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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 QDP
↳5 Instantiation (⇔)
↳6 QDP
↳7 DependencyGraphProof (⇔)
↳8 QDP
↳9 QDPOrderProof (⇔)
↳10 QDP
↳11 QDPOrderProof (⇔)
↳12 QDP
↳13 QDPOrderProof (⇔)
↳14 QDP
↳15 DependencyGraphProof (⇔)
↳16 AND
  ↳17 QDP
    ↳18 QDPOrderProof (⇔)
    ↳19 QDP
    ↳20 DependencyGraphProof (⇔)
    ↳21 AND
      ↳22 QDP
        ↳23 QDPOrderProof (⇔)
        ↳24 QDP
        ↳25 Instantiation (⇔)
        ↳26 QDP
        ↳27 DependencyGraphProof (⇔)
        ↳28 TRUE
      ↳29 QDP
        ↳30 QDPOrderProof (⇔)
        ↳31 QDP
        ↳32 PisEmptyProof (⇔)
        ↳33 TRUE
  ↳34 QDP
    ↳35 QDPOrderProof (⇔)
    ↳36 QDP
    ↳37 PisEmptyProof (⇔)
    ↳38 TRUE

(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) Instantiation (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)

(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].


The following pairs can be oriented strictly and are deleted.


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)) = 1A + 0A·x1

POL(n__s(x1)) = -I + 0A·x1

POL(n__div(x1, x2)) = 2A + 0A·x1 + 1A·x2

POL(DIV(x1, x2)) = 2A + 0A·x1 + 1A·x2

POL(activate(x1)) = 0A + 0A·x1

POL(s(x1)) = -I + 0A·x1

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

POL(geq(x1, x2)) = 0A + -I·x1 + 0A·x2

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

POL(n__0) = 0A

POL(true) = 0A

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

POL(GEQ(x1, x2)) = 2A + 0A·x1 + 0A·x2

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

POL(0) = 0A

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

POL(false) = 0A

POL(div(x1, x2)) = 2A + 0A·x1 + 1A·x2

The following usable rules [FROCOS05] were oriented:

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

(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__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)

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].


The following pairs can be oriented strictly and are deleted.


MINUS(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)) = -I + 0A·x1

POL(n__s(x1)) = 0A + 0A·x1

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

POL(DIV(x1, x2)) = -I + 0A·x1 + 1A·x2

POL(activate(x1)) = 0A + 0A·x1

POL(s(x1)) = 0A + 0A·x1

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

POL(geq(x1, x2)) = -I + 0A·x1 + 0A·x2

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

POL(n__0) = 0A

POL(true) = 0A

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

POL(GEQ(x1, x2)) = -I + 0A·x1 + 0A·x2

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

POL(0) = 0A

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

POL(false) = 0A

POL(div(x1, x2)) = 0A + 0A·x1 + 1A·x2

The following usable rules [FROCOS05] were oriented:

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

(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__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)
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.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__div(X1, X2)) → ACTIVATE(X1)
DIV(s(X), n__s(Y)) → GEQ(X, activate(Y))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(DIV(x1, x2)) = 1 + x1 + x2   
POL(GEQ(x1, x2)) = x1 + x2   
POL(IF(x1, x2, x3)) = x2   
POL(MINUS(x1, x2)) = x1   
POL(activate(x1)) = x1   
POL(div(x1, x2)) = 1 + x1 + x2   
POL(false) = 0   
POL(geq(x1, x2)) = 0   
POL(if(x1, x2, x3)) = x2 + x3   
POL(minus(x1, x2)) = x1   
POL(n__0) = 0   
POL(n__div(x1, x2)) = 1 + x1 + x2   
POL(n__minus(x1, x2)) = x1   
POL(n__s(x1)) = x1   
POL(s(x1)) = x1   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

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

(14) 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)
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.

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) Complex Obligation (AND)

(17) 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)

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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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 interpretation [POLO]:

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(DIV(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = x2   
POL(MINUS(x1, x2)) = x1   
POL(activate(x1)) = x1   
POL(div(x1, x2)) = 0   
POL(false) = 0   
POL(geq(x1, x2)) = 0   
POL(if(x1, x2, x3)) = x2 + x3   
POL(minus(x1, x2)) = 1 + x1   
POL(n__0) = 0   
POL(n__div(x1, x2)) = 0   
POL(n__minus(x1, x2)) = 1 + x1   
POL(n__s(x1)) = x1   
POL(s(x1)) = x1   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

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

(19) 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)

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.

(20) DependencyGraphProof (EQUIVALENT transformation)

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

(21) Complex Obligation (AND)

(22) 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.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVATE(n__s(X)) → ACTIVATE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(DIV(x1, x2)) = x1   
POL(IF(x1, x2, x3)) = x2   
POL(activate(x1)) = x1   
POL(div(x1, x2)) = x1   
POL(false) = 0   
POL(geq(x1, x2)) = 0   
POL(if(x1, x2, x3)) = x2 + x3   
POL(minus(x1, x2)) = x1   
POL(n__0) = 0   
POL(n__div(x1, x2)) = x1   
POL(n__minus(x1, x2)) = x1   
POL(n__s(x1)) = 1 + x1   
POL(s(x1)) = 1 + x1   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

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

(24) 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__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.

(25) Instantiation (EQUIVALENT transformation)

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

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

(26) 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)
ACTIVATE(n__div(X1, X2)) → DIV(activate(X1), X2)
IF(true, n__s(n__div(n__minus(y_4, y_6), n__s(y_8))), n__0) → ACTIVATE(n__s(n__div(n__minus(y_4, y_6), n__s(y_8))))

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.

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(28) TRUE

(29) 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.

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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 interpretation [POLO]:

POL(0) = 0   
POL(MINUS(x1, x2)) = x2   
POL(activate(x1)) = x1   
POL(div(x1, x2)) = x1   
POL(false) = 0   
POL(geq(x1, x2)) = x2   
POL(if(x1, x2, x3)) = x2 + x3   
POL(minus(x1, x2)) = x1   
POL(n__0) = 0   
POL(n__div(x1, x2)) = x1   
POL(n__minus(x1, x2)) = x1   
POL(n__s(x1)) = 1 + x1   
POL(s(x1)) = 1 + x1   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

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

(31) 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.

(32) PisEmptyProof (EQUIVALENT transformation)

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

(33) TRUE

(34) 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.

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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 interpretation [POLO]:

POL(0) = 0   
POL(GEQ(x1, x2)) = x2   
POL(activate(x1)) = x1   
POL(div(x1, x2)) = x1   
POL(false) = 0   
POL(geq(x1, x2)) = x2   
POL(if(x1, x2, x3)) = x2 + x3   
POL(minus(x1, x2)) = x1   
POL(n__0) = 0   
POL(n__div(x1, x2)) = x1   
POL(n__minus(x1, x2)) = x1   
POL(n__s(x1)) = 1 + x1   
POL(s(x1)) = 1 + x1   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

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

(36) 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.

(37) PisEmptyProof (EQUIVALENT transformation)

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

(38) TRUE