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:
lif(TRUE, x, y) → +@z(1@z, log(/@z(x, y), y))
lif(FALSE, x, y) → 0@z
log(x, y) → lif(&&(>=@z(x, y), >@z(y, 1@z)), x, y)
The set Q consists of the following terms:
lif(TRUE, x0, x1)
lif(FALSE, x0, x1)
log(x0, x1)
Represented integers and predefined function symbols by Terms
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → PLUS_INT(pos(s(0)), log(div_int(x, y), y))
LIF(true, x, y) → LOG(div_int(x, y), y)
LIF(true, x, y) → DIV_INT(x, y)
LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
LOG(x, y) → AND(greatereq_int(x, y), greater_int(y, pos(s(0))))
LOG(x, y) → GREATEREQ_INT(x, y)
LOG(x, y) → GREATER_INT(y, pos(s(0)))
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)
DIV_INT(pos(x), pos(s(y))) → DIV_NAT(x, s(y))
DIV_INT(pos(x), neg(s(y))) → DIV_NAT(x, s(y))
DIV_INT(neg(x), pos(s(y))) → DIV_NAT(x, s(y))
DIV_INT(neg(x), neg(s(y))) → DIV_NAT(x, s(y))
DIV_NAT(s(x), s(y)) → IF(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
DIV_NAT(s(x), s(y)) → GREATEREQ_INT(pos(x), pos(y))
DIV_NAT(s(x), s(y)) → DIV_NAT(minus_nat_s(x, y), s(y))
DIV_NAT(s(x), s(y)) → MINUS_NAT_S(x, y)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
MINUS_NAT_S(s(x), s(y)) → MINUS_NAT_S(x, y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → PLUS_INT(pos(s(0)), log(div_int(x, y), y))
LIF(true, x, y) → LOG(div_int(x, y), y)
LIF(true, x, y) → DIV_INT(x, y)
LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
LOG(x, y) → AND(greatereq_int(x, y), greater_int(y, pos(s(0))))
LOG(x, y) → GREATEREQ_INT(x, y)
LOG(x, y) → GREATER_INT(y, pos(s(0)))
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)
DIV_INT(pos(x), pos(s(y))) → DIV_NAT(x, s(y))
DIV_INT(pos(x), neg(s(y))) → DIV_NAT(x, s(y))
DIV_INT(neg(x), pos(s(y))) → DIV_NAT(x, s(y))
DIV_INT(neg(x), neg(s(y))) → DIV_NAT(x, s(y))
DIV_NAT(s(x), s(y)) → IF(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
DIV_NAT(s(x), s(y)) → GREATEREQ_INT(pos(x), pos(y))
DIV_NAT(s(x), s(y)) → DIV_NAT(minus_nat_s(x, y), s(y))
DIV_NAT(s(x), s(y)) → MINUS_NAT_S(x, y)
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
MINUS_NAT_S(s(x), s(y)) → MINUS_NAT_S(x, y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 9 SCCs with 16 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:
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
R is empty.
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(GREATER_INT(x1, x2)) = 2·x1 + x2
POL(neg(x1)) = x1
POL(s(x1)) = 2·x1
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
R is empty.
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(GREATER_INT(x1, x2)) = 2·x1 + x2
POL(pos(x1)) = x1
POL(s(x1)) = 2·x1
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ 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:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ 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:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ 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
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
R is empty.
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(GREATEREQ_INT(x1, x2)) = 2·x1 + x2
POL(neg(x1)) = x1
POL(s(x1)) = 2·x1
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ 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:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
R is empty.
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QDP
↳ 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
↳ 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:
DIV_NAT(s(x), s(y)) → DIV_NAT(minus_nat_s(x, y), s(y))
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
DIV_NAT(s(x), s(y)) → DIV_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:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
DIV_NAT(s(x), s(y)) → DIV_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.
DIV_NAT(s(x), s(y)) → DIV_NAT(minus_nat_s(x, y), s(y))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(DIV_NAT(x1, x2)) = x1
POL(minus_nat_s(x1, x2)) = x1
POL(s(x1)) = 1 + x1
The following usable rules [FROCOS05] were oriented:
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)
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ 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
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
R is empty.
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- MINUS_NAT(s(x), s(y)) → MINUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 > 2
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
R is empty.
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- PLUS_NAT(s(x), y) → PLUS_NAT(x, y)
The graph contains the following edges 1 > 1, 2 >= 2
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
The TRS R consists of the following rules:
lif(true, x, y) → plus_int(pos(s(0)), log(div_int(x, y), y))
lif(false, x, y) → pos(0)
log(x, y) → lif(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
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)
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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))
if(true, x, y) → x
if(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)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
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
↳ RemovalProof
↳ QDP
↳ RemovalProof
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y, x_removed) → LOG(div_int(x, y), y, x_removed)
LOG(x, y, x_removed) → LIF(and(greatereq_int(x, y), greater_int(y, x_removed)), x, y, x_removed)
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
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
↳ RemovalProof
↳ RemovalProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y, x_removed) → LOG(div_int(x, y), y, x_removed)
LOG(x, y, x_removed) → LIF(and(greatereq_int(x, y), greater_int(y, x_removed)), x, y, x_removed)
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y) at position [0] we obtained the following new rules [LPAR04]:
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(s(x1)), pos(s(0)))), neg(x0), pos(s(x1)))
LOG(pos(x0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), pos(x0), pos(0))
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(neg(s(x0)), pos(0)) → LIF(and(false, greater_int(pos(0), pos(s(0)))), neg(s(x0)), pos(0))
LOG(neg(s(x0)), neg(0)) → LIF(and(false, greater_int(neg(0), pos(s(0)))), neg(s(x0)), neg(0))
LOG(neg(0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), neg(0), pos(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(s(0)))), neg(s(x0)), neg(s(x1)))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(s(x1)), pos(s(0)))), neg(x0), pos(s(x1)))
LOG(pos(x0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), pos(x0), pos(0))
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(neg(s(x0)), pos(0)) → LIF(and(false, greater_int(pos(0), pos(s(0)))), neg(s(x0)), pos(0))
LOG(neg(s(x0)), neg(0)) → LIF(and(false, greater_int(neg(0), pos(s(0)))), neg(s(x0)), neg(0))
LOG(neg(0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), neg(0), pos(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(s(0)))), neg(s(x0)), neg(s(x1)))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(s(x1)), pos(s(0)))), neg(x0), pos(s(x1)))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(pos(x0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), pos(x0), pos(0))
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(neg(0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), neg(0), pos(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(s(0)))), neg(s(x0)), neg(s(x1)))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(s(x1)), pos(s(0)))), neg(x0), pos(s(x1))) at position [0,1] we obtained the following new rules [LPAR04]:
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(pos(x0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), pos(x0), pos(0))
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(neg(0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), neg(0), pos(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(s(0)))), neg(s(x0)), neg(s(x1)))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LOG(pos(x0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), pos(x0), pos(0)) at position [0,1] we obtained the following new rules [LPAR04]:
LOG(pos(x0), pos(0)) → LIF(and(true, false), pos(x0), pos(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(neg(0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), neg(0), pos(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(s(0)))), neg(s(x0)), neg(s(x1)))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(pos(x0), pos(0)) → LIF(and(true, false), pos(x0), pos(0))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(neg(0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), neg(0), pos(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(s(0)))), neg(s(x0)), neg(s(x1)))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LOG(neg(0), pos(0)) → LIF(and(true, greater_int(pos(0), pos(s(0)))), neg(0), pos(0)) at position [0,1] we obtained the following new rules [LPAR04]:
LOG(neg(0), pos(0)) → LIF(and(true, false), neg(0), pos(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(s(0)))), neg(s(x0)), neg(s(x1)))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(0), pos(0)) → LIF(and(true, false), neg(0), pos(0))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(s(0)))), neg(s(x0)), neg(s(x1)))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), greater_int(neg(s(x1)), pos(s(0)))), neg(s(x0)), neg(s(x1))) at position [0,1] we obtained the following new rules [LPAR04]:
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(s(x0)), pos(s(0)))), pos(0), pos(s(x0))) at position [0,1] we obtained the following new rules [LPAR04]:
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(s(x1)), pos(s(0)))), pos(s(x0)), pos(s(x1))) at position [0,1] we obtained the following new rules [LPAR04]:
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0))
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
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))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(y0, neg(0)) → LIF(and(greatereq_int(y0, neg(0)), false), y0, neg(0)) at position [0] we obtained the following new rules [LPAR04]:
LOG(neg(s(x0)), neg(0)) → LIF(and(false, false), neg(s(x0)), neg(0))
LOG(pos(x0), neg(0)) → LIF(and(true, false), pos(x0), neg(0))
LOG(neg(0), neg(0)) → LIF(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
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LOG(neg(s(x0)), neg(0)) → LIF(and(false, false), neg(s(x0)), neg(0))
LOG(pos(x0), neg(0)) → LIF(and(true, false), pos(x0), neg(0))
LOG(neg(0), neg(0)) → LIF(and(true, false), neg(0), neg(0))
The TRS R consists of the following rules:
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
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))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
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))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(pos(x0), neg(x1)) → LIF(and(true, greater_int(neg(x1), pos(s(0)))), pos(x0), neg(x1)) at position [0] we obtained the following new rules [LPAR04]:
LOG(pos(y0), neg(s(x0))) → LIF(and(true, false), pos(y0), neg(s(x0)))
LOG(pos(y0), neg(0)) → LIF(and(true, false), pos(y0), neg(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LOG(pos(y0), neg(s(x0))) → LIF(and(true, false), pos(y0), neg(s(x0)))
LOG(pos(y0), neg(0)) → LIF(and(true, false), pos(y0), neg(0))
The TRS R consists of the following rules:
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
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))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
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))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(neg(0), neg(x0)) → LIF(and(true, greater_int(neg(x0), pos(s(0)))), neg(0), neg(x0)) at position [0] we obtained the following new rules [LPAR04]:
LOG(neg(0), neg(s(x0))) → LIF(and(true, false), neg(0), neg(s(x0)))
LOG(neg(0), neg(0)) → LIF(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
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LOG(neg(0), neg(s(x0))) → LIF(and(true, false), neg(0), neg(s(x0)))
LOG(neg(0), neg(0)) → LIF(and(true, false), neg(0), neg(0))
The TRS R consists of the following rules:
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
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))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
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))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
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))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(y0, pos(0)) → LIF(and(greatereq_int(y0, pos(0)), false), y0, pos(0)) at position [0] we obtained the following new rules [LPAR04]:
LOG(neg(0), pos(0)) → LIF(and(true, false), neg(0), pos(0))
LOG(neg(s(x0)), pos(0)) → LIF(and(false, false), neg(s(x0)), pos(0))
LOG(pos(x0), pos(0)) → LIF(and(true, false), pos(x0), pos(0))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LOG(neg(0), pos(0)) → LIF(and(true, false), neg(0), pos(0))
LOG(neg(s(x0)), pos(0)) → LIF(and(false, false), neg(s(x0)), pos(0))
LOG(pos(x0), pos(0)) → LIF(and(true, false), pos(x0), pos(0))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
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))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
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))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
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))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(y0, neg(s(x0))) → LIF(and(greatereq_int(y0, neg(s(x0))), false), y0, neg(s(x0))) at position [0] we obtained the following new rules [LPAR04]:
LOG(neg(0), neg(s(y1))) → LIF(and(true, false), neg(0), neg(s(y1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(x0), neg(s(y1))) → LIF(and(true, false), pos(x0), neg(s(y1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LOG(neg(0), neg(s(y1))) → LIF(and(true, false), neg(0), neg(s(y1)))
LOG(pos(x0), neg(s(y1))) → LIF(and(true, false), pos(x0), neg(s(y1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
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))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
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))
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greatereq_int(neg(s(x)), neg(0)) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(neg(x0), pos(s(x1))) → LIF(and(false, greater_int(pos(x1), pos(0))), neg(x0), pos(s(x1))) at position [0] we obtained the following new rules [LPAR04]:
LOG(neg(y0), pos(s(0))) → LIF(and(false, false), neg(y0), pos(s(0)))
LOG(neg(y0), pos(s(s(x0)))) → LIF(and(false, true), neg(y0), pos(s(s(x0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LOG(neg(y0), pos(s(0))) → LIF(and(false, false), neg(y0), pos(s(0)))
LOG(neg(y0), pos(s(s(x0)))) → LIF(and(false, true), neg(y0), pos(s(s(x0))))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule LOG(pos(0), pos(s(x0))) → LIF(and(false, greater_int(pos(x0), pos(0))), pos(0), pos(s(x0))) at position [0] we obtained the following new rules [LPAR04]:
LOG(pos(0), pos(s(0))) → LIF(and(false, false), pos(0), pos(s(0)))
LOG(pos(0), pos(s(s(x0)))) → LIF(and(false, true), pos(0), pos(s(s(x0))))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LOG(pos(0), pos(s(0))) → LIF(and(false, false), pos(0), pos(s(0)))
LOG(pos(0), pos(s(s(x0)))) → LIF(and(false, true), pos(0), pos(s(s(x0))))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule LIF(true, x, y) → LOG(div_int(x, y), y) we obtained the following new rules [LPAR04]:
LIF(true, neg(s(z0)), neg(s(z1))) → LOG(div_int(neg(s(z0)), neg(s(z1))), neg(s(z1)))
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(div_int(pos(s(z0)), pos(s(z1))), pos(s(z1)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LIF(true, neg(s(z0)), neg(s(z1))) → LOG(div_int(neg(s(z0)), neg(s(z1))), neg(s(z1)))
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(div_int(pos(s(z0)), pos(s(z1))), pos(s(z1)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, neg(s(z0)), neg(s(z1))) → LOG(div_int(neg(s(z0)), neg(s(z1))), neg(s(z1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, neg(s(z0)), neg(s(z1))) → LOG(div_int(neg(s(z0)), neg(s(z1))), neg(s(z1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
The TRS R consists of the following rules:
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, neg(s(z0)), neg(s(z1))) → LOG(div_int(neg(s(z0)), neg(s(z1))), neg(s(z1)))
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
The TRS R consists of the following rules:
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LIF(true, neg(s(z0)), neg(s(z1))) → LOG(div_int(neg(s(z0)), neg(s(z1))), neg(s(z1))) at position [0] we obtained the following new rules [LPAR04]:
LIF(true, neg(s(z0)), neg(s(z1))) → LOG(pos(div_nat(s(z0), s(z1))), neg(s(z1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(neg(s(x0)), neg(s(x1))) → LIF(and(greatereq_int(neg(x0), neg(x1)), false), neg(s(x0)), neg(s(x1)))
LIF(true, neg(s(z0)), neg(s(z1))) → LOG(pos(div_nat(s(z0), s(z1))), neg(s(z1)))
The TRS R consists of the following rules:
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(true, false) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(div_int(pos(s(z0)), pos(s(z1))), pos(s(z1)))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(x), pos(s(y))) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(div_int(pos(s(z0)), pos(s(z1))), pos(s(z1)))
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LIF(true, pos(s(z0)), pos(s(z1))) → LOG(div_int(pos(s(z0)), pos(s(z1))), pos(s(z1))) at position [0] we obtained the following new rules [LPAR04]:
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(pos(div_nat(s(z0), s(z1))), pos(s(z1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(pos(div_nat(s(z0), s(z1))), pos(s(z1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule LIF(true, pos(s(z0)), pos(s(z1))) → LOG(pos(div_nat(s(z0), s(z1))), pos(s(z1))) at position [0,0] we obtained the following new rules [LPAR04]:
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(pos(if(greatereq_int(pos(z0), pos(z1)), div_nat(minus_nat_s(z0, z1), s(z1)), 0)), pos(s(z1)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(pos(if(greatereq_int(pos(z0), pos(z1)), div_nat(minus_nat_s(z0, z1), s(z1)), 0)), pos(s(z1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
LOG(pos(s(x0)), pos(s(x1))) → LIF(and(greatereq_int(pos(x0), pos(x1)), greater_int(pos(x1), pos(0))), pos(s(x0)), pos(s(x1)))
The remaining pairs can at least be oriented weakly.
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(pos(if(greatereq_int(pos(z0), pos(z1)), div_nat(minus_nat_s(z0, z1), s(z1)), 0)), pos(s(z1)))
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(LIF(x1, x2, x3)) = 0
POL(LOG(x1, x2)) = x1
POL(and(x1, x2)) = 0
POL(div_int(x1, x2)) = 0
POL(div_nat(x1, x2)) = 0
POL(false) = 0
POL(greater_int(x1, x2)) = 0
POL(greatereq_int(x1, x2)) = 0
POL(if(x1, x2, x3)) = x2 + x3
POL(minus_nat_s(x1, x2)) = 0
POL(neg(x1)) = 0
POL(pos(x1)) = x1
POL(s(x1)) = 1
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
if(false, x, y) → y
if(true, x, y) → x
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(pos(if(greatereq_int(pos(z0), pos(z1)), div_nat(minus_nat_s(z0, z1), s(z1)), 0)), pos(s(z1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
LIF(true, pos(s(z0)), pos(s(z1))) → LOG(pos(if(greatereq_int(pos(z0), pos(z1)), div_nat(minus_nat_s(z0, z1), s(z1)), 0)), pos(s(z1)))
The remaining pairs can at least be oriented weakly.
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
Used ordering: Matrix interpretation [MATRO]:
| POL(LOG(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(LIF(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(and(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greatereq_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(greater_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(div_int(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(if(x1, x2, x3)) = | | · | x1 | + | | + | | · | x2 | + | | · | x3 |
| POL(div_nat(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
| POL(minus_nat_s(x1, x2)) = | | · | x1 | + | | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
if(false, x, y) → y
if(true, x, y) → x
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RemovalProof
↳ RemovalProof
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
LOG(y0, pos(s(x0))) → LIF(and(greatereq_int(y0, pos(s(x0))), greater_int(pos(x0), pos(0))), y0, pos(s(x0)))
LIF(true, z0, pos(s(z1))) → LOG(div_int(z0, pos(s(z1))), pos(s(z1)))
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(neg(x), pos(s(y))) → false
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
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))
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
lif(true, x0, x1)
lif(false, x0, x1)
log(x0, x1)
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
↳ ITRS
↳ ITRStoQTRSProof
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LIF(true, x, y) → LOG(div_int(x, y), y)
LOG(x, y) → LIF(and(greatereq_int(x, y), greater_int(y, pos(s(0)))), x, y)
The TRS R consists of the following rules:
div_int(pos(x), pos(s(y))) → pos(div_nat(x, s(y)))
div_int(pos(x), neg(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), pos(s(y))) → neg(div_nat(x, s(y)))
div_int(neg(x), neg(s(y))) → pos(div_nat(x, s(y)))
div_nat(0, s(y)) → 0
div_nat(s(x), s(y)) → if(greatereq_int(pos(x), pos(y)), div_nat(minus_nat_s(x, y), s(y)), 0)
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)
if(true, x, y) → x
if(false, x, y) → y
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greater_int(pos(0), pos(0)) → false
greater_int(pos(s(x)), pos(0)) → true
The set Q consists of the following terms:
div_int(pos(x0), pos(s(x1)))
div_int(pos(x0), neg(s(x1)))
div_int(neg(x0), pos(s(x1)))
div_int(neg(x0), neg(s(x1)))
div_nat(0, s(x0))
div_nat(s(x0), s(x1))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
if(true, x0, x1)
if(false, x0, x1)
minus_nat_s(x0, 0)
minus_nat_s(0, s(x0))
minus_nat_s(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
We have to consider all minimal (P,Q,R)-chains.