Termination Proof using AProVE

Term Rewriting System R:
[Y, X, X1, X2, X3]
a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_MINUS}(s(X), s(Y)) -> A_{_MINUS}(X, Y)
A_{_GEQ}(s(X), s(Y)) -> A_{_GEQ}(X, Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
A_{_DIV}(s(X), s(Y)) -> A_{_GEQ}(X, Y)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_IF}(false, X, Y) -> MARK(Y)
MARK(minus(X1, X2)) -> A_{_MINUS}(X1, X2)
MARK(geq(X1, X2)) -> A_{_GEQ}(X1, X2)
MARK(div(X1, X2)) -> A_{_DIV}(mark(X1), X2)
MARK(div(X1, X2)) -> MARK(X1)
MARK(if(X1, X2, X3)) -> A_{_IF}(mark(X1), X2, X3)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

Dependency Pair:

A_{_MINUS}(s(X), s(Y)) -> A_{_MINUS}(X, Y)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

The following dependency pair can be strictly oriented:

A_{_MINUS}(s(X), s(Y)) -> A_{_MINUS}(X, Y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(A__MINUS(x₁, x₂)) = x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 4

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Nar

Dependency Pair:

A_{_GEQ}(s(X), s(Y)) -> A_{_GEQ}(X, Y)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

The following dependency pair can be strictly oriented:

A_{_GEQ}(s(X), s(Y)) -> A_{_GEQ}(X, Y)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(A__GEQ(x₁, x₂)) = x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Narrowing Transformation

Dependency Pairs:

MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
A_{_IF}(false, X, Y) -> MARK(Y)
MARK(if(X1, X2, X3)) -> A_{_IF}(mark(X1), X2, X3)
MARK(div(X1, X2)) -> MARK(X1)
MARK(div(X1, X2)) -> A_{_DIV}(mark(X1), X2)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(div(X1, X2)) -> A_{_DIV}(mark(X1), X2)

eight new Dependency Pairs are created:

MARK(div(minus(X1'', X2''), X2)) -> A_{_DIV}(a_{_minus}(X1'', X2''), X2)
MARK(div(geq(X1'', X2''), X2)) -> A_{_DIV}(a_{_geq}(X1'', X2''), X2)
MARK(div(div(X1'', X2''), X2)) -> A_{_DIV}(a_{_div}(mark(X1''), X2''), X2)
MARK(div(if(X1'', X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(mark(X1''), X2'', X3'), X2)
MARK(div(0, X2)) -> A_{_DIV}(0, X2)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(div(true, X2)) -> A_{_DIV}(true, X2)
MARK(div(false, X2)) -> A_{_DIV}(false, X2)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Narrowing Transformation

Dependency Pairs:

MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(div(if(X1'', X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(mark(X1''), X2'', X3'), X2)
MARK(div(div(X1'', X2''), X2)) -> A_{_DIV}(a_{_div}(mark(X1''), X2''), X2)
MARK(div(geq(X1'', X2''), X2)) -> A_{_DIV}(a_{_geq}(X1'', X2''), X2)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(minus(X1'', X2''), X2)) -> A_{_DIV}(a_{_minus}(X1'', X2''), X2)
MARK(if(X1, X2, X3)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
MARK(if(X1, X2, X3)) -> A_{_IF}(mark(X1), X2, X3)
MARK(div(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(if(X1, X2, X3)) -> A_{_IF}(mark(X1), X2, X3)

eight new Dependency Pairs are created:

MARK(if(minus(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_minus}(X1'', X2''), X2, X3)
MARK(if(geq(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_geq}(X1'', X2''), X2, X3)
MARK(if(div(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_div}(mark(X1''), X2''), X2, X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)
MARK(if(0, X2, X3)) -> A_{_IF}(0, X2, X3)
MARK(if(s(X'), X2, X3)) -> A_{_IF}(s(mark(X')), X2, X3)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)
MARK(if(div(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_div}(mark(X1''), X2''), X2, X3)
MARK(if(geq(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_geq}(X1'', X2''), X2, X3)
A_{_IF}(false, X, Y) -> MARK(Y)
MARK(if(minus(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_minus}(X1'', X2''), X2, X3)
MARK(div(if(X1'', X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(mark(X1''), X2'', X3'), X2)
MARK(div(div(X1'', X2''), X2)) -> A_{_DIV}(a_{_div}(mark(X1''), X2''), X2)
MARK(div(geq(X1'', X2''), X2)) -> A_{_DIV}(a_{_geq}(X1'', X2''), X2)
MARK(div(minus(X1'', X2''), X2)) -> A_{_DIV}(a_{_minus}(X1'', X2''), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(div(minus(X1'', X2''), X2)) -> A_{_DIV}(a_{_minus}(X1'', X2''), X2)

three new Dependency Pairs are created:

MARK(div(minus(0, X2'''), X2)) -> A_{_DIV}(0, X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(div(minus(X1''', X2'''), X2)) -> A_{_DIV}(minus(X1''', X2'''), X2)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)
MARK(if(div(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_div}(mark(X1''), X2''), X2, X3)
MARK(if(geq(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_geq}(X1'', X2''), X2, X3)
MARK(if(minus(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_minus}(X1'', X2''), X2, X3)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(div(if(X1'', X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(mark(X1''), X2'', X3'), X2)
MARK(div(div(X1'', X2''), X2)) -> A_{_DIV}(a_{_div}(mark(X1''), X2''), X2)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(geq(X1'', X2''), X2)) -> A_{_DIV}(a_{_geq}(X1'', X2''), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(false, X, Y) -> MARK(Y)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(div(geq(X1'', X2''), X2)) -> A_{_DIV}(a_{_geq}(X1'', X2''), X2)

four new Dependency Pairs are created:

MARK(div(geq(X1''', 0), X2)) -> A_{_DIV}(true, X2)
MARK(div(geq(0, s(Y')), X2)) -> A_{_DIV}(false, X2)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)
MARK(div(geq(X1''', X2'''), X2)) -> A_{_DIV}(geq(X1''', X2'''), X2)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)
MARK(if(div(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_div}(mark(X1''), X2''), X2, X3)
MARK(if(geq(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_geq}(X1'', X2''), X2, X3)
A_{_IF}(false, X, Y) -> MARK(Y)
MARK(if(minus(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_minus}(X1'', X2''), X2, X3)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(div(if(X1'', X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(mark(X1''), X2'', X3'), X2)
MARK(div(div(X1'', X2''), X2)) -> A_{_DIV}(a_{_div}(mark(X1''), X2''), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(div(div(X1'', X2''), X2)) -> A_{_DIV}(a_{_div}(mark(X1''), X2''), X2)

nine new Dependency Pairs are created:

MARK(div(div(X1''', X2'''), X2)) -> A_{_DIV}(div(mark(X1'''), X2'''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(true, X2''), X2)) -> A_{_DIV}(a_{_div}(true, X2''), X2)
MARK(div(div(false, X2''), X2)) -> A_{_DIV}(a_{_div}(false, X2''), X2)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 10

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(div(div(false, X2''), X2)) -> A_{_DIV}(a_{_div}(false, X2''), X2)
MARK(div(div(true, X2''), X2)) -> A_{_DIV}(a_{_div}(true, X2''), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)
MARK(if(div(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_div}(mark(X1''), X2''), X2, X3)
MARK(if(geq(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_geq}(X1'', X2''), X2, X3)
A_{_IF}(false, X, Y) -> MARK(Y)
MARK(if(minus(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_minus}(X1'', X2''), X2, X3)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(div(if(X1'', X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(mark(X1''), X2'', X3'), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(div(if(X1'', X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(mark(X1''), X2'', X3'), X2)

nine new Dependency Pairs are created:

MARK(div(if(X1''', X2''', X3''), X2)) -> A_{_DIV}(if(mark(X1'''), X2''', X3''), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(if(geq(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(true, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(true, X2'', X3'), X2)
MARK(div(if(false, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(false, X2'', X3'), X2)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(div(if(false, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(false, X2'', X3'), X2)
MARK(div(if(true, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(true, X2'', X3'), X2)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(geq(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3'), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(true, X2''), X2)) -> A_{_DIV}(a_{_div}(true, X2''), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)
MARK(if(div(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_div}(mark(X1''), X2''), X2, X3)
MARK(if(geq(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_geq}(X1'', X2''), X2, X3)
A_{_IF}(false, X, Y) -> MARK(Y)
MARK(if(minus(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_minus}(X1'', X2''), X2, X3)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(div(false, X2''), X2)) -> A_{_DIV}(a_{_div}(false, X2''), X2)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(if(minus(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_minus}(X1'', X2''), X2, X3)

three new Dependency Pairs are created:

MARK(if(minus(0, X2'''), X2, X3)) -> A_{_IF}(0, X2, X3)
MARK(if(minus(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_minus}(X', Y'), X2, X3)
MARK(if(minus(X1''', X2'''), X2, X3)) -> A_{_IF}(minus(X1''', X2'''), X2, X3)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(if(minus(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_minus}(X', Y'), X2, X3)
MARK(div(if(true, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(true, X2'', X3'), X2)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(geq(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3'), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(false, X2''), X2)) -> A_{_DIV}(a_{_div}(false, X2''), X2)
MARK(div(div(true, X2''), X2)) -> A_{_DIV}(a_{_div}(true, X2''), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)
MARK(if(div(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_div}(mark(X1''), X2''), X2, X3)
A_{_IF}(false, X, Y) -> MARK(Y)
MARK(if(geq(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_geq}(X1'', X2''), X2, X3)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(if(false, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(false, X2'', X3'), X2)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(if(geq(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_geq}(X1'', X2''), X2, X3)

four new Dependency Pairs are created:

MARK(if(geq(X1''', 0), X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(geq(0, s(Y')), X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(geq(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_geq}(X', Y'), X2, X3)
MARK(if(geq(X1''', X2'''), X2, X3)) -> A_{_IF}(geq(X1''', X2'''), X2, X3)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(if(geq(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_geq}(X', Y'), X2, X3)
MARK(if(geq(0, s(Y')), X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(geq(X1''', 0), X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(div(if(false, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(false, X2'', X3'), X2)
MARK(div(if(true, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(true, X2'', X3'), X2)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(geq(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3'), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(false, X2''), X2)) -> A_{_DIV}(a_{_div}(false, X2''), X2)
MARK(div(div(true, X2''), X2)) -> A_{_DIV}(a_{_div}(true, X2''), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)
MARK(if(div(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_div}(mark(X1''), X2''), X2, X3)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
MARK(if(minus(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_minus}(X', Y'), X2, X3)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(if(div(X1'', X2''), X2, X3)) -> A_{_IF}(a_{_div}(mark(X1''), X2''), X2, X3)

nine new Dependency Pairs are created:

MARK(if(div(X1''', X2'''), X2, X3)) -> A_{_IF}(div(mark(X1'''), X2'''), X2, X3)
MARK(if(div(minus(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2, X3)
MARK(if(div(geq(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2, X3)
MARK(if(div(div(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2, X3)
MARK(if(div(if(X1', X2''', X3''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_if}(mark(X1'), X2''', X3''), X2''), X2, X3)
MARK(if(div(0, X2''), X2, X3)) -> A_{_IF}(a_{_div}(0, X2''), X2, X3)
MARK(if(div(s(X'), X2''), X2, X3)) -> A_{_IF}(a_{_div}(s(mark(X')), X2''), X2, X3)
MARK(if(div(true, X2''), X2, X3)) -> A_{_IF}(a_{_div}(true, X2''), X2, X3)
MARK(if(div(false, X2''), X2, X3)) -> A_{_IF}(a_{_div}(false, X2''), X2, X3)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 14

                 ↳Narrowing Transformation

Dependency Pairs:

MARK(if(div(false, X2''), X2, X3)) -> A_{_IF}(a_{_div}(false, X2''), X2, X3)
MARK(if(div(true, X2''), X2, X3)) -> A_{_IF}(a_{_div}(true, X2''), X2, X3)
MARK(if(div(s(X'), X2''), X2, X3)) -> A_{_IF}(a_{_div}(s(mark(X')), X2''), X2, X3)
MARK(if(div(0, X2''), X2, X3)) -> A_{_IF}(a_{_div}(0, X2''), X2, X3)
MARK(if(div(if(X1', X2''', X3''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_if}(mark(X1'), X2''', X3''), X2''), X2, X3)
MARK(if(div(div(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2, X3)
MARK(if(div(geq(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2, X3)
MARK(if(div(minus(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2, X3)
MARK(if(geq(0, s(Y')), X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(geq(X1''', 0), X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(minus(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_minus}(X', Y'), X2, X3)
MARK(div(if(false, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(false, X2'', X3'), X2)
MARK(div(if(true, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(true, X2'', X3'), X2)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(geq(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3'), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(false, X2''), X2)) -> A_{_DIV}(a_{_div}(false, X2''), X2)
MARK(div(div(true, X2''), X2)) -> A_{_DIV}(a_{_div}(true, X2''), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
MARK(if(geq(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_geq}(X', Y'), X2, X3)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(if(if(X1'', X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(mark(X1''), X2'', X3''), X2, X3)

nine new Dependency Pairs are created:

MARK(if(if(X1''', X2''', X3'''), X2, X3)) -> A_{_IF}(if(mark(X1'''), X2''', X3'''), X2, X3)
MARK(if(if(minus(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(if(geq(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(if(div(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3''), X2, X3)
MARK(if(if(if(X1', X2''', X3'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_if}(mark(X1'), X2''', X3'''), X2'', X3''), X2, X3)
MARK(if(if(0, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(0, X2'', X3''), X2, X3)
MARK(if(if(s(X'), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(s(mark(X')), X2'', X3''), X2, X3)
MARK(if(if(true, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(true, X2'', X3''), X2, X3)
MARK(if(if(false, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(false, X2'', X3''), X2, X3)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 15

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(if(if(false, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(false, X2'', X3''), X2, X3)
MARK(if(if(true, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(true, X2'', X3''), X2, X3)
MARK(if(if(s(X'), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(s(mark(X')), X2'', X3''), X2, X3)
MARK(if(if(0, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(0, X2'', X3''), X2, X3)
MARK(if(if(if(X1', X2''', X3'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_if}(mark(X1'), X2''', X3'''), X2'', X3''), X2, X3)
MARK(if(if(div(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3''), X2, X3)
MARK(if(if(geq(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(if(minus(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(div(true, X2''), X2, X3)) -> A_{_IF}(a_{_div}(true, X2''), X2, X3)
MARK(if(div(s(X'), X2''), X2, X3)) -> A_{_IF}(a_{_div}(s(mark(X')), X2''), X2, X3)
MARK(if(div(0, X2''), X2, X3)) -> A_{_IF}(a_{_div}(0, X2''), X2, X3)
MARK(if(div(if(X1', X2''', X3''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_if}(mark(X1'), X2''', X3''), X2''), X2, X3)
MARK(if(div(div(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2, X3)
MARK(if(div(geq(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2, X3)
MARK(if(div(minus(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2, X3)
MARK(if(geq(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_geq}(X', Y'), X2, X3)
MARK(if(geq(0, s(Y')), X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(geq(X1''', 0), X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(minus(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_minus}(X', Y'), X2, X3)
MARK(div(if(false, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(false, X2'', X3'), X2)
MARK(div(if(true, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(true, X2'', X3'), X2)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(geq(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3'), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(false, X2''), X2)) -> A_{_DIV}(a_{_div}(false, X2''), X2)
MARK(div(div(true, X2''), X2)) -> A_{_DIV}(a_{_div}(true, X2''), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
MARK(if(div(false, X2''), X2, X3)) -> A_{_IF}(a_{_div}(false, X2''), X2, X3)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

The following dependency pairs can be strictly oriented:

MARK(if(if(true, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(true, X2'', X3''), X2, X3)
MARK(if(div(true, X2''), X2, X3)) -> A_{_IF}(a_{_div}(true, X2''), X2, X3)
MARK(div(if(true, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(true, X2'', X3'), X2)
MARK(div(div(true, X2''), X2)) -> A_{_DIV}(a_{_div}(true, X2''), X2)
MARK(if(true, X2, X3)) -> A_{_IF}(true, X2, X3)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MARK(x₁)) = x₁

POL(geq(x₁, x₂)) = 0

POL(false) = 0

POL(minus(x₁, x₂)) = 0

POL(true) = 1

POL(mark(x₁)) = 0

POL(a__div(x₁, x₂)) = 0

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(0) = 0

POL(a__if(x₁, x₂, x₃)) = 0

POL(a__minus(x₁, x₂)) = 0

POL(s(x₁)) = x₁

POL(a__geq(x₁, x₂)) = 0

POL(div(x₁, x₂)) = x₁

POL(A__DIV(x₁, x₂)) = 0

POL(A__IF(x₁, x₂, x₃)) = x₂ + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 16

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(if(if(false, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(false, X2'', X3''), X2, X3)
MARK(if(if(s(X'), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(s(mark(X')), X2'', X3''), X2, X3)
MARK(if(if(0, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(0, X2'', X3''), X2, X3)
MARK(if(if(if(X1', X2''', X3'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_if}(mark(X1'), X2''', X3'''), X2'', X3''), X2, X3)
MARK(if(if(div(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3''), X2, X3)
MARK(if(if(geq(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(if(minus(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(div(s(X'), X2''), X2, X3)) -> A_{_IF}(a_{_div}(s(mark(X')), X2''), X2, X3)
MARK(if(div(0, X2''), X2, X3)) -> A_{_IF}(a_{_div}(0, X2''), X2, X3)
MARK(if(div(if(X1', X2''', X3''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_if}(mark(X1'), X2''', X3''), X2''), X2, X3)
MARK(if(div(div(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2, X3)
MARK(if(div(geq(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2, X3)
MARK(if(div(minus(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2, X3)
MARK(if(geq(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_geq}(X', Y'), X2, X3)
MARK(if(geq(0, s(Y')), X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(geq(X1''', 0), X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(minus(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_minus}(X', Y'), X2, X3)
MARK(div(if(false, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(false, X2'', X3'), X2)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(geq(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3'), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(false, X2''), X2)) -> A_{_DIV}(a_{_div}(false, X2''), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)
MARK(if(div(false, X2''), X2, X3)) -> A_{_IF}(a_{_div}(false, X2''), X2, X3)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

The following dependency pairs can be strictly oriented:

MARK(if(if(false, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(false, X2'', X3''), X2, X3)
MARK(div(if(false, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(false, X2'', X3'), X2)
MARK(div(div(false, X2''), X2)) -> A_{_DIV}(a_{_div}(false, X2''), X2)
MARK(if(false, X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(div(false, X2''), X2, X3)) -> A_{_IF}(a_{_div}(false, X2''), X2, X3)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MARK(x₁)) = x₁

POL(geq(x₁, x₂)) = 0

POL(false) = 1

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = 0

POL(a__div(x₁, x₂)) = 0

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(0) = 0

POL(a__if(x₁, x₂, x₃)) = 0

POL(a__minus(x₁, x₂)) = 0

POL(s(x₁)) = x₁

POL(a__geq(x₁, x₂)) = 0

POL(div(x₁, x₂)) = x₁

POL(A__DIV(x₁, x₂)) = 0

POL(A__IF(x₁, x₂, x₃)) = x₂ + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 17

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(if(if(s(X'), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(s(mark(X')), X2'', X3''), X2, X3)
MARK(if(if(0, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(0, X2'', X3''), X2, X3)
MARK(if(if(if(X1', X2''', X3'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_if}(mark(X1'), X2''', X3'''), X2'', X3''), X2, X3)
MARK(if(if(div(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3''), X2, X3)
MARK(if(if(geq(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(if(minus(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(div(s(X'), X2''), X2, X3)) -> A_{_IF}(a_{_div}(s(mark(X')), X2''), X2, X3)
MARK(if(div(0, X2''), X2, X3)) -> A_{_IF}(a_{_div}(0, X2''), X2, X3)
MARK(if(div(if(X1', X2''', X3''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_if}(mark(X1'), X2''', X3''), X2''), X2, X3)
MARK(if(div(div(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2, X3)
MARK(if(div(geq(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2, X3)
MARK(if(div(minus(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2, X3)
MARK(if(geq(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_geq}(X', Y'), X2, X3)
MARK(if(geq(0, s(Y')), X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(geq(X1''', 0), X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(if(minus(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_minus}(X', Y'), X2, X3)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(geq(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3'), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

The following dependency pairs can be strictly oriented:

MARK(if(if(geq(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(div(geq(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2, X3)
MARK(if(geq(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_geq}(X', Y'), X2, X3)
MARK(if(geq(0, s(Y')), X2, X3)) -> A_{_IF}(false, X2, X3)
MARK(if(geq(X1''', 0), X2, X3)) -> A_{_IF}(true, X2, X3)
MARK(div(if(geq(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_geq}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(geq(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_geq}(X1', X2'''), X2''), X2)
MARK(div(geq(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_geq}(X', Y'), X2)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MARK(x₁)) = x₁

POL(geq(x₁, x₂)) = 1

POL(false) = 0

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = 0

POL(a__div(x₁, x₂)) = 0

POL(if(x₁, x₂, x₃)) = x₁ + x₂ + x₃

POL(0) = 0

POL(a__if(x₁, x₂, x₃)) = 0

POL(a__minus(x₁, x₂)) = 0

POL(s(x₁)) = x₁

POL(a__geq(x₁, x₂)) = 0

POL(div(x₁, x₂)) = x₁

POL(A__DIV(x₁, x₂)) = 0

POL(A__IF(x₁, x₂, x₃)) = x₂ + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 18

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(if(if(s(X'), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(s(mark(X')), X2'', X3''), X2, X3)
MARK(if(if(0, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(0, X2'', X3''), X2, X3)
MARK(if(if(if(X1', X2''', X3'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_if}(mark(X1'), X2''', X3'''), X2'', X3''), X2, X3)
MARK(if(if(div(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3''), X2, X3)
MARK(if(if(minus(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(div(s(X'), X2''), X2, X3)) -> A_{_IF}(a_{_div}(s(mark(X')), X2''), X2, X3)
MARK(if(div(0, X2''), X2, X3)) -> A_{_IF}(a_{_div}(0, X2''), X2, X3)
MARK(if(div(if(X1', X2''', X3''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_if}(mark(X1'), X2''', X3''), X2''), X2, X3)
MARK(if(div(div(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2, X3)
MARK(if(div(minus(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2, X3)
MARK(if(minus(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_minus}(X', Y'), X2, X3)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(if(X1, X2, X3)) -> MARK(X1)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

The following dependency pairs can be strictly oriented:

MARK(if(if(s(X'), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(s(mark(X')), X2'', X3''), X2, X3)
MARK(if(if(0, X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(0, X2'', X3''), X2, X3)
MARK(if(if(if(X1', X2''', X3'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_if}(mark(X1'), X2''', X3'''), X2'', X3''), X2, X3)
MARK(if(if(div(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3''), X2, X3)
MARK(if(if(minus(X1', X2'''), X2'', X3''), X2, X3)) -> A_{_IF}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3''), X2, X3)
MARK(if(div(s(X'), X2''), X2, X3)) -> A_{_IF}(a_{_div}(s(mark(X')), X2''), X2, X3)
MARK(if(div(0, X2''), X2, X3)) -> A_{_IF}(a_{_div}(0, X2''), X2, X3)
MARK(if(div(if(X1', X2''', X3''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_if}(mark(X1'), X2''', X3''), X2''), X2, X3)
MARK(if(div(div(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2, X3)
MARK(if(div(minus(X1', X2'''), X2''), X2, X3)) -> A_{_IF}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2, X3)
MARK(if(minus(s(X'), s(Y')), X2, X3)) -> A_{_IF}(a_{_minus}(X', Y'), X2, X3)
MARK(div(if(s(X'), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(s(mark(X')), X2'', X3'), X2)
MARK(div(if(0, X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(0, X2'', X3'), X2)
MARK(div(if(if(X1', X2''', X3''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_if}(mark(X1'), X2''', X3''), X2'', X3'), X2)
MARK(div(if(div(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_div}(mark(X1'), X2'''), X2'', X3'), X2)
MARK(div(if(minus(X1', X2'''), X2'', X3'), X2)) -> A_{_DIV}(a_{_if}(a_{_minus}(X1', X2'''), X2'', X3'), X2)
MARK(div(div(if(X1', X2''', X3'), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_if}(mark(X1'), X2''', X3'), X2''), X2)
MARK(if(X1, X2, X3)) -> MARK(X1)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MARK(x₁)) = x₁

POL(geq(x₁, x₂)) = 0

POL(false) = 0

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = 0

POL(a__div(x₁, x₂)) = 0

POL(if(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃

POL(0) = 0

POL(a__if(x₁, x₂, x₃)) = 0

POL(a__minus(x₁, x₂)) = 0

POL(s(x₁)) = x₁

POL(a__geq(x₁, x₂)) = 0

POL(div(x₁, x₂)) = x₁

POL(A__DIV(x₁, x₂)) = 0

POL(A__IF(x₁, x₂, x₃)) = x₂ + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 19

                 ↳Instantiation Transformation

Dependency Pairs:

MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(true, X, Y) -> MARK(X)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

A_{_IF}(true, X, Y) -> MARK(X)

one new Dependency Pair is created:

A_{_IF}(true, s(div(minus(X'', Y'0), s(Y'''))), 0) -> MARK(s(div(minus(X'', Y'0), s(Y'''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 20

                 ↳Instantiation Transformation

Dependency Pairs:

A_{_IF}(true, s(div(minus(X'', Y'0), s(Y'''))), 0) -> MARK(s(div(minus(X'', Y'0), s(Y'''))))
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
MARK(div(X1, X2)) -> MARK(X1)
A_{_IF}(false, X, Y) -> MARK(Y)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

A_{_IF}(false, X, Y) -> MARK(Y)

one new Dependency Pair is created:

A_{_IF}(false, s(div(minus(X'', Y'0), s(Y'''))), 0) -> MARK(0)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 21

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(div(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
A_{_IF}(true, s(div(minus(X'', Y'0), s(Y'''))), 0) -> MARK(s(div(minus(X'', Y'0), s(Y'''))))

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

The following dependency pair can be strictly oriented:

MARK(div(div(0, X2''), X2)) -> A_{_DIV}(a_{_div}(0, X2''), X2)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MARK(x₁)) = x₁

POL(geq(x₁, x₂)) = 0

POL(false) = 0

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = 0

POL(a__div(x₁, x₂)) = 0

POL(if(x₁, x₂, x₃)) = 0

POL(0) = 1

POL(a__if(x₁, x₂, x₃)) = 0

POL(a__minus(x₁, x₂)) = 0

POL(s(x₁)) = x₁

POL(a__geq(x₁, x₂)) = 0

POL(div(x₁, x₂)) = x₁

POL(A__DIV(x₁, x₂)) = 0

POL(A__IF(x₁, x₂, x₃)) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 22

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(div(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)
A_{_IF}(true, s(div(minus(X'', Y'0), s(Y'''))), 0) -> MARK(s(div(minus(X'', Y'0), s(Y'''))))

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

The following dependency pairs can be strictly oriented:

MARK(div(div(s(X'), X2''), X2)) -> A_{_DIV}(a_{_div}(s(mark(X')), X2''), X2)
MARK(div(div(div(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_div}(mark(X1'), X2'''), X2''), X2)
MARK(div(div(minus(X1', X2'''), X2''), X2)) -> A_{_DIV}(a_{_div}(a_{_minus}(X1', X2'''), X2''), X2)
MARK(div(X1, X2)) -> MARK(X1)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MARK(x₁)) = x₁

POL(geq(x₁, x₂)) = 0

POL(false) = 0

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = 0

POL(a__div(x₁, x₂)) = 0

POL(if(x₁, x₂, x₃)) = 0

POL(0) = 1

POL(a__if(x₁, x₂, x₃)) = 0

POL(a__minus(x₁, x₂)) = 0

POL(s(x₁)) = x₁

POL(a__geq(x₁, x₂)) = 0

POL(div(x₁, x₂)) = 1 + x₁

POL(A__DIV(x₁, x₂)) = 1

POL(A__IF(x₁, x₂, x₃)) = x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 23

                 ↳Polynomial Ordering

Dependency Pairs:

MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
MARK(s(X)) -> MARK(X)
A_{_IF}(true, s(div(minus(X'', Y'0), s(Y'''))), 0) -> MARK(s(div(minus(X'', Y'0), s(Y'''))))

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

The following dependency pair can be strictly oriented:

MARK(s(X)) -> MARK(X)

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MARK(x₁)) = x₁

POL(geq(x₁, x₂)) = 0

POL(false) = 0

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = x₁

POL(a__div(x₁, x₂)) = x₁

POL(if(x₁, x₂, x₃)) = x₂ + x₃

POL(0) = 0

POL(a__if(x₁, x₂, x₃)) = x₂ + x₃

POL(a__minus(x₁, x₂)) = 0

POL(s(x₁)) = 1 + x₁

POL(a__geq(x₁, x₂)) = 0

POL(div(x₁, x₂)) = x₁

POL(A__DIV(x₁, x₂)) = x₁

POL(A__IF(x₁, x₂, x₃)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 24

                 ↳Dependency Graph

Dependency Pairs:

MARK(div(minus(s(X'), s(Y')), X2)) -> A_{_DIV}(a_{_minus}(X', Y'), X2)
A_{_DIV}(s(X), s(Y)) -> A_{_IF}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
MARK(div(s(X'), X2)) -> A_{_DIV}(s(mark(X')), X2)
A_{_IF}(true, s(div(minus(X'', Y'0), s(Y'''))), 0) -> MARK(s(div(minus(X'', Y'0), s(Y'''))))

Rules:

a_{_minus}(0, Y) -> 0
a_{_minus}(s(X), s(Y)) -> a_{_minus}(X, Y)
a_{_minus}(X1, X2) -> minus(X1, X2)
a_{_geq}(X, 0) -> true
a_{_geq}(0, s(Y)) -> false
a_{_geq}(s(X), s(Y)) -> a_{_geq}(X, Y)
a_{_geq}(X1, X2) -> geq(X1, X2)
a_{_div}(0, s(Y)) -> 0
a_{_div}(s(X), s(Y)) -> a_{_if}(a_{_geq}(X, Y), s(div(minus(X, Y), s(Y))), 0)
a_{_div}(X1, X2) -> div(X1, X2)
a_{_if}(true, X, Y) -> mark(X)
a_{_if}(false, X, Y) -> mark(Y)
a_{_if}(X1, X2, X3) -> if(X1, X2, X3)
mark(minus(X1, X2)) -> a_{_minus}(X1, X2)
mark(geq(X1, X2)) -> a_{_geq}(X1, X2)
mark(div(X1, X2)) -> a_{_div}(mark(X1), X2)
mark(if(X1, X2, X3)) -> a_{_if}(mark(X1), X2, X3)
mark(0) -> 0
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:12 minutes

POL(A__MINUS(x₁, x₂))	= x₁
POL(s(x₁))	= 1 + x₁

POL(MARK(x₁))	= x₁
POL(geq(x₁, x₂))	= 0
POL(false)	= 0
POL(minus(x₁, x₂))	= 0
POL(true)	= 1
POL(mark(x₁))	= 0
POL(a__div(x₁, x₂))	= 0
POL(if(x₁, x₂, x₃))	= x₁ + x₂ + x₃
POL(0)	= 0
POL(a__if(x₁, x₂, x₃))	= 0
POL(a__minus(x₁, x₂))	= 0
POL(s(x₁))	= x₁
POL(a__geq(x₁, x₂))	= 0
POL(div(x₁, x₂))	= x₁
POL(A__DIV(x₁, x₂))	= 0
POL(A__IF(x₁, x₂, x₃))	= x₂ + x₃