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:
f(TRUE, x, y, z) → f(&&(>@z(x, y), >@z(x, z)), x, y, +@z(z, 1@z))
f(TRUE, x, y, z) → f(&&(>@z(x, y), >@z(x, z)), x, +@z(y, 1@z), z)
The set Q consists of the following terms:
f(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:
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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))
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, x, y, z) → AND(greater_int(x, y), greater_int(x, z))
F(true, x, y, z) → GREATER_INT(x, y)
F(true, x, y, z) → GREATER_INT(x, z)
F(true, x, y, z) → PLUS_INT(pos(s(0)), z)
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
F(true, x, y, z) → PLUS_INT(pos(s(0)), y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
The TRS R consists of the following rules:
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, x, y, z) → AND(greater_int(x, y), greater_int(x, z))
F(true, x, y, z) → GREATER_INT(x, y)
F(true, x, y, z) → GREATER_INT(x, z)
F(true, x, y, z) → PLUS_INT(pos(s(0)), z)
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
F(true, x, y, z) → PLUS_INT(pos(s(0)), y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
PLUS_INT(pos(x), neg(y)) → MINUS_NAT(x, y)
PLUS_INT(neg(x), pos(y)) → MINUS_NAT(y, x)
PLUS_INT(neg(x), neg(y)) → PLUS_NAT(x, y)
PLUS_INT(pos(x), pos(y)) → PLUS_NAT(x, y)
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
The TRS R consists of the following rules:
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 9 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:
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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
↳ 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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ 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:
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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
↳ 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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ 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:
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
The TRS R consists of the following rules:
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
R is empty.
The set Q consists of the following terms:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(GREATER_INT(x1, x2)) = 2·x1 + x2
POL(neg(x1)) = x1
POL(s(x1)) = 2·x1
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
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:
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
The TRS R consists of the following rules:
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
R is empty.
The set Q consists of the following terms:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(GREATER_INT(x1, x2)) = 2·x1 + x2
POL(pos(x1)) = x1
POL(s(x1)) = 2·x1
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
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
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
The TRS R consists of the following rules:
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
f(true, x, y, z) → f(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
f(true, x0, x1, x2)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(true, x0, x1, x2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
Positions in right side of the pair: Pair: F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ QDP
↳ RemovalProof
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z, x_removed) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(x_removed, y), z, x_removed)
F(true, x, y, z, x_removed) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(x_removed, z), x_removed)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z)
Positions in right side of the pair: Pair: F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z, x_removed) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(x_removed, y), z, x_removed)
F(true, x, y, z, x_removed) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(x_removed, z), x_removed)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, plus_int(pos(s(0)), y), z) at position [0] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(s(x1))), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(s(x1))), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(s(x0))), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(s(x0))), y2)
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(s(x1))), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(s(x1))), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(s(x0))), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(s(x0))), y2)
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(s(x1))), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(s(x1))), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(s(x0))), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(s(x0))), y2)
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(s(x1))), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(s(x0))), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(s(x0))), y2)
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(s(x0))), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(s(x0))), y2)
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(s(x0))), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), neg(0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), neg(0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), neg(0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), plus_int(pos(s(0)), pos(s(x0))), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), plus_int(pos(s(0)), pos(s(x0))), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), plus_int(pos(s(0)), pos(0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), neg(0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), plus_int(pos(s(0)), pos(s(x1))), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), s(x1)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), s(x1)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), s(x0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), s(x0)), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(s(0), 0), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), 0)), y2) at position [2,0] we obtained the following new rules [LPAR04]:
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), minus_nat(s(0), 0), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), 0)), y2) at position [2,0] we obtained the following new rules [LPAR04]:
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), s(x1))), y2) at position [2,0] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), minus_nat(s(0), 0), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(plus_nat(s(0), s(x0))), y2) at position [2,0] we obtained the following new rules [LPAR04]:
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), 0)), y2) at position [2,0] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(plus_nat(s(0), s(x0))), y2) at position [2,0] we obtained the following new rules [LPAR04]:
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(plus_nat(s(0), 0)), y2) at position [2,0] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(s(0), 0), y2) at position [2] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2)
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(plus_nat(s(0), s(x1))), y2) at position [2,0] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, 0))), y2) at position [2,0,0] we obtained the following new rules [LPAR04]:
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, 0))), y2) at position [2,0,0] we obtained the following new rules [LPAR04]:
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, s(x1)))), y2) at position [2,0,0] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(plus_nat(0, s(x0)))), y2) at position [2,0,0] we obtained the following new rules [LPAR04]:
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, 0))), y2) at position [2,0,0] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(plus_nat(0, s(x0)))), y2) at position [2,0,0] we obtained the following new rules [LPAR04]:
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(plus_nat(0, 0))), y2) at position [2,0,0] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(plus_nat(0, s(x1)))), y2) at position [2,0,0] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule F(true, x, y, z) → F(and(greater_int(x, y), greater_int(x, z)), x, y, plus_int(pos(s(0)), z)) at position [0] we obtained the following new rules [LPAR04]:
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(s(x0))))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(s(x0))))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1))))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1))))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(s(x0))))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(s(x0))))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1))))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1))))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(s(x0))))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(s(x0))))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(true, false) → false
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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(s(x0)))) at position [3] we obtained the following new rules [LPAR04]:
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, pos(plus_nat(s(0), s(x0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, pos(plus_nat(s(0), s(x0))))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(true, false) → false
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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), neg(0))) at position [3] we obtained the following new rules [LPAR04]:
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, minus_nat(s(0), 0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, pos(plus_nat(s(0), s(x0))))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, minus_nat(s(0), 0))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(true, false) → false
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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(0)) at position [0] we obtained the following new rules [LPAR04]:
F(true, pos(0), pos(s(x0)), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(s(x0))), pos(0))
F(true, pos(0), pos(0), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(0)), pos(0))
F(true, pos(0), neg(s(x0)), pos(0)) → F(and(true, false), pos(0), plus_int(pos(s(0)), neg(s(x0))), pos(0))
F(true, pos(0), neg(0), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), neg(0)), pos(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, pos(plus_nat(s(0), s(x0))))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, minus_nat(s(0), 0))
F(true, pos(0), pos(s(x0)), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(s(x0))), pos(0))
F(true, pos(0), pos(0), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(0)), pos(0))
F(true, pos(0), neg(s(x0)), pos(0)) → F(and(true, false), pos(0), plus_int(pos(s(0)), neg(s(x0))), pos(0))
F(true, pos(0), neg(0), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), neg(0)), pos(0))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(true, false) → false
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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, pos(plus_nat(s(0), s(x0))))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, minus_nat(s(0), 0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(0), pos(s(x0)), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(s(x0))), pos(0))
F(true, pos(0), pos(0), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(0)), pos(0))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(true, false) → false
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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), plus_int(pos(s(0)), y1), neg(s(x0))) at position [0] we obtained the following new rules [LPAR04]:
F(true, pos(0), neg(0), neg(s(y1))) → F(and(false, true), pos(0), plus_int(pos(s(0)), neg(0)), neg(s(y1)))
F(true, pos(0), pos(0), neg(s(y1))) → F(and(false, true), pos(0), plus_int(pos(s(0)), pos(0)), neg(s(y1)))
F(true, pos(0), neg(s(x0)), neg(s(y1))) → F(and(true, true), pos(0), plus_int(pos(s(0)), neg(s(x0))), neg(s(y1)))
F(true, pos(0), pos(s(x0)), neg(s(y1))) → F(and(false, true), pos(0), plus_int(pos(s(0)), pos(s(x0))), neg(s(y1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, pos(plus_nat(s(0), s(x0))))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, minus_nat(s(0), 0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(0), pos(s(x0)), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(s(x0))), pos(0))
F(true, pos(0), pos(0), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(0)), pos(0))
F(true, pos(0), neg(0), neg(s(y1))) → F(and(false, true), pos(0), plus_int(pos(s(0)), neg(0)), neg(s(y1)))
F(true, pos(0), pos(0), neg(s(y1))) → F(and(false, true), pos(0), plus_int(pos(s(0)), pos(0)), neg(s(y1)))
F(true, pos(0), neg(s(x0)), neg(s(y1))) → F(and(true, true), pos(0), plus_int(pos(s(0)), neg(s(x0))), neg(s(y1)))
F(true, pos(0), pos(s(x0)), neg(s(y1))) → F(and(false, true), pos(0), plus_int(pos(s(0)), pos(s(x0))), neg(s(y1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(true, false) → false
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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), minus_nat(0, x0), y2)
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(s(x0))), y2)
F(true, pos(0), y1, pos(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(0)), y2)
F(true, pos(0), pos(s(x0)), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(s(x0))) → F(and(greater_int(pos(0), y1), true), pos(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, pos(0), pos(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, pos(s(x0))) → F(and(greater_int(pos(0), y1), false), pos(0), y1, pos(plus_nat(s(0), s(x0))))
F(true, pos(0), neg(s(x0)), y2) → F(and(true, greater_int(pos(0), y2)), pos(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, pos(0), y1, neg(0)) → F(and(greater_int(pos(0), y1), false), pos(0), y1, minus_nat(s(0), 0))
F(true, pos(0), neg(0), y2) → F(and(false, greater_int(pos(0), y2)), pos(0), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(0), neg(s(x0)), neg(s(y1))) → F(and(true, true), pos(0), plus_int(pos(s(0)), neg(s(x0))), neg(s(y1)))
F(true, pos(0), pos(0), neg(s(y1))) → F(and(false, true), pos(0), plus_int(pos(s(0)), pos(0)), neg(s(y1)))
F(true, pos(0), pos(s(x0)), neg(s(y1))) → F(and(false, true), pos(0), plus_int(pos(s(0)), pos(s(x0))), neg(s(y1)))
F(true, pos(0), pos(s(x0)), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(s(x0))), pos(0))
F(true, pos(0), pos(0), pos(0)) → F(and(false, false), pos(0), plus_int(pos(s(0)), pos(0)), pos(0))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(true, false) → false
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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1))))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1))))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), neg(0))) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, minus_nat(s(0), 0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1))))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, minus_nat(s(0), 0))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1)))) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, minus_nat(s(0), 0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1)))) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(s(0), s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, minus_nat(s(0), 0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(s(0), s(x1)))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, plus_int(pos(s(0)), pos(0))) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), 0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, minus_nat(s(0), 0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(s(0), s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), 0)))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, minus_nat(s(0), 0)) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(s(0), s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), 0)))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), s(x1)))) at position [3,0] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(plus_nat(0, s(x1)))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(s(0), s(x1)))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), 0)))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(plus_nat(0, s(x1)))))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(s(0), s(x1))) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), 0)))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(plus_nat(0, s(x1)))))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(plus_nat(s(0), 0))) at position [3,0] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(plus_nat(0, 0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(plus_nat(0, s(x1)))))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(plus_nat(0, 0))))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(plus_nat(0, s(x1))))) at position [3,0,0] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(plus_nat(0, 0))))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(plus_nat(0, 0)))) at position [3,0,0] we obtained the following new rules [LPAR04]:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), minus_nat(0, x1), y2)
The remaining pairs can at least be oriented weakly.
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
Used ordering: Matrix interpretation [MATRO]:
| POL(F(x1, x2, x3, x4)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(and(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greater_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, 0) → pos(0)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
The remaining pairs can at least be oriented weakly.
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(F(x1, x2, x3, x4)) = x4
POL(and(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(minus_nat(x1, x2)) = 1
POL(neg(x1)) = 1
POL(plus_int(x1, x2)) = x2
POL(plus_nat(x1, x2)) = 1 + x2
POL(pos(x1)) = 0
POL(s(x1)) = 0
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, 0) → pos(0)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(0)))
The remaining pairs can at least be oriented weakly.
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 1
POL(F(x1, x2, x3, x4)) = x4
POL(and(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(minus_nat(x1, x2)) = 1
POL(neg(x1)) = 1
POL(plus_int(x1, x2)) = x1 + x2
POL(plus_nat(x1, x2)) = x2
POL(pos(x1)) = x1
POL(s(x1)) = 0
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_int(pos(x), neg(y)) → minus_nat(x, y)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
The remaining pairs can at least be oriented weakly.
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 1
POL(F(x1, x2, x3, x4)) = x3
POL(and(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(minus_nat(x1, x2)) = 0
POL(neg(x1)) = x1
POL(plus_int(x1, x2)) = x2
POL(plus_nat(x1, x2)) = 1 + x2
POL(pos(x1)) = 0
POL(s(x1)) = 0
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, 0) → pos(0)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(s(x1))), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(0)), y2)
F(true, neg(s(x0)), pos(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), neg(0), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), pos(s(x1)), y2) → F(and(false, greater_int(neg(s(x0)), y2)), neg(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
The remaining pairs can at least be oriented weakly.
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
Used ordering: Matrix interpretation [MATRO]:
| POL(F(x1, x2, x3, x4)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(and(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greater_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
greater_int(neg(0), neg(0)) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(true, false) → false
and(false, true) → false
and(true, true) → true
greater_int(neg(s(x)), pos(s(y))) → false
and(false, false) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), y1, minus_nat(0, x1))
The remaining pairs can at least be oriented weakly.
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
Used ordering: Matrix interpretation [MATRO]:
| POL(F(x1, x2, x3, x4)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(and(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greater_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(0, 0) → pos(0)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(true, neg(s(x0)), y1, pos(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(s(x0)), y1, neg(0)) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, neg(s(x0)), y1, pos(s(x1))) → F(and(greater_int(neg(s(x0)), y1), false), neg(s(x0)), y1, pos(s(s(x1))))
The remaining pairs can at least be oriented weakly.
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
Used ordering: Matrix interpretation [MATRO]:
| POL(F(x1, x2, x3, x4)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| POL(and(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greater_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(plus_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
greater_int(neg(0), neg(0)) → false
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(true, false) → false
and(false, true) → false
and(true, true) → true
greater_int(neg(s(x)), pos(s(y))) → false
and(false, false) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
Positions in right side of the pair: Pair: F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ RemovalProof
↳ QDP
↳ RemovalProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, neg(s(x1)), x_removed) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(x_removed, y1), neg(s(x1)), x_removed)
F(true, neg(s(x0)), neg(s(x1)), y2, x_removed) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(x_removed, y2), x_removed)
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: F(true, neg(s(x0)), y1, neg(s(x1))) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
Positions in right side of the pair: Pair: F(true, neg(s(x0)), neg(s(x1)), y2) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(s(x0)), y1, neg(s(x1)), x_removed) → F(and(greater_int(neg(s(x0)), y1), greater_int(neg(x0), neg(x1))), neg(s(x0)), plus_int(x_removed, y1), neg(s(x1)), x_removed)
F(true, neg(s(x0)), neg(s(x1)), y2, x_removed) → F(and(greater_int(neg(x0), neg(x1)), greater_int(neg(s(x0)), y2)), neg(s(x0)), neg(s(x1)), plus_int(x_removed, y2), x_removed)
The TRS R consists of the following rules:
greater_int(neg(s(x)), pos(0)) → false
greater_int(neg(s(x)), neg(0)) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), neg(s(y))) → true
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1))))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1))))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), 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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, plus_int(pos(s(0)), pos(s(x1)))) at position [3] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), pos(0)))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), 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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), pos(0))) at position [3] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, pos(plus_nat(s(0), 0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(0)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, pos(plus_nat(s(0), 0)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), 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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(0))) at position [3] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, minus_nat(s(0), 0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1))))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, pos(plus_nat(s(0), 0)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, minus_nat(s(0), 0))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), 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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, plus_int(pos(s(0)), neg(s(x1)))) at position [3] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, minus_nat(s(0), s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, pos(plus_nat(s(0), s(x1))))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, pos(plus_nat(s(0), 0)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, minus_nat(s(0), 0))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, minus_nat(s(0), s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), 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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, pos(plus_nat(s(0), s(x1)))) at position [3,0] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, pos(s(plus_nat(0, s(x1)))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, pos(plus_nat(s(0), 0)))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, minus_nat(s(0), 0))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, minus_nat(s(0), s(x1)))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, pos(s(plus_nat(0, s(x1)))))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), 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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, pos(plus_nat(s(0), 0))) at position [3,0] we obtained the following new rules [LPAR04]:
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, pos(s(plus_nat(0, 0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), plus_int(pos(s(0)), y1), pos(s(x1)))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), minus_nat(0, x1), y2)
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), pos(0))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(0))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(0)), y2)
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), plus_int(pos(s(0)), y1), neg(s(x1)))
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(s(x1))), y2)
F(true, pos(s(x0)), pos(s(x1)), y2) → F(and(greater_int(pos(x0), pos(x1)), greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), pos(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), pos(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(0), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(0), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), neg(s(x1)), y2) → F(and(true, greater_int(pos(s(x0)), y2)), pos(s(x0)), neg(s(x1)), plus_int(pos(s(0)), y2))
F(true, pos(s(x0)), y1, neg(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, minus_nat(s(0), 0))
F(true, pos(s(x0)), y1, neg(s(x1))) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, minus_nat(s(0), s(x1)))
F(true, pos(s(x0)), y1, pos(s(x1))) → F(and(greater_int(pos(s(x0)), y1), greater_int(pos(x0), pos(x1))), pos(s(x0)), y1, pos(s(plus_nat(0, s(x1)))))
F(true, pos(s(x0)), y1, pos(0)) → F(and(greater_int(pos(s(x0)), y1), true), pos(s(x0)), y1, pos(s(plus_nat(0, 0))))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(pos(s(x)), pos(0)) → true
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(pos(s(x)), neg(0)) → true
greater_int(pos(s(x)), neg(s(y))) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), 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:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(s(x0))))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), neg(0)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(s(x0))))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), neg(0)))
The TRS R consists of the following rules:
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(0), neg(s(y))) → true
and(false, false) → false
and(true, false) → false
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(s(x0)))) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), s(x0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(0)))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), s(x0))))
The TRS R consists of the following rules:
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(0), neg(s(y))) → true
and(false, false) → false
and(true, false) → false
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), pos(0))) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), 0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, plus_int(pos(s(0)), neg(s(x0))))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), s(x0))))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), 0)))
The TRS R consists of the following rules:
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(0), neg(s(y))) → true
and(false, false) → false
and(true, false) → false
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, plus_int(pos(s(0)), neg(s(x0)))) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, minus_nat(s(0), s(x0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), neg(0)))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), s(x0))))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), 0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, minus_nat(s(0), s(x0)))
The TRS R consists of the following rules:
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(0), neg(s(y))) → true
and(false, false) → false
and(true, false) → false
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, plus_int(pos(s(0)), neg(0))) at position [3] we obtained the following new rules [LPAR04]:
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, minus_nat(s(0), 0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), s(x0))))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), 0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, minus_nat(s(0), s(x0)))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, minus_nat(s(0), 0))
The TRS R consists of the following rules:
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(0), neg(s(y))) → true
and(false, false) → false
and(true, false) → false
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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 rewriting [LPAR04] the rule F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), s(x0)))) at position [3,0] we obtained the following new rules [LPAR04]:
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(s(plus_nat(0, s(x0)))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), 0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, minus_nat(s(0), s(x0)))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, minus_nat(s(0), 0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(s(plus_nat(0, s(x0)))))
The TRS R consists of the following rules:
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(0), neg(s(y))) → true
and(false, false) → false
and(true, false) → false
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), neg(0)) at position [0] we obtained the following new rules [LPAR04]:
F(true, neg(0), neg(s(x0)), neg(0)) → F(and(true, false), neg(0), plus_int(pos(s(0)), neg(s(x0))), neg(0))
F(true, neg(0), neg(0), neg(0)) → F(and(false, false), neg(0), plus_int(pos(s(0)), neg(0)), neg(0))
F(true, neg(0), pos(0), neg(0)) → F(and(false, false), neg(0), plus_int(pos(s(0)), pos(0)), neg(0))
F(true, neg(0), pos(s(x0)), neg(0)) → F(and(false, false), neg(0), plus_int(pos(s(0)), pos(s(x0))), neg(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), 0)))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, minus_nat(s(0), s(x0)))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, minus_nat(s(0), 0))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(s(plus_nat(0, s(x0)))))
F(true, neg(0), neg(s(x0)), neg(0)) → F(and(true, false), neg(0), plus_int(pos(s(0)), neg(s(x0))), neg(0))
F(true, neg(0), neg(0), neg(0)) → F(and(false, false), neg(0), plus_int(pos(s(0)), neg(0)), neg(0))
F(true, neg(0), pos(0), neg(0)) → F(and(false, false), neg(0), plus_int(pos(s(0)), pos(0)), neg(0))
F(true, neg(0), pos(s(x0)), neg(0)) → F(and(false, false), neg(0), plus_int(pos(s(0)), pos(s(x0))), neg(0))
The TRS R consists of the following rules:
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(0), neg(s(y))) → true
and(false, false) → false
and(true, false) → false
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(0))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), minus_nat(0, x0), y2)
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), plus_int(pos(s(0)), y1), pos(s(x0)))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), plus_int(pos(s(0)), y1), neg(s(x0)))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(0)), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(s(x0))), y2)
F(true, neg(0), pos(s(x0)), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(plus_nat(s(0), 0)))
F(true, neg(0), neg(s(x0)), y2) → F(and(true, greater_int(neg(0), y2)), neg(0), neg(s(x0)), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(s(x0))) → F(and(greater_int(neg(0), y1), true), neg(0), y1, minus_nat(s(0), s(x0)))
F(true, neg(0), neg(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), neg(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, neg(0)) → F(and(greater_int(neg(0), y1), false), neg(0), y1, minus_nat(s(0), 0))
F(true, neg(0), pos(0), y2) → F(and(false, greater_int(neg(0), y2)), neg(0), pos(0), plus_int(pos(s(0)), y2))
F(true, neg(0), y1, pos(s(x0))) → F(and(greater_int(neg(0), y1), false), neg(0), y1, pos(s(plus_nat(0, s(x0)))))
F(true, neg(0), pos(0), neg(0)) → F(and(false, false), neg(0), plus_int(pos(s(0)), pos(0)), neg(0))
F(true, neg(0), pos(s(x0)), neg(0)) → F(and(false, false), neg(0), plus_int(pos(s(0)), pos(s(x0))), neg(0))
The TRS R consists of the following rules:
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(0), neg(s(y))) → true
and(false, false) → false
and(true, false) → false
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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, true) → false
and(true, true) → true
The set Q consists of the following terms:
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
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.