(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(minus(0, Y)) → MARK(0)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
ACTIVE(minus(s(X), s(Y))) → MINUS(X, Y)
ACTIVE(geq(X, 0)) → MARK(true)
ACTIVE(geq(0, s(Y))) → MARK(false)
ACTIVE(geq(s(X), s(Y))) → MARK(geq(X, Y))
ACTIVE(geq(s(X), s(Y))) → GEQ(X, Y)
ACTIVE(div(0, s(Y))) → MARK(0)
ACTIVE(div(s(X), s(Y))) → MARK(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
ACTIVE(div(s(X), s(Y))) → IF(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
ACTIVE(div(s(X), s(Y))) → GEQ(X, Y)
ACTIVE(div(s(X), s(Y))) → S(div(minus(X, Y), s(Y)))
ACTIVE(div(s(X), s(Y))) → DIV(minus(X, Y), s(Y))
ACTIVE(div(s(X), s(Y))) → MINUS(X, Y)
ACTIVE(if(true, X, Y)) → MARK(X)
ACTIVE(if(false, X, Y)) → MARK(Y)
MARK(minus(X1, X2)) → ACTIVE(minus(X1, X2))
MARK(0) → ACTIVE(0)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(geq(X1, X2)) → ACTIVE(geq(X1, X2))
MARK(true) → ACTIVE(true)
MARK(false) → ACTIVE(false)
MARK(div(X1, X2)) → ACTIVE(div(mark(X1), X2))
MARK(div(X1, X2)) → DIV(mark(X1), X2)
MARK(div(X1, X2)) → MARK(X1)
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))
MARK(if(X1, X2, X3)) → IF(mark(X1), X2, X3)
MARK(if(X1, X2, X3)) → MARK(X1)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MINUS(X1, mark(X2)) → MINUS(X1, X2)
MINUS(active(X1), X2) → MINUS(X1, X2)
MINUS(X1, active(X2)) → MINUS(X1, X2)
S(mark(X)) → S(X)
S(active(X)) → S(X)
GEQ(mark(X1), X2) → GEQ(X1, X2)
GEQ(X1, mark(X2)) → GEQ(X1, X2)
GEQ(active(X1), X2) → GEQ(X1, X2)
GEQ(X1, active(X2)) → GEQ(X1, X2)
DIV(mark(X1), X2) → DIV(X1, X2)
DIV(X1, mark(X2)) → DIV(X1, X2)
DIV(active(X1), X2) → DIV(X1, X2)
DIV(X1, active(X2)) → DIV(X1, X2)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 17 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


IF(X1, mark(X2), X3) → IF(X1, X2, X3)
IF(X1, active(X2), X3) → IF(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3)  =  IF(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
minus(x1, x2)  =  minus
0  =  0
s(x1)  =  s
geq(x1, x2)  =  geq
true  =  true
false  =  false
div(x1, x2)  =  div
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
div > [IF1, mark1, active1, minus] > s > [0, geq, true, false, if2]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


IF(mark(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, X2, mark(X3)) → IF(X1, X2, X3)
IF(active(X1), X2, X3) → IF(X1, X2, X3)
IF(X1, X2, active(X3)) → IF(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3)  =  IF(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
minus(x1, x2)  =  minus
0  =  0
s(x1)  =  s
geq(x1, x2)  =  geq
true  =  true
false  =  false
div(x1, x2)  =  div
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
div > s > if2 > [mark1, active1] > [IF3, minus, 0, geq, true, false]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DIV(X1, mark(X2)) → DIV(X1, X2)
DIV(mark(X1), X2) → DIV(X1, X2)
DIV(active(X1), X2) → DIV(X1, X2)
DIV(X1, active(X2)) → DIV(X1, X2)

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


DIV(X1, mark(X2)) → DIV(X1, X2)
DIV(mark(X1), X2) → DIV(X1, X2)
DIV(active(X1), X2) → DIV(X1, X2)
DIV(X1, active(X2)) → DIV(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DIV(x1, x2)  =  DIV(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
minus(x1, x2)  =  minus
0  =  0
s(x1)  =  s
geq(x1, x2)  =  geq
true  =  true
false  =  false
div(x1, x2)  =  div
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
div > [mark1, active1, s] > minus > 0 > [DIV2, geq, true, false, if2]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(14) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GEQ(X1, mark(X2)) → GEQ(X1, X2)
GEQ(mark(X1), X2) → GEQ(X1, X2)
GEQ(active(X1), X2) → GEQ(X1, X2)
GEQ(X1, active(X2)) → GEQ(X1, X2)

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


GEQ(X1, mark(X2)) → GEQ(X1, X2)
GEQ(mark(X1), X2) → GEQ(X1, X2)
GEQ(active(X1), X2) → GEQ(X1, X2)
GEQ(X1, active(X2)) → GEQ(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
GEQ(x1, x2)  =  GEQ(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
minus(x1, x2)  =  minus
0  =  0
s(x1)  =  s
geq(x1, x2)  =  geq
true  =  true
false  =  false
div(x1, x2)  =  div
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
div > [mark1, active1, s] > minus > 0 > [GEQ2, geq, true, false, if2]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(19) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(21) TRUE

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(active(X)) → S(X)
S(mark(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
minus(x1, x2)  =  minus
0  =  0
s(x1)  =  s
geq(x1, x2)  =  geq
true  =  true
false  =  false
div(x1, x2)  =  div(x2)
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
[minus, s, geq] > div1 > [S1, active1, mark1] > if2 > [0, true, false]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(24) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(25) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(26) TRUE

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MINUS(X1, mark(X2)) → MINUS(X1, X2)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MINUS(active(X1), X2) → MINUS(X1, X2)
MINUS(X1, active(X2)) → MINUS(X1, X2)

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MINUS(X1, mark(X2)) → MINUS(X1, X2)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MINUS(active(X1), X2) → MINUS(X1, X2)
MINUS(X1, active(X2)) → MINUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MINUS(x1, x2)  =  MINUS(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
minus(x1, x2)  =  minus
0  =  0
s(x1)  =  s
geq(x1, x2)  =  geq
true  =  true
false  =  false
div(x1, x2)  =  div
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
div > [mark1, active1, s] > minus > 0 > [MINUS2, geq, true, false, if2]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(29) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(30) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(31) TRUE

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(minus(X1, X2)) → ACTIVE(minus(X1, X2))
ACTIVE(geq(s(X), s(Y))) → MARK(geq(X, Y))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(div(s(X), s(Y))) → MARK(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
MARK(s(X)) → MARK(X)
MARK(geq(X1, X2)) → ACTIVE(geq(X1, X2))
ACTIVE(if(true, X, Y)) → MARK(X)
MARK(div(X1, X2)) → ACTIVE(div(mark(X1), X2))
ACTIVE(if(false, X, Y)) → MARK(Y)
MARK(div(X1, X2)) → MARK(X1)
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))
MARK(if(X1, X2, X3)) → MARK(X1)

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(div(s(X), s(Y))) → MARK(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
MARK(s(X)) → MARK(X)
ACTIVE(if(true, X, Y)) → MARK(X)
ACTIVE(if(false, X, Y)) → MARK(Y)
MARK(div(X1, X2)) → MARK(X1)
MARK(if(X1, X2, X3)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
minus(x1, x2)  =  minus
s(x1)  =  s(x1)
MARK(x1)  =  MARK(x1)
geq(x1, x2)  =  geq
mark(x1)  =  x1
div(x1, x2)  =  div(x1)
if(x1, x2, x3)  =  if(x1, x2, x3)
0  =  0
true  =  true
false  =  false
active(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
div1 > if3 > [ACTIVE1, minus, s1, MARK1, geq, 0, true, false]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(34) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(minus(X1, X2)) → ACTIVE(minus(X1, X2))
ACTIVE(geq(s(X), s(Y))) → MARK(geq(X, Y))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(geq(X1, X2)) → ACTIVE(geq(X1, X2))
MARK(div(X1, X2)) → ACTIVE(div(mark(X1), X2))
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(div(X1, X2)) → ACTIVE(div(mark(X1), X2))
MARK(if(X1, X2, X3)) → ACTIVE(if(mark(X1), X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE
minus(x1, x2)  =  minus
s(x1)  =  s
MARK(x1)  =  x1
geq(x1, x2)  =  geq
mark(x1)  =  mark
div(x1, x2)  =  div
if(x1, x2, x3)  =  if
active(x1)  =  active
0  =  0
true  =  true
false  =  false

Recursive Path Order [RPO].
Precedence:
[s, div] > [ACTIVE, minus, geq, mark, active, 0, true, false]
if > [ACTIVE, minus, geq, mark, active, 0, true, false]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(minus(X1, X2)) → ACTIVE(minus(X1, X2))
ACTIVE(geq(s(X), s(Y))) → MARK(geq(X, Y))
MARK(geq(X1, X2)) → ACTIVE(geq(X1, X2))

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(geq(s(X), s(Y))) → MARK(geq(X, Y))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
minus(x1, x2)  =  minus
s(x1)  =  s(x1)
MARK(x1)  =  MARK(x1)
geq(x1, x2)  =  geq(x2)
active(x1)  =  x1
0  =  0
mark(x1)  =  x1
true  =  true
false  =  false
div(x1, x2)  =  div(x1, x2)
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
[ACTIVE1, MARK1] > [geq1, false] > [minus, 0, true, if2]
div2 > s1 > [geq1, false] > [minus, 0, true, if2]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(minus(X1, X2)) → ACTIVE(minus(X1, X2))
MARK(geq(X1, X2)) → ACTIVE(geq(X1, X2))

The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(minus(X1, X2)) → ACTIVE(minus(X1, X2))
MARK(geq(X1, X2)) → ACTIVE(geq(X1, X2))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
minus(x1, x2)  =  minus(x1)
s(x1)  =  s(x1)
MARK(x1)  =  MARK(x1)
geq(x1, x2)  =  x1
active(x1)  =  x1
0  =  0
mark(x1)  =  x1
true  =  true
false  =  false
div(x1, x2)  =  div(x1, x2)
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
div2 > s1 > if2 > [minus1, MARK1, 0, true, false]


The following usable rules [FROCOS05] were oriented:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

(40) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(minus(X1, X2)) → active(minus(X1, X2))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(geq(X1, X2)) → active(geq(X1, X2))
mark(true) → active(true)
mark(false) → active(false)
mark(div(X1, X2)) → active(div(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
geq(mark(X1), X2) → geq(X1, X2)
geq(X1, mark(X2)) → geq(X1, X2)
geq(active(X1), X2) → geq(X1, X2)
geq(X1, active(X2)) → geq(X1, X2)
div(mark(X1), X2) → div(X1, X2)
div(X1, mark(X2)) → div(X1, X2)
div(active(X1), X2) → div(X1, X2)
div(X1, active(X2)) → div(X1, X2)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(41) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(42) TRUE