and(x1, x2)) = x2
POL(false) = 0
POL(greater_int(x1, x2)) = x1
POL(greatereq_int(x1, x2)) = 0
POL(minus_int(x1, x2)) = 1
POL(minus_nat(x1, x2)) = 1
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = x1
POL(s(x1)) = 1
POL(true) = 1
The following usable rules [FROCOS05] were oriented:
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(0)) → false
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
and(false, false) → false
and(true, true) → true
and(false, true) → false
and(true, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
EVAL(y0, neg(0)) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(0)), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(0))
EVAL(y0, neg(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, neg(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, neg(s(x0)))
The remaining pairs can at least be oriented weakly.
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(COND_EVAL(x1, x2, x3)) = x1
POL(COND_EVAL1(x1, x2, x3)) = x3
POL(COND_EVAL2(x1, x2, x3)) = x3
POL(EVAL(x1, x2)) = x2
POL(and(x1, x2)) = x2
POL(false) = 0
POL(greater_int(x1, x2)) = x1
POL(greatereq_int(x1, x2)) = 0
POL(minus_int(x1, x2)) = 1
POL(minus_nat(x1, x2)) = 1
POL(neg(x1)) = 1
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = x1
POL(s(x1)) = 1
POL(true) = 1
The following usable rules [FROCOS05] were oriented:
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(0)) → false
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
and(false, false) → false
and(true, true) → true
and(false, true) → false
and(true, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
EVAL(y0, pos(s(x0))) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), false), y0, pos(s(x0)))
The remaining pairs can at least be oriented weakly.
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(COND_EVAL(x1, x2, x3)) = 1
POL(COND_EVAL1(x1, x2, x3)) = x1
POL(COND_EVAL2(x1, x2, x3)) = x1
POL(EVAL(x1, x2)) = 1
POL(and(x1, x2)) = x2
POL(false) = 0
POL(greater_int(x1, x2)) = 1
POL(greatereq_int(x1, x2)) = 1
POL(minus_int(x1, x2)) = 0
POL(minus_nat(x1, x2)) = 0
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 0
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = 0
POL(s(x1)) = 0
POL(true) = 1
The following usable rules [FROCOS05] were oriented:
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(0), pos(0)) → false
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
and(true, true) → true
and(false, true) → false
and(true, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
EVAL(neg(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), true), neg(x0), neg(x1))
EVAL(neg(x0), neg(x1)) → COND_EVAL(and(and(greater_int(neg(plus_nat(x0, x1)), pos(0)), true), greater_int(neg(x1), pos(0))), neg(x0), neg(x1))
The remaining pairs can at least be oriented weakly.
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(COND_EVAL(x1, x2, x3)) = x1
POL(COND_EVAL1(x1, x2, x3)) = x1
POL(COND_EVAL2(x1, x2, x3)) = 1
POL(EVAL(x1, x2)) = 1
POL(and(x1, x2)) = x1
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)) = 1
POL(neg(x1)) = 0
POL(plus_int(x1, x2)) = 1
POL(plus_nat(x1, x2)) = 0
POL(pos(x1)) = 1
POL(s(x1)) = 0
POL(true) = 1
The following usable rules [FROCOS05] were oriented:
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(0)) → false
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
and(false, false) → false
and(true, true) → true
and(false, true) → false
and(true, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
EVAL(pos(s(x0)), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greater_int(y1, pos(0))), pos(s(x0)), y1)
The remaining pairs can at least be oriented weakly.
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(COND_EVAL1(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(EVAL(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(and(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greater_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greatereq_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(COND_EVAL2(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(COND_EVAL(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(minus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(0)) → false
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
and(true, true) → true
and(false, true) → false
and(true, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
COND_EVAL1(true, neg(z0), neg(z1)) → EVAL(neg(z0), neg(z1))
The remaining pairs can at least be oriented weakly.
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(COND_EVAL1(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(EVAL(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(and(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greater_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greatereq_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(COND_EVAL2(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(COND_EVAL(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(minus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(0)) → false
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
and(true, true) → true
and(false, true) → false
and(true, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
EVAL(pos(x0), neg(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(neg(x1), pos(0))), pos(x0), neg(x1))
The remaining pairs can at least be oriented weakly.
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), pos(x1))
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(COND_EVAL1(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(EVAL(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(and(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greater_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greatereq_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(COND_EVAL2(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(COND_EVAL(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(minus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(0), pos(0)) → false
greater_int(pos(0), pos(0)) → false
plus_nat(s(x), y) → s(plus_nat(x, y))
plus_nat(0, x) → x
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(0, 0) → pos(0)
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
and(true, true) → true
and(false, true) → false
and(true, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, neg(z0), z1) → EVAL(neg(z0), z1)
EVAL(y0, neg(x1)) → COND_EVAL1(and(and(greater_int(plus_int(y0, neg(x1)), pos(0)), greatereq_int(pos(0), y0)), true), y0, neg(x1))
COND_EVAL1(true, pos(z0), neg(z1)) → EVAL(pos(z0), neg(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), false), greatereq_int(pos(0), y1)), pos(s(x0)), y1)
COND_EVAL1(true, z0, pos(s(z1))) → EVAL(z0, pos(s(z1)))
EVAL(neg(x1), y1) → COND_EVAL1(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greatereq_int(pos(0), y1)), neg(x1), y1)
COND_EVAL1(true, neg(z0), pos(z1)) → EVAL(neg(z0), pos(z1))
EVAL(y0, pos(0)) → COND_EVAL1(and(and(greater_int(plus_int(y0, pos(0)), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(0))
COND_EVAL1(true, pos(0), z0) → EVAL(pos(0), z0)
EVAL(pos(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greatereq_int(pos(0), pos(x1))), pos(x0), pos(x1))
COND_EVAL1(true, z0, pos(0)) → EVAL(z0, pos(0))
EVAL(pos(0), y1) → COND_EVAL1(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greatereq_int(pos(0), y1)), pos(0), y1)
COND_EVAL1(true, z0, neg(z1)) → EVAL(z0, neg(z1))
EVAL(pos(x0), neg(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x0, x1), pos(0)), greatereq_int(pos(0), pos(x0))), true), pos(x0), neg(x1))
COND_EVAL1(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), z1)
COND_EVAL1(true, pos(z0), pos(z1)) → EVAL(pos(z0), pos(z1))
EVAL(pos(s(x0)), y1) → COND_EVAL2(and(greater_int(plus_int(pos(s(x0)), y1), pos(0)), true), pos(s(x0)), y1)
COND_EVAL2(true, pos(z0), neg(z1)) → EVAL(minus_nat(z0, s(0)), neg(z1))
COND_EVAL(true, pos(0), z0) → EVAL(pos(0), minus_int(z0, pos(s(0))))
EVAL(pos(x0), pos(x1)) → COND_EVAL2(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), pos(x1))
COND_EVAL2(true, pos(z0), pos(z1)) → EVAL(minus_nat(z0, s(0)), pos(z1))
EVAL(neg(x0), pos(x1)) → COND_EVAL1(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greatereq_int(pos(0), pos(x1))), neg(x0), pos(x1))
EVAL(y0, pos(s(x0))) → COND_EVAL(and(and(greater_int(plus_int(y0, pos(s(x0))), pos(0)), greatereq_int(pos(0), y0)), true), y0, pos(s(x0)))
COND_EVAL(true, pos(s(z0)), z1) → EVAL(pos(s(z0)), minus_int(z1, pos(s(0))))
EVAL(pos(x0), neg(x1)) → COND_EVAL2(and(greater_int(minus_nat(x0, x1), pos(0)), greater_int(pos(x0), pos(0))), pos(x0), neg(x1))
COND_EVAL(true, neg(z0), z1) → EVAL(neg(z0), minus_int(z1, pos(s(0))))
EVAL(neg(x1), y1) → COND_EVAL(and(and(greater_int(plus_int(neg(x1), y1), pos(0)), true), greater_int(y1, pos(0))), neg(x1), y1)
COND_EVAL(true, neg(z0), pos(z1)) → EVAL(neg(z0), minus_nat(z1, s(0)))
EVAL(neg(x0), pos(x1)) → COND_EVAL(and(and(greater_int(minus_nat(x1, x0), pos(0)), true), greater_int(pos(x1), pos(0))), neg(x0), pos(x1))
COND_EVAL(true, pos(z0), pos(z1)) → EVAL(pos(z0), minus_nat(z1, s(0)))
EVAL(pos(0), y1) → COND_EVAL(and(and(greater_int(plus_int(pos(0), y1), pos(0)), true), greater_int(y1, pos(0))), pos(0), y1)
EVAL(pos(x0), pos(x1)) → COND_EVAL(and(and(greater_int(pos(plus_nat(x0, x1)), pos(0)), greatereq_int(pos(0), pos(x0))), greater_int(pos(x1), pos(0))), pos(x0), 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)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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))
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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)))
We have to consider all minimal (P,Q,R)-chains.