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:
if(FALSE, u, v) → v
minusNat(TRUE, x, y) → minus(x, round(x))
if(TRUE, u, v) → u
minus(x, y) → minusNat(&&(>=@z(y, 0@z), =@z(x, +@z(y, 1@z))), x, y)
round(x) → if(=@z(%@z(x, 2@z), 0@z), x, +@z(x, 1@z))
The set Q consists of the following terms:
if(FALSE, x0, x1)
minusNat(TRUE, x0, x1)
if(TRUE, x0, x1)
minus(x0, x1)
round(x0)
Represented integers and predefined function symbols by Terms
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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:
MINUSNAT(true, x, y) → MINUS(x, round(x))
MINUSNAT(true, x, y) → ROUND(x)
MINUS(x, y) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
MINUS(x, y) → AND(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y)))
MINUS(x, y) → GREATEREQ_INT(y, pos(0))
MINUS(x, y) → EQUAL_INT(x, plus_int(pos(s(0)), y))
MINUS(x, y) → PLUS_INT(pos(s(0)), y)
ROUND(x) → IF(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
ROUND(x) → EQUAL_INT(mod_int(x, pos(s(s(0)))), pos(0))
ROUND(x) → MOD_INT(x, pos(s(s(0))))
ROUND(x) → PLUS_INT(pos(s(0)), x)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_INT(pos(x), pos(y))
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_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)
MOD_INT(pos(x), pos(y)) → MOD_NAT(x, y)
MOD_INT(pos(x), neg(y)) → MOD_NAT(x, y)
MOD_INT(neg(x), pos(y)) → MOD_NAT(x, y)
MOD_INT(neg(x), neg(y)) → MOD_NAT(x, y)
MOD_NAT(s(x), s(y)) → IF1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
MOD_NAT(s(x), s(y)) → GREATEREQ_INT(pos(x), pos(y))
MOD_NAT(s(x), s(y)) → MOD_NAT(minus_nat_s(x, y), s(y))
MOD_NAT(s(x), s(y)) → MINUS_NAT_S(x, y)
MINUS_NAT_S(s(x), s(y)) → MINUS_NAT_S(x, y)
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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:
MINUSNAT(true, x, y) → MINUS(x, round(x))
MINUSNAT(true, x, y) → ROUND(x)
MINUS(x, y) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
MINUS(x, y) → AND(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y)))
MINUS(x, y) → GREATEREQ_INT(y, pos(0))
MINUS(x, y) → EQUAL_INT(x, plus_int(pos(s(0)), y))
MINUS(x, y) → PLUS_INT(pos(s(0)), y)
ROUND(x) → IF(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
ROUND(x) → EQUAL_INT(mod_int(x, pos(s(s(0)))), pos(0))
ROUND(x) → MOD_INT(x, pos(s(s(0))))
ROUND(x) → PLUS_INT(pos(s(0)), x)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_INT(pos(x), pos(y))
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_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)
MOD_INT(pos(x), pos(y)) → MOD_NAT(x, y)
MOD_INT(pos(x), neg(y)) → MOD_NAT(x, y)
MOD_INT(neg(x), pos(y)) → MOD_NAT(x, y)
MOD_INT(neg(x), neg(y)) → MOD_NAT(x, y)
MOD_NAT(s(x), s(y)) → IF1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
MOD_NAT(s(x), s(y)) → GREATEREQ_INT(pos(x), pos(y))
MOD_NAT(s(x), s(y)) → MOD_NAT(minus_nat_s(x, y), s(y))
MOD_NAT(s(x), s(y)) → MINUS_NAT_S(x, y)
MINUS_NAT_S(s(x), s(y)) → MINUS_NAT_S(x, y)
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 9 SCCs with 20 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT_S(s(x), s(y)) → MINUS_NAT_S(x, y)
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT_S(s(x), s(y)) → MINUS_NAT_S(x, y)
R is empty.
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT_S(s(x), s(y)) → MINUS_NAT_S(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(s(x), s(y)) → MINUS_NAT_S(x, y)
The graph contains the following edges 1 > 1, 2 > 2
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ 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:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ 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:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 > 2
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
R is empty.
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 >= 2
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_INT(neg(x), neg(y))
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_INT(neg(x), neg(y))
R is empty.
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_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:
EQUAL_INT(neg(s(x)), neg(s(y))) → EQUAL_INT(neg(x), neg(y))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(EQUAL_INT(x1, x2)) = 2·x1 + x2
POL(neg(x1)) = x1
POL(s(x1)) = 2·x1
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_INT(pos(x), pos(y))
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_INT(pos(x), pos(y))
R is empty.
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_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:
EQUAL_INT(pos(s(x)), pos(s(y))) → EQUAL_INT(pos(x), pos(y))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(EQUAL_INT(x1, x2)) = 2·x1 + x2
POL(pos(x1)) = x1
POL(s(x1)) = 2·x1
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
R is empty.
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(GREATEREQ_INT(x1, x2)) = 2·x1 + x2
POL(neg(x1)) = x1
POL(s(x1)) = 2·x1
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
R is empty.
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(GREATEREQ_INT(x1, x2)) = 2·x1 + x2
POL(pos(x1)) = x1
POL(s(x1)) = 2·x1
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MOD_NAT(s(x), s(y)) → MOD_NAT(minus_nat_s(x, y), s(y))
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MOD_NAT(s(x), s(y)) → MOD_NAT(minus_nat_s(x, y), s(y))
The TRS R consists of the following rules:
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MOD_NAT(s(x), s(y)) → MOD_NAT(minus_nat_s(x, y), s(y))
The TRS R consists of the following rules:
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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.
MOD_NAT(s(x), s(y)) → MOD_NAT(minus_nat_s(x, y), s(y))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [MATRO]:
| POL(MOD_NAT(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat_s(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
minus_nat_s(0, s(y)) → 0
minus_nat_s(x, 0) → x
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(x, y) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
MINUSNAT(true, x, y) → MINUS(x, round(x))
The TRS R consists of the following rules:
if(false, u, v) → v
minusNat(true, x, y) → minus(x, round(x))
if(true, u, v) → u
minus(x, y) → minusNat(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_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)
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(pos(x), neg(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
mod_int(neg(x), neg(y)) → neg(mod_nat(x, y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if1(true, x, y) → x
if1(false, x, y) → y
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(x, y) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
MINUSNAT(true, x, y) → MINUS(x, round(x))
The TRS R consists of the following rules:
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
minusNat(true, x0, x1)
if(true, x0, x1)
minus(x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
minusNat(true, x0, x1)
minus(x0, x1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUS(x, y) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
MINUSNAT(true, x, y) → MINUS(x, round(x))
The TRS R consists of the following rules:
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUSNAT(true, x, y) → MINUS(x, round(x)) at position [1] we obtained the following new rules [LPAR04]:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(x, y) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
The TRS R consists of the following rules:
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), pos(0)) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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)
round(x) → if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(x, y) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
round(x0)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
round(x0)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MINUS(x, y) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y)
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(s(0))) is replaced by the fresh variable x_removed.
Pair: MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ QDP
↳ RemovalProof
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MINUS(x, y, x_removed) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y, x_removed)
MINUSNAT(true, x, y, x_removed) → MINUS(x, if(equal_int(mod_int(x, x_removed), pos(0)), x, plus_int(pos(s(0)), x)), x_removed)
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(s(0))) is replaced by the fresh variable x_removed.
Pair: MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MINUS(x, y, x_removed) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y, x_removed)
MINUSNAT(true, x, y, x_removed) → MINUS(x, if(equal_int(mod_int(x, x_removed), pos(0)), x, plus_int(pos(s(0)), x)), x_removed)
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule MINUS(x, y) → MINUSNAT(and(greatereq_int(y, pos(0)), equal_int(x, plus_int(pos(s(0)), y))), x, y) at position [0] we obtained the following new rules [LPAR04]:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, plus_int(pos(s(0)), neg(s(x0))))), y0, neg(s(x0)))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, plus_int(pos(s(0)), pos(x0)))), y0, pos(x0))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, plus_int(pos(s(0)), neg(0)))), y0, neg(0))
MINUS(y0, pos(x1)) → MINUSNAT(and(greatereq_int(pos(x1), pos(0)), equal_int(y0, pos(plus_nat(s(0), x1)))), y0, pos(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, plus_int(pos(s(0)), neg(s(x0))))), y0, neg(s(x0)))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, plus_int(pos(s(0)), pos(x0)))), y0, pos(x0))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, plus_int(pos(s(0)), neg(0)))), y0, neg(0))
MINUS(y0, pos(x1)) → MINUSNAT(and(greatereq_int(pos(x1), pos(0)), equal_int(y0, pos(plus_nat(s(0), x1)))), y0, pos(x1))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, plus_int(pos(s(0)), neg(s(x0))))), y0, neg(s(x0))) at position [0,1,1] we obtained the following new rules [LPAR04]:
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(s(0), s(x0)))), y0, neg(s(x0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, plus_int(pos(s(0)), pos(x0)))), y0, pos(x0))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, plus_int(pos(s(0)), neg(0)))), y0, neg(0))
MINUS(y0, pos(x1)) → MINUSNAT(and(greatereq_int(pos(x1), pos(0)), equal_int(y0, pos(plus_nat(s(0), x1)))), y0, pos(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(s(0), s(x0)))), y0, neg(s(x0)))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, plus_int(pos(s(0)), pos(x0)))), y0, pos(x0)) at position [0,1,1] we obtained the following new rules [LPAR04]:
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(plus_nat(s(0), x0)))), y0, pos(x0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, plus_int(pos(s(0)), neg(0)))), y0, neg(0))
MINUS(y0, pos(x1)) → MINUSNAT(and(greatereq_int(pos(x1), pos(0)), equal_int(y0, pos(plus_nat(s(0), x1)))), y0, pos(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(s(0), s(x0)))), y0, neg(s(x0)))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(plus_nat(s(0), x0)))), y0, pos(x0))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, plus_int(pos(s(0)), neg(0)))), y0, neg(0)) at position [0,1,1] we obtained the following new rules [LPAR04]:
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, minus_nat(s(0), 0))), y0, neg(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, pos(x1)) → MINUSNAT(and(greatereq_int(pos(x1), pos(0)), equal_int(y0, pos(plus_nat(s(0), x1)))), y0, pos(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(s(0), s(x0)))), y0, neg(s(x0)))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(plus_nat(s(0), x0)))), y0, pos(x0))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, minus_nat(s(0), 0))), y0, neg(0))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUS(y0, pos(x1)) → MINUSNAT(and(greatereq_int(pos(x1), pos(0)), equal_int(y0, pos(plus_nat(s(0), x1)))), y0, pos(x1)) at position [0,0] we obtained the following new rules [LPAR04]:
MINUS(y0, pos(x1)) → MINUSNAT(and(true, equal_int(y0, pos(plus_nat(s(0), x1)))), y0, pos(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(s(0), s(x0)))), y0, neg(s(x0)))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(plus_nat(s(0), x0)))), y0, pos(x0))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, minus_nat(s(0), 0))), y0, neg(0))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(s(0), s(x0)))), y0, neg(s(x0))) at position [0,1,1] we obtained the following new rules [LPAR04]:
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(plus_nat(s(0), x0)))), y0, pos(x0))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, minus_nat(s(0), 0))), y0, neg(0))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(plus_nat(s(0), x0)))), y0, pos(x0)) at position [0,1,1,0] we obtained the following new rules [LPAR04]:
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(plus_nat(0, x0))))), y0, pos(x0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, minus_nat(s(0), 0))), y0, neg(0))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(plus_nat(0, x0))))), y0, pos(x0))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, minus_nat(s(0), 0))), y0, neg(0)) at position [0,1,1] we obtained the following new rules [LPAR04]:
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(plus_nat(0, x0))))), y0, pos(x0))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(plus_nat(0, x0))))), y0, pos(x0)) at position [0,1,1,0,0] we obtained the following new rules [LPAR04]:
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x)))
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule MINUSNAT(true, x, y) → MINUS(x, if(equal_int(mod_int(x, pos(s(s(0)))), pos(0)), x, plus_int(pos(s(0)), x))) at position [1] we obtained the following new rules [LPAR04]:
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), plus_int(pos(s(0)), neg(x0))))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(mod_int(pos(x1), pos(s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
MINUSNAT(true, neg(x1), y1) → MINUS(neg(x1), if(equal_int(mod_int(neg(x1), pos(s(s(0)))), pos(0)), neg(x1), minus_nat(s(0), x1)))
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), plus_int(pos(s(0)), neg(x0))))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(mod_int(pos(x1), pos(s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
MINUSNAT(true, neg(x1), y1) → MINUS(neg(x1), if(equal_int(mod_int(neg(x1), pos(s(s(0)))), pos(0)), neg(x1), minus_nat(s(0), x1)))
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0))))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), plus_int(pos(s(0)), neg(x0)))) at position [1,2] we obtained the following new rules [LPAR04]:
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(mod_int(pos(x1), pos(s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
MINUSNAT(true, neg(x1), y1) → MINUS(neg(x1), if(equal_int(mod_int(neg(x1), pos(s(s(0)))), pos(0)), neg(x1), minus_nat(s(0), x1)))
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0))))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
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)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(mod_int(pos(x1), pos(s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
MINUSNAT(true, neg(x1), y1) → MINUS(neg(x1), if(equal_int(mod_int(neg(x1), pos(s(s(0)))), pos(0)), neg(x1), minus_nat(s(0), x1)))
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0))))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_nat(s(x), y) → s(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(mod_int(pos(x1), pos(s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1)))) at position [1,0,0] we obtained the following new rules [LPAR04]:
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x1), y1) → MINUS(neg(x1), if(equal_int(mod_int(neg(x1), pos(s(s(0)))), pos(0)), neg(x1), minus_nat(s(0), x1)))
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0))))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
The TRS R consists of the following rules:
mod_int(pos(x), pos(y)) → pos(mod_nat(x, y))
equal_int(pos(0), pos(0)) → true
equal_int(neg(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(s(x)), pos(0)) → false
plus_nat(s(x), y) → s(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x1), y1) → MINUS(neg(x1), if(equal_int(mod_int(neg(x1), pos(s(s(0)))), pos(0)), neg(x1), minus_nat(s(0), x1)))
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0))))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
The TRS R consists of the following rules:
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUSNAT(true, neg(x1), y1) → MINUS(neg(x1), if(equal_int(mod_int(neg(x1), pos(s(s(0)))), pos(0)), neg(x1), minus_nat(s(0), x1))) at position [1,0,0] we obtained the following new rules [LPAR04]:
MINUSNAT(true, neg(x1), y1) → MINUS(neg(x1), if(equal_int(neg(mod_nat(x1, s(s(0)))), pos(0)), neg(x1), minus_nat(s(0), x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0))))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
The TRS R consists of the following rules:
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
mod_int(neg(x), pos(y)) → neg(mod_nat(x, y))
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0))))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
The TRS R consists of the following rules:
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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].
mod_int(pos(x0), pos(x1))
mod_int(pos(x0), neg(x1))
mod_int(neg(x0), pos(x1))
mod_int(neg(x0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0))))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
The TRS R consists of the following rules:
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), plus_int(pos(s(0)), pos(x0)))) at position [1,2] we obtained the following new rules [LPAR04]:
MINUSNAT(true, pos(x0), y1) → MINUS(pos(x0), if(equal_int(pos(mod_nat(x0, s(s(0)))), pos(0)), pos(x0), pos(plus_nat(s(0), x0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
The TRS R consists of the following rules:
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
The TRS R consists of the following rules:
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
plus_nat(s(x), y) → s(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
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))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1))))
The TRS R consists of the following rules:
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
plus_nat(s(x), y) → s(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(plus_nat(s(0), x1)))) at position [1,2,0] we obtained the following new rules [LPAR04]:
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(plus_nat(0, x1)))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(plus_nat(0, x1)))))
The TRS R consists of the following rules:
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
plus_nat(s(x), y) → s(plus_nat(x, y))
if(false, u, v) → v
if(true, u, v) → u
plus_nat(0, x) → x
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(plus_nat(0, x1)))))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
plus_nat(0, x) → x
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(plus_nat(0, x1))))) at position [1,2,0,0] we obtained the following new rules [LPAR04]:
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
plus_nat(0, x) → x
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
plus_nat(0, x0)
plus_nat(s(x0), x1)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1))
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule MINUS(y0, neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(y0, minus_nat(s(0), x1))), y0, neg(x1)) we obtained the following new rules [LPAR04]:
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0)))
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule MINUS(y0, neg(s(x0))) → MINUSNAT(and(false, equal_int(y0, minus_nat(0, x0))), y0, neg(s(x0))) we obtained the following new rules [LPAR04]:
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule MINUS(y0, neg(0)) → MINUSNAT(and(true, equal_int(y0, pos(s(0)))), y0, neg(0)) we obtained the following new rules [LPAR04]:
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(neg(z0), neg(0)) → MINUSNAT(and(true, equal_int(neg(z0), pos(s(0)))), neg(z0), neg(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(neg(z0), neg(0)) → MINUSNAT(and(true, equal_int(neg(z0), pos(s(0)))), neg(z0), neg(0))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule MINUS(neg(z0), neg(0)) → MINUSNAT(and(true, equal_int(neg(z0), pos(s(0)))), neg(z0), neg(0)) at position [0] we obtained the following new rules [LPAR04]:
MINUS(neg(s(x0)), neg(0)) → MINUSNAT(and(true, false), neg(s(x0)), neg(0))
MINUS(neg(0), neg(0)) → MINUSNAT(and(true, false), neg(0), neg(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(neg(s(x0)), neg(0)) → MINUSNAT(and(true, false), neg(s(x0)), neg(0))
MINUS(neg(0), neg(0)) → MINUSNAT(and(true, false), neg(0), neg(0))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule MINUSNAT(true, neg(x0), y1) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0))) we obtained the following new rules [LPAR04]:
MINUSNAT(true, neg(z0), neg(z1)) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUSNAT(true, neg(x0), pos(z1)) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, neg(z0), neg(s(z1))) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
MINUSNAT(true, neg(z0), neg(z1)) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUSNAT(true, neg(x0), pos(z1)) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, neg(z0), neg(s(z1))) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule MINUS(y0, pos(x0)) → MINUSNAT(and(true, equal_int(y0, pos(s(x0)))), y0, pos(x0)) at position [0] we obtained the following new rules [LPAR04]:
MINUS(neg(s(x0)), pos(x1)) → MINUSNAT(and(true, false), neg(s(x0)), pos(x1))
MINUS(pos(0), pos(x0)) → MINUSNAT(and(true, false), pos(0), pos(x0))
MINUS(pos(s(x0)), pos(x1)) → MINUSNAT(and(true, equal_int(pos(x0), pos(x1))), pos(s(x0)), pos(x1))
MINUS(neg(0), pos(x0)) → MINUSNAT(and(true, false), neg(0), pos(x0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
MINUSNAT(true, neg(z0), neg(z1)) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUSNAT(true, neg(x0), pos(z1)) → MINUS(neg(x0), if(equal_int(neg(mod_nat(x0, s(s(0)))), pos(0)), neg(x0), minus_nat(s(0), x0)))
MINUSNAT(true, neg(z0), neg(s(z1))) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUS(neg(s(x0)), pos(x1)) → MINUSNAT(and(true, false), neg(s(x0)), pos(x1))
MINUS(pos(0), pos(x0)) → MINUSNAT(and(true, false), pos(0), pos(x0))
MINUS(pos(s(x0)), pos(x1)) → MINUSNAT(and(true, equal_int(pos(x0), pos(x1))), pos(s(x0)), pos(x1))
MINUS(neg(0), pos(x0)) → MINUSNAT(and(true, false), neg(0), pos(x0))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, neg(z0), neg(z1)) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUSNAT(true, neg(z0), neg(s(z1))) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, neg(z0), neg(z1)) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUSNAT(true, neg(z0), neg(s(z1))) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(neg(0), neg(s(y))) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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.
MINUS(neg(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(neg(z0), minus_nat(0, x1))), neg(z0), neg(s(x1)))
The remaining pairs can at least be oriented weakly.
MINUSNAT(true, neg(z0), neg(z1)) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUSNAT(true, neg(z0), neg(s(z1))) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(MINUS(x1, x2)) = 1
POL(MINUSNAT(x1, x2, x3)) = x1
POL(and(x1, x2)) = x1
POL(equal_int(x1, x2)) = 0
POL(false) = 0
POL(greatereq_int(x1, x2)) = 1
POL(if(x1, x2, x3)) = 0
POL(if1(x1, x2, x3)) = 0
POL(minus_nat(x1, x2)) = 0
POL(minus_nat_s(x1, x2)) = 0
POL(mod_nat(x1, x2)) = 0
POL(neg(x1)) = 0
POL(pos(x1)) = 0
POL(s(x1)) = 0
POL(true) = 1
The following usable rules [FROCOS05] were oriented:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
and(true, true) → true
and(true, false) → false
and(false, true) → false
and(false, false) → false
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, neg(z0), neg(z1)) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
MINUS(neg(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1))
MINUSNAT(true, neg(z0), neg(s(z1))) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(neg(0), neg(s(y))) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables s(s(0)) is replaced by the fresh variable x_removed.
Pair: MINUSNAT(true, neg(z0), neg(z1)) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
Positions in right side of the pair: Pair: MINUSNAT(true, neg(z0), neg(s(z1))) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ RemovalProof
↳ QDP
↳ RemovalProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, neg(z0), neg(z1), x_removed) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, x_removed)), pos(0)), neg(z0), minus_nat(s(0), z0)), x_removed)
MINUS(neg(z0), neg(x1), x_removed) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1), x_removed)
MINUSNAT(true, neg(z0), neg(s(z1)), x_removed) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, x_removed)), pos(0)), neg(z0), minus_nat(s(0), z0)), x_removed)
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(neg(0), neg(s(y))) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables s(s(0)) is replaced by the fresh variable x_removed.
Pair: MINUSNAT(true, neg(z0), neg(z1)) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
Positions in right side of the pair: Pair: MINUSNAT(true, neg(z0), neg(s(z1))) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, s(s(0)))), pos(0)), neg(z0), minus_nat(s(0), z0)))
Positions in right side of the pair: The new variable was added to all pairs as a new argument[CONREM].
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUSNAT(true, neg(z0), neg(z1), x_removed) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, x_removed)), pos(0)), neg(z0), minus_nat(s(0), z0)), x_removed)
MINUS(neg(z0), neg(x1), x_removed) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(neg(z0), minus_nat(s(0), x1))), neg(z0), neg(x1), x_removed)
MINUSNAT(true, neg(z0), neg(s(z1)), x_removed) → MINUS(neg(z0), if(equal_int(neg(mod_nat(z0, x_removed)), pos(0)), neg(z0), minus_nat(s(0), z0)), x_removed)
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(neg(0), neg(s(y))) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(pos(s(x0)), pos(x1)) → MINUSNAT(and(true, equal_int(pos(x0), pos(x1))), pos(s(x0)), pos(x1))
The TRS R consists of the following rules:
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), pos(s(y))) → false
equal_int(neg(0), pos(s(y))) → false
equal_int(neg(s(x)), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
equal_int(neg(0), pos(0)) → true
equal_int(neg(s(x)), pos(0)) → false
equal_int(neg(0), neg(0)) → true
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(neg(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(neg(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
equal_int(neg(s(x)), neg(s(y))) → equal_int(neg(x), neg(y))
and(true, false) → false
and(true, true) → true
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1))))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(pos(s(x0)), pos(x1)) → MINUSNAT(and(true, equal_int(pos(x0), pos(x1))), pos(s(x0)), pos(x1))
The TRS R consists of the following rules:
equal_int(pos(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule MINUSNAT(true, pos(x1), y1) → MINUS(pos(x1), if(equal_int(pos(mod_nat(x1, s(s(0)))), pos(0)), pos(x1), pos(s(x1)))) we obtained the following new rules [LPAR04]:
MINUSNAT(true, pos(z0), neg(s(z1))) → MINUS(pos(z0), if(equal_int(pos(mod_nat(z0, s(s(0)))), pos(0)), pos(z0), pos(s(z0))))
MINUSNAT(true, pos(z0), neg(z1)) → MINUS(pos(z0), if(equal_int(pos(mod_nat(z0, s(s(0)))), pos(0)), pos(z0), pos(s(z0))))
MINUSNAT(true, pos(s(z0)), pos(z1)) → MINUS(pos(s(z0)), if(equal_int(pos(mod_nat(s(z0), s(s(0)))), pos(0)), pos(s(z0)), pos(s(s(z0)))))
MINUSNAT(true, pos(z0), neg(0)) → MINUS(pos(z0), if(equal_int(pos(mod_nat(z0, s(s(0)))), pos(0)), pos(z0), pos(s(z0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(pos(s(x0)), pos(x1)) → MINUSNAT(and(true, equal_int(pos(x0), pos(x1))), pos(s(x0)), pos(x1))
MINUSNAT(true, pos(z0), neg(s(z1))) → MINUS(pos(z0), if(equal_int(pos(mod_nat(z0, s(s(0)))), pos(0)), pos(z0), pos(s(z0))))
MINUSNAT(true, pos(z0), neg(z1)) → MINUS(pos(z0), if(equal_int(pos(mod_nat(z0, s(s(0)))), pos(0)), pos(z0), pos(s(z0))))
MINUSNAT(true, pos(s(z0)), pos(z1)) → MINUS(pos(s(z0)), if(equal_int(pos(mod_nat(s(z0), s(s(0)))), pos(0)), pos(s(z0)), pos(s(s(z0)))))
MINUSNAT(true, pos(z0), neg(0)) → MINUS(pos(z0), if(equal_int(pos(mod_nat(z0, s(s(0)))), pos(0)), pos(z0), pos(s(z0))))
The TRS R consists of the following rules:
equal_int(pos(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule MINUSNAT(true, pos(s(z0)), pos(z1)) → MINUS(pos(s(z0)), if(equal_int(pos(mod_nat(s(z0), s(s(0)))), pos(0)), pos(s(z0)), pos(s(s(z0))))) at position [1,0,0,0] we obtained the following new rules [LPAR04]:
MINUSNAT(true, pos(s(z0)), pos(z1)) → MINUS(pos(s(z0)), if(equal_int(pos(if1(greatereq_int(pos(z0), pos(s(0))), mod_nat(minus_nat_s(z0, s(0)), s(s(0))), s(z0))), pos(0)), pos(s(z0)), pos(s(s(z0)))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ 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
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS(pos(z0), neg(x1)) → MINUSNAT(and(greatereq_int(neg(x1), pos(0)), equal_int(pos(z0), minus_nat(s(0), x1))), pos(z0), neg(x1))
MINUS(pos(z0), neg(s(x1))) → MINUSNAT(and(false, equal_int(pos(z0), minus_nat(0, x1))), pos(z0), neg(s(x1)))
MINUS(pos(z0), neg(0)) → MINUSNAT(and(true, equal_int(pos(z0), pos(s(0)))), pos(z0), neg(0))
MINUS(pos(s(x0)), pos(x1)) → MINUSNAT(and(true, equal_int(pos(x0), pos(x1))), pos(s(x0)), pos(x1))
MINUSNAT(true, pos(z0), neg(s(z1))) → MINUS(pos(z0), if(equal_int(pos(mod_nat(z0, s(s(0)))), pos(0)), pos(z0), pos(s(z0))))
MINUSNAT(true, pos(z0), neg(z1)) → MINUS(pos(z0), if(equal_int(pos(mod_nat(z0, s(s(0)))), pos(0)), pos(z0), pos(s(z0))))
MINUSNAT(true, pos(z0), neg(0)) → MINUS(pos(z0), if(equal_int(pos(mod_nat(z0, s(s(0)))), pos(0)), pos(z0), pos(s(z0))))
MINUSNAT(true, pos(s(z0)), pos(z1)) → MINUS(pos(s(z0)), if(equal_int(pos(if1(greatereq_int(pos(z0), pos(s(0))), mod_nat(minus_nat_s(z0, s(0)), s(s(0))), s(z0))), pos(0)), pos(s(z0)), pos(s(s(z0)))))
The TRS R consists of the following rules:
equal_int(pos(0), pos(s(y))) → false
equal_int(pos(s(x)), pos(s(y))) → equal_int(pos(x), pos(y))
and(true, false) → false
and(true, true) → true
equal_int(pos(0), pos(0)) → true
equal_int(pos(s(x)), pos(0)) → false
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
equal_int(pos(0), neg(0)) → true
equal_int(pos(0), neg(s(y))) → false
equal_int(pos(s(x)), neg(0)) → false
equal_int(pos(s(x)), neg(s(y))) → false
and(false, false) → false
and(false, true) → false
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
mod_nat(0, s(x)) → 0
mod_nat(s(x), s(y)) → if1(greatereq_int(pos(x), pos(y)), mod_nat(minus_nat_s(x, y), s(y)), s(x))
if(false, u, v) → v
if(true, u, v) → u
greatereq_int(pos(x), pos(0)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
minus_nat_s(x, 0) → x
minus_nat_s(0, s(y)) → 0
minus_nat_s(s(x), s(y)) → minus_nat_s(x, y)
if1(true, x, y) → x
if1(false, x, y) → y
The set Q consists of the following terms:
if(false, x0, x1)
if(true, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
equal_int(pos(0), pos(0))
equal_int(neg(0), pos(0))
equal_int(neg(0), neg(0))
equal_int(pos(0), neg(0))
equal_int(pos(0), pos(s(x0)))
equal_int(neg(0), pos(s(x0)))
equal_int(pos(0), neg(s(x0)))
equal_int(neg(0), neg(s(x0)))
equal_int(pos(s(x0)), pos(0))
equal_int(pos(s(x0)), neg(0))
equal_int(neg(s(x0)), pos(0))
equal_int(neg(s(x0)), neg(0))
equal_int(pos(s(x0)), neg(s(x1)))
equal_int(neg(s(x0)), pos(s(x1)))
equal_int(pos(s(x0)), pos(s(x1)))
equal_int(neg(s(x0)), neg(s(x1)))
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
mod_nat(0, s(x0))
mod_nat(s(x0), s(x1))
if1(true, x0, x1)
if1(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.