Termination of the given ITRSProblem could not be shown:
↳ ITRS
↳ ITRStoQTRSProof
ITRS problem:
The following domains are used:
z
The TRS R consists of the following rules:
eval(x, y, z) → Cond_eval1(>=@z(x, 0@z), x, y, z)
Cond_eval1(TRUE, x, y, z) → eval(+@z(x, y), -@z(y, 2@z), z)
Cond_eval(TRUE, x, y, z) → eval(+@z(x, z), +@z(y, 1@z), -@z(z, 2@z))
eval(x, y, z) → Cond_eval(>=@z(x, 0@z), x, y, z)
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(TRUE, x0, x1, x2)
Cond_eval(TRUE, x0, x1, x2)
Represented integers and predefined function symbols by Terms
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
eval(x, y, z) → Cond_eval1(greatereq_int(x, pos(0)), x, y, z)
Cond_eval1(true, x, y, z) → eval(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
Cond_eval(true, x, y, z) → eval(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
eval(x, y, z) → Cond_eval(greatereq_int(x, pos(0)), x, y, z)
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))
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))
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)
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))
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
EVAL(x, y, z) → COND_EVAL1(greatereq_int(x, pos(0)), x, y, z)
EVAL(x, y, z) → GREATEREQ_INT(x, pos(0))
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
COND_EVAL1(true, x, y, z) → PLUS_INT(x, y)
COND_EVAL1(true, x, y, z) → MINUS_INT(y, pos(s(s(0))))
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
COND_EVAL(true, x, y, z) → PLUS_INT(x, z)
COND_EVAL(true, x, y, z) → PLUS_INT(pos(s(0)), y)
COND_EVAL(true, x, y, z) → MINUS_INT(z, pos(s(s(0))))
EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z)
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))
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)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(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)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
The TRS R consists of the following rules:
eval(x, y, z) → Cond_eval1(greatereq_int(x, pos(0)), x, y, z)
Cond_eval1(true, x, y, z) → eval(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
Cond_eval(true, x, y, z) → eval(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
eval(x, y, z) → Cond_eval(greatereq_int(x, pos(0)), x, y, z)
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))
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))
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)
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))
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(x, y, z) → COND_EVAL1(greatereq_int(x, pos(0)), x, y, z)
EVAL(x, y, z) → GREATEREQ_INT(x, pos(0))
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
COND_EVAL1(true, x, y, z) → PLUS_INT(x, y)
COND_EVAL1(true, x, y, z) → MINUS_INT(y, pos(s(s(0))))
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
COND_EVAL(true, x, y, z) → PLUS_INT(x, z)
COND_EVAL(true, x, y, z) → PLUS_INT(pos(s(0)), y)
COND_EVAL(true, x, y, z) → MINUS_INT(z, pos(s(s(0))))
EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z)
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))
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)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(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)) → PLUS_NAT(x, y)
MINUS_INT(pos(x), neg(y)) → PLUS_NAT(x, y)
The TRS R consists of the following rules:
eval(x, y, z) → Cond_eval1(greatereq_int(x, pos(0)), x, y, z)
Cond_eval1(true, x, y, z) → eval(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
Cond_eval(true, x, y, z) → eval(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
eval(x, y, z) → Cond_eval(greatereq_int(x, pos(0)), x, y, z)
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))
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))
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)
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))
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 14 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
The TRS R consists of the following rules:
eval(x, y, z) → Cond_eval1(greatereq_int(x, pos(0)), x, y, z)
Cond_eval1(true, x, y, z) → eval(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
Cond_eval(true, x, y, z) → eval(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
eval(x, y, z) → Cond_eval(greatereq_int(x, pos(0)), x, y, z)
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))
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))
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)
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))
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
R is empty.
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 > 2
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
The TRS R consists of the following rules:
eval(x, y, z) → Cond_eval1(greatereq_int(x, pos(0)), x, y, z)
Cond_eval1(true, x, y, z) → eval(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
Cond_eval(true, x, y, z) → eval(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
eval(x, y, z) → Cond_eval(greatereq_int(x, pos(0)), x, y, z)
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))
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))
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)
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))
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
R is empty.
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
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
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
The TRS R consists of the following rules:
eval(x, y, z) → Cond_eval1(greatereq_int(x, pos(0)), x, y, z)
Cond_eval1(true, x, y, z) → eval(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
Cond_eval(true, x, y, z) → eval(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
eval(x, y, z) → Cond_eval(greatereq_int(x, pos(0)), x, y, z)
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))
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))
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)
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))
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
R is empty.
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
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
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
The TRS R consists of the following rules:
eval(x, y, z) → Cond_eval1(greatereq_int(x, pos(0)), x, y, z)
Cond_eval1(true, x, y, z) → eval(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
Cond_eval(true, x, y, z) → eval(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
eval(x, y, z) → Cond_eval(greatereq_int(x, pos(0)), x, y, z)
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))
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))
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)
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))
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
R is empty.
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
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
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(x, y, z) → COND_EVAL1(greatereq_int(x, pos(0)), x, y, z)
EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
The TRS R consists of the following rules:
eval(x, y, z) → Cond_eval1(greatereq_int(x, pos(0)), x, y, z)
Cond_eval1(true, x, y, z) → eval(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
Cond_eval(true, x, y, z) → eval(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
eval(x, y, z) → Cond_eval(greatereq_int(x, pos(0)), x, y, z)
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))
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))
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)
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))
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(x, y, z) → COND_EVAL1(greatereq_int(x, pos(0)), x, y, z)
EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_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)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(x, y, z) → COND_EVAL1(greatereq_int(x, pos(0)), x, y, z)
EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_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)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(x, y, z) → COND_EVAL1(greatereq_int(x, pos(0)), x, y, z) at position [0] we obtained the following new rules [LPAR04]:
EVAL(neg(0), y1, y2) → COND_EVAL1(true, neg(0), y1, y2)
EVAL(neg(s(x0)), y1, y2) → COND_EVAL1(false, neg(s(x0)), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(neg(0), y1, y2) → COND_EVAL1(true, neg(0), y1, y2)
EVAL(neg(s(x0)), y1, y2) → COND_EVAL1(false, neg(s(x0)), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_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)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(neg(0), y1, y2) → COND_EVAL1(true, neg(0), y1, y2)
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_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)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z) at position [0] we obtained the following new rules [LPAR04]:
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(false, neg(s(x0)), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL(true, neg(0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(neg(0), y1, y2) → COND_EVAL1(true, neg(0), y1, y2)
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
EVAL(neg(s(x0)), y1, y2) → COND_EVAL(false, neg(s(x0)), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL(true, neg(0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_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)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(neg(0), y1, y2) → COND_EVAL1(true, neg(0), y1, y2)
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL(true, neg(0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_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)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(neg(0), y1, y2) → COND_EVAL1(true, neg(0), y1, y2)
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL(true, neg(0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(neg(0), y1, y2) → COND_EVAL1(true, neg(0), y1, y2)
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL(true, neg(0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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, y2) → COND_EVAL1(true, neg(0), y1, y2)
The remaining pairs can at least be oriented weakly.
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL(true, neg(0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
Used ordering: Matrix interpretation [MATRO]:
| POL(EVAL(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(COND_EVAL1(x1, x2, x3, x4)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(plus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(COND_EVAL(x1, x2, x3, x4)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(plus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat(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)
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), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL(true, neg(0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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, y2) → COND_EVAL(true, neg(0), y1, y2)
The remaining pairs can at least be oriented weakly.
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
Used ordering: Matrix interpretation [MATRO]:
| POL(COND_EVAL1(x1, x2, x3, x4)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(EVAL(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(plus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(COND_EVAL(x1, x2, x3, x4)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(plus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat(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)
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), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL1(true, neg(x0), pos(x1), y2) → EVAL(minus_nat(x1, x0), minus_int(pos(x1), pos(s(s(0)))), y2)
COND_EVAL1(true, neg(x0), neg(x1), y2) → EVAL(neg(plus_nat(x0, x1)), minus_int(neg(x1), pos(s(s(0)))), y2)
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), minus_int(neg(x1), pos(s(s(0)))), y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_int(pos(x1), pos(s(s(0)))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL1(true, neg(x0), pos(x1), y2) → EVAL(minus_nat(x1, x0), minus_int(pos(x1), pos(s(s(0)))), y2)
COND_EVAL1(true, neg(x0), neg(x1), y2) → EVAL(neg(plus_nat(x0, x1)), minus_int(neg(x1), pos(s(s(0)))), y2)
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), minus_int(neg(x1), pos(s(s(0)))), y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_int(pos(x1), pos(s(s(0)))), y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), minus_int(neg(x1), pos(s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_int(pos(x1), pos(s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), minus_int(neg(x1), pos(s(s(0)))), y2) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_int(pos(x1), pos(s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_int(pos(x1), pos(s(s(0)))), y2) at position [1] we obtained the following new rules [LPAR04]:
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0))))) at position [0] we obtained the following new rules [LPAR04]:
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_int(pos(x1), pos(s(s(0)))))
COND_EVAL(true, neg(x0), y1, pos(x1)) → EVAL(minus_nat(x1, x0), plus_int(pos(s(0)), y1), minus_int(pos(x1), pos(s(s(0)))))
COND_EVAL(true, neg(x0), y1, neg(x1)) → EVAL(neg(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_int(neg(x1), pos(s(s(0)))))
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), minus_int(neg(x1), pos(s(s(0)))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_int(pos(x1), pos(s(s(0)))))
COND_EVAL(true, neg(x0), y1, pos(x1)) → EVAL(minus_nat(x1, x0), plus_int(pos(s(0)), y1), minus_int(pos(x1), pos(s(s(0)))))
COND_EVAL(true, neg(x0), y1, neg(x1)) → EVAL(neg(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_int(neg(x1), pos(s(s(0)))))
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), minus_int(neg(x1), pos(s(s(0)))))
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_int(pos(x1), pos(s(s(0)))))
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), minus_int(neg(x1), pos(s(s(0)))))
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_int(pos(x1), pos(s(s(0)))))
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), minus_int(neg(x1), pos(s(s(0)))))
The TRS R consists of the following rules:
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_int(pos(x1), pos(s(s(0))))) at position [2] we obtained the following new rules [LPAR04]:
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_nat(x1, s(s(0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), minus_int(neg(x1), pos(s(s(0)))))
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_nat(x1, s(s(0))))
The TRS R consists of the following rules:
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), minus_int(neg(x1), pos(s(s(0)))))
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_nat(x1, s(s(0))))
The TRS R consists of the following rules:
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), minus_int(neg(x1), pos(s(s(0))))) at position [2] we obtained the following new rules [LPAR04]:
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), neg(plus_nat(x1, s(s(0)))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_nat(x1, s(s(0))))
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), neg(plus_nat(x1, s(s(0)))))
The TRS R consists of the following rules:
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(s(x), 0) → pos(s(x))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_nat(x1, s(s(0))))
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), neg(plus_nat(x1, s(s(0)))))
The TRS R consists of the following rules:
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ ForwardInstantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_nat(x1, s(s(0))))
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), neg(plus_nat(x1, s(s(0)))))
The TRS R consists of the following rules:
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL(pos(x0), y1, y2) → COND_EVAL1(true, pos(x0), y1, y2) we obtained the following new rules [LPAR04]:
EVAL(pos(x0), pos(y_1), x2) → COND_EVAL1(true, pos(x0), pos(y_1), x2)
EVAL(pos(x0), neg(y_1), x2) → COND_EVAL1(true, pos(x0), neg(y_1), x2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2)
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_nat(x1, s(s(0))))
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), neg(plus_nat(x1, s(s(0)))))
EVAL(pos(x0), pos(y_1), x2) → COND_EVAL1(true, pos(x0), pos(y_1), x2)
EVAL(pos(x0), neg(y_1), x2) → COND_EVAL1(true, pos(x0), neg(y_1), x2)
The TRS R consists of the following rules:
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [JAR06] the rule EVAL(pos(x0), y1, y2) → COND_EVAL(true, pos(x0), y1, y2) we obtained the following new rules [LPAR04]:
EVAL(pos(x0), x1, neg(y_2)) → COND_EVAL(true, pos(x0), x1, neg(y_2))
EVAL(pos(x0), x1, pos(y_2)) → COND_EVAL(true, pos(x0), x1, pos(y_2))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, pos(x0), neg(x1), y2) → EVAL(minus_nat(x0, x1), neg(plus_nat(x1, s(s(0)))), y2)
COND_EVAL1(true, pos(x0), pos(x1), y2) → EVAL(pos(plus_nat(x0, x1)), minus_nat(x1, s(s(0))), y2)
COND_EVAL(true, pos(x0), y1, pos(x1)) → EVAL(pos(plus_nat(x0, x1)), plus_int(pos(s(0)), y1), minus_nat(x1, s(s(0))))
COND_EVAL(true, pos(x0), y1, neg(x1)) → EVAL(minus_nat(x0, x1), plus_int(pos(s(0)), y1), neg(plus_nat(x1, s(s(0)))))
EVAL(pos(x0), pos(y_1), x2) → COND_EVAL1(true, pos(x0), pos(y_1), x2)
EVAL(pos(x0), neg(y_1), x2) → COND_EVAL1(true, pos(x0), neg(y_1), x2)
EVAL(pos(x0), x1, neg(y_2)) → COND_EVAL(true, pos(x0), x1, neg(y_2))
EVAL(pos(x0), x1, pos(y_2)) → COND_EVAL(true, pos(x0), x1, pos(y_2))
The TRS R consists of the following rules:
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
The set Q consists of the following terms:
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
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
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(x, y, z) → COND_EVAL1(greatereq_int(x, pos(0)), x, y, z)
EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_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)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
COND_EVAL1(true, x, y, z) → EVAL(plus_int(x, y), minus_int(y, pos(s(s(0)))), z)
EVAL(x, y, z) → COND_EVAL1(greatereq_int(x, pos(0)), x, y, z)
EVAL(x, y, z) → COND_EVAL(greatereq_int(x, pos(0)), x, y, z)
COND_EVAL(true, x, y, z) → EVAL(plus_int(x, z), plus_int(pos(s(0)), y), minus_int(z, pos(s(s(0)))))
The TRS R consists of the following rules:
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_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)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(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))
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
We have to consider all minimal (P,Q,R)-chains.