YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex49_GM04_FR.ari

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔, 0 ms)
↳2 QDP
↳3 DependencyGraphProof (⇔, 3 ms)
↳4 QDP
↳5 TransformationProof (⇔, 0 ms)
↳6 QDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 QDP
↳9 QDPOrderProof (⇔, 128 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 73 ms)
↳12 QDP
↳13 DependencyGraphProof (⇔, 0 ms)
↳14 AND
    ↳15 QDP
        ↳16 QDPOrderProof (⇔, 8 ms)
        ↳17 QDP
        ↳18 PisEmptyProof (⇔, 0 ms)
        ↳19 YES
    ↳20 QDP
        ↳21 QDPOrderProof (⇔, 0 ms)
        ↳22 QDP
        ↳23 DependencyGraphProof (⇔, 0 ms)
        ↳24 AND
            ↳25 QDP
                ↳26 QDPOrderProof (⇔, 57 ms)
                ↳27 QDP
                ↳28 DependencyGraphProof (⇔, 0 ms)
                ↳29 TRUE
            ↳30 QDP
                ↳31 QDPOrderProof (⇔, 26 ms)
                ↳32 QDP
                ↳33 PisEmptyProof (⇔, 0 ms)
                ↳34 YES

(0) Obligation:

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

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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:

MINUS(n__0, Y) → 01
MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))
MINUS(n__s(X), n__s(Y)) → ACTIVATE(X)
MINUS(n__s(X), n__s(Y)) → ACTIVATE(Y)
GEQ(n__s(X), n__s(Y)) → GEQ(activate(X), activate(Y))
GEQ(n__s(X), n__s(Y)) → ACTIVATE(X)
GEQ(n__s(X), n__s(Y)) → ACTIVATE(Y)
DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
DIV(s(X), n__s(Y)) → GEQ(X, activate(Y))
DIV(s(X), n__s(Y)) → ACTIVATE(Y)
IF(true, X, Y) → ACTIVATE(X)
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__0) → 01
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
ACTIVATE(n__div(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__minus(X1, X2)) → MINUS(X1, X2)

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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 1 SCC with 3 less nodes.

(4) Obligation:

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

MINUS(n__s(X), n__s(Y)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
IF(true, X, Y) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__minus(X1, X2)) → MINUS(X1, X2)
MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))
MINUS(n__s(X), n__s(Y)) → ACTIVATE(Y)
IF(false, X, Y) → ACTIVATE(Y)
DIV(s(X), n__s(Y)) → GEQ(X, activate(Y))
GEQ(n__s(X), n__s(Y)) → GEQ(activate(X), activate(Y))
GEQ(n__s(X), n__s(Y)) → ACTIVATE(X)
GEQ(n__s(X), n__s(Y)) → ACTIVATE(Y)
DIV(s(X), n__s(Y)) → ACTIVATE(Y)

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(5) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IF(false, X, Y) → ACTIVATE(Y) we obtained the following new rules [LPAR04]:

IF(false, n__s(n__div(n__minus(y_4, y_6), n__s(y_8))), n__0) → ACTIVATE(n__0) → IF(false, n__s(n__div(n__minus(y_4, y_6), n__s(y_8))), n__0) → ACTIVATE(n__0)

(6) Obligation:

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

MINUS(n__s(X), n__s(Y)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
IF(true, X, Y) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__minus(X1, X2)) → MINUS(X1, X2)
MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))
MINUS(n__s(X), n__s(Y)) → ACTIVATE(Y)
DIV(s(X), n__s(Y)) → GEQ(X, activate(Y))
GEQ(n__s(X), n__s(Y)) → GEQ(activate(X), activate(Y))
GEQ(n__s(X), n__s(Y)) → ACTIVATE(X)
GEQ(n__s(X), n__s(Y)) → ACTIVATE(Y)
DIV(s(X), n__s(Y)) → ACTIVATE(Y)
IF(false, n__s(n__div(n__minus(y_4, y_6), n__s(y_8))), n__0) → ACTIVATE(n__0)

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(8) Obligation:

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

ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
IF(true, X, Y) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__minus(X1, X2)) → MINUS(X1, X2)
MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))
MINUS(n__s(X), n__s(Y)) → ACTIVATE(X)
MINUS(n__s(X), n__s(Y)) → ACTIVATE(Y)
DIV(s(X), n__s(Y)) → GEQ(X, activate(Y))
GEQ(n__s(X), n__s(Y)) → GEQ(activate(X), activate(Y))
GEQ(n__s(X), n__s(Y)) → ACTIVATE(X)
GEQ(n__s(X), n__s(Y)) → ACTIVATE(Y)
DIV(s(X), n__s(Y)) → ACTIVATE(Y)

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__div(X1, X2)) → ACTIVATE(X1)
DIV(s(X), n__s(Y)) → ACTIVATE(Y)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x1)) = 0A + 1A·x1

POL(n__s(x1)) = -I + 0A·x1

POL(n__div(x1, x2)) = 0A + 4A·x1 + 4A·x2

POL(DIV(x1, x2)) = 1A + 5A·x1 + 5A·x2

POL(activate(x1)) = -I + 0A·x1

POL(s(x1)) = -I + 0A·x1

POL(IF(x1, x2, x3)) = -I + 0A·x1 + 1A·x2 + 1A·x3

POL(geq(x1, x2)) = -I + 0A·x1 + 2A·x2

POL(n__minus(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(n__0) = 0A

POL(true) = 2A

POL(MINUS(x1, x2)) = 0A + 1A·x1 + 1A·x2

POL(GEQ(x1, x2)) = 1A + 1A·x1 + 1A·x2

POL(0) = 0A

POL(div(x1, x2)) = 0A + 4A·x1 + 4A·x2

POL(if(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + 0A·x3

POL(false) = 0A

POL(minus(x1, x2)) = -I + 0A·x1 + 0A·x2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
s(X) → n__s(X)
div(0, n__s(Y)) → 0
div(X1, X2) → n__div(X1, X2)
minus(n__0, Y) → 0
minus(X1, X2) → n__minus(X1, X2)
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
0 → n__0

(10) Obligation:

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

ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
IF(true, X, Y) → ACTIVATE(X)
ACTIVATE(n__minus(X1, X2)) → MINUS(X1, X2)
MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))
MINUS(n__s(X), n__s(Y)) → ACTIVATE(X)
MINUS(n__s(X), n__s(Y)) → ACTIVATE(Y)
DIV(s(X), n__s(Y)) → GEQ(X, activate(Y))
GEQ(n__s(X), n__s(Y)) → GEQ(activate(X), activate(Y))
GEQ(n__s(X), n__s(Y)) → ACTIVATE(X)
GEQ(n__s(X), n__s(Y)) → ACTIVATE(Y)

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


GEQ(n__s(X), n__s(Y)) → ACTIVATE(X)
GEQ(n__s(X), n__s(Y)) → ACTIVATE(Y)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x1)) = 0A + 0A·x1

POL(n__s(x1)) = -I + 0A·x1

POL(n__div(x1, x2)) = 5A + 3A·x1 + 3A·x2

POL(DIV(x1, x2)) = 5A + 3A·x1 + 3A·x2

POL(activate(x1)) = 1A + 0A·x1

POL(s(x1)) = 1A + 0A·x1

POL(IF(x1, x2, x3)) = 1A + -I·x1 + 0A·x2 + -I·x3

POL(geq(x1, x2)) = 1A + -I·x1 + 0A·x2

POL(n__minus(x1, x2)) = 1A + 0A·x1 + 0A·x2

POL(n__0) = 3A

POL(true) = 3A

POL(MINUS(x1, x2)) = 1A + 0A·x1 + 0A·x2

POL(GEQ(x1, x2)) = 5A + 2A·x1 + 1A·x2

POL(0) = 3A

POL(div(x1, x2)) = 5A + 3A·x1 + 3A·x2

POL(if(x1, x2, x3)) = -I + 0A·x1 + 0A·x2 + 2A·x3

POL(false) = 1A

POL(minus(x1, x2)) = 1A + 0A·x1 + 0A·x2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
s(X) → n__s(X)
div(0, n__s(Y)) → 0
div(X1, X2) → n__div(X1, X2)
minus(n__0, Y) → 0
minus(X1, X2) → n__minus(X1, X2)
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
0 → n__0

(12) Obligation:

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

ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
IF(true, X, Y) → ACTIVATE(X)
ACTIVATE(n__minus(X1, X2)) → MINUS(X1, X2)
MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))
MINUS(n__s(X), n__s(Y)) → ACTIVATE(X)
MINUS(n__s(X), n__s(Y)) → ACTIVATE(Y)
DIV(s(X), n__s(Y)) → GEQ(X, activate(Y))
GEQ(n__s(X), n__s(Y)) → GEQ(activate(X), activate(Y))

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(14) Complex Obligation (AND)

(15) Obligation:

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

GEQ(n__s(X), n__s(Y)) → GEQ(activate(X), activate(Y))

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


GEQ(n__s(X), n__s(Y)) → GEQ(activate(X), activate(Y))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( GEQ(x1, x2) ) = 2x2 + 2

POL( activate(x1) ) = x1

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__s(x1) ) = 2x1 + 1

POL( s(x1) ) = 2x1 + 1

POL( n__div(x1, x2) ) = x1

POL( div(x1, x2) ) = x1

POL( if(x1, ..., x3) ) = max{0, x1 + x2 + 2x3 - 2}

POL( geq(x1, x2) ) = 2x1 + 2

POL( n__minus(x1, x2) ) = max{0, -2}

POL( true ) = 2

POL( false ) = 2

POL( minus(x1, x2) ) = 0


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X
s(X) → n__s(X)
div(0, n__s(Y)) → 0
div(X1, X2) → n__div(X1, X2)
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
minus(n__0, Y) → 0
minus(X1, X2) → n__minus(X1, X2)
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
0 → n__0

(17) Obligation:

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

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(18) PisEmptyProof (EQUIVALENT transformation)

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

(19) YES

(20) Obligation:

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

ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
IF(true, X, Y) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__minus(X1, X2)) → MINUS(X1, X2)
MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))
MINUS(n__s(X), n__s(Y)) → ACTIVATE(X)
MINUS(n__s(X), n__s(Y)) → ACTIVATE(Y)

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__minus(X1, X2)) → MINUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( DIV(x1, x2) ) = 2

POL( IF(x1, ..., x3) ) = 2x1 + x2

POL( MINUS(x1, x2) ) = 2x1 + x2 + 2

POL( n__div(x1, x2) ) = 0

POL( n__s(x1) ) = 2x1

POL( n__minus(x1, x2) ) = 2x1 + x2 + 2

POL( activate(x1) ) = 2x1

POL( n__0 ) = 0

POL( 0 ) = 0

POL( s(x1) ) = 2x1

POL( div(x1, x2) ) = 0

POL( if(x1, ..., x3) ) = max{0, 2x1 + 2x2 + 2x3 - 2}

POL( geq(x1, x2) ) = 1

POL( true ) = 1

POL( false ) = 1

POL( minus(x1, x2) ) = 2x1 + x2 + 2

POL( ACTIVATE(x1) ) = x1 + 2


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
s(X) → n__s(X)
div(0, n__s(Y)) → 0
div(X1, X2) → n__div(X1, X2)
minus(n__0, Y) → 0
minus(X1, X2) → n__minus(X1, X2)
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
0 → n__0

(22) Obligation:

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

ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
IF(true, X, Y) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))
MINUS(n__s(X), n__s(Y)) → ACTIVATE(X)
MINUS(n__s(X), n__s(Y)) → ACTIVATE(Y)

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.

(24) Complex Obligation (AND)

(25) Obligation:

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

DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
IF(true, X, Y) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


DIV(s(X), n__s(Y)) → IF(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( DIV(x1, x2) ) = 2x1 + 1

POL( IF(x1, ..., x3) ) = 2x2 + x3 + 2

POL( n__div(x1, x2) ) = 2x1

POL( n__s(x1) ) = x1 + 1

POL( n__minus(x1, x2) ) = 0

POL( activate(x1) ) = 2x1

POL( n__0 ) = 0

POL( 0 ) = 0

POL( s(x1) ) = x1 + 2

POL( div(x1, x2) ) = 2x1

POL( if(x1, ..., x3) ) = x1 + 2x2 + 2x3

POL( geq(x1, x2) ) = 2

POL( true ) = 1

POL( false ) = 0

POL( minus(x1, x2) ) = 0

POL( ACTIVATE(x1) ) = 2x1 + 2


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
s(X) → n__s(X)
div(0, n__s(Y)) → 0
div(X1, X2) → n__div(X1, X2)
minus(n__0, Y) → 0
minus(X1, X2) → n__minus(X1, X2)
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
0 → n__0

(27) Obligation:

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

IF(true, X, Y) → ACTIVATE(X)

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(29) TRUE

(30) Obligation:

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

MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))

The TRS R consists of the following rules:

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


MINUS(n__s(X), n__s(Y)) → MINUS(activate(X), activate(Y))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( MINUS(x1, x2) ) = 2x2 + 2

POL( activate(x1) ) = x1

POL( n__0 ) = 0

POL( 0 ) = 0

POL( n__s(x1) ) = 2x1 + 1

POL( s(x1) ) = 2x1 + 1

POL( n__div(x1, x2) ) = x1

POL( div(x1, x2) ) = x1

POL( if(x1, ..., x3) ) = max{0, x1 + x2 + 2x3 - 2}

POL( geq(x1, x2) ) = 2x1 + 2

POL( n__minus(x1, x2) ) = max{0, -2}

POL( true ) = 2

POL( false ) = 2

POL( minus(x1, x2) ) = 0


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X
s(X) → n__s(X)
div(0, n__s(Y)) → 0
div(X1, X2) → n__div(X1, X2)
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
minus(n__0, Y) → 0
minus(X1, X2) → n__minus(X1, X2)
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
0 → n__0

(32) Obligation:

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

minus(n__0, Y) → 0
minus(n__s(X), n__s(Y)) → minus(activate(X), activate(Y))
geq(X, n__0) → true
geq(n__0, n__s(Y)) → false
geq(n__s(X), n__s(Y)) → geq(activate(X), activate(Y))
div(0, n__s(Y)) → 0
div(s(X), n__s(Y)) → if(geq(X, activate(Y)), n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))), n__0)
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
0 → n__0
s(X) → n__s(X)
div(X1, X2) → n__div(X1, X2)
minus(X1, X2) → n__minus(X1, X2)
activate(n__0) → 0
activate(n__s(X)) → s(activate(X))
activate(n__div(X1, X2)) → div(activate(X1), X2)
activate(n__minus(X1, X2)) → minus(X1, X2)
activate(X) → X

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

(33) PisEmptyProof (EQUIVALENT transformation)

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

(34) YES