Termination Proof using AProVE

Term Rewriting System R:
[y, x, z]
minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

MINUS_ACTIVE(s(x), s(y)) -> MINUS_ACTIVE(x, y)
MARK(s(x)) -> MARK(x)
MARK(minus(x, y)) -> MINUS_ACTIVE(x, y)
MARK(ge(x, y)) -> GE_ACTIVE(x, y)
MARK(div(x, y)) -> DIV_ACTIVE(mark(x), y)
MARK(div(x, y)) -> MARK(x)
MARK(if(x, y, z)) -> IF_ACTIVE(mark(x), y, z)
MARK(if(x, y, z)) -> MARK(x)
GE_ACTIVE(s(x), s(y)) -> GE_ACTIVE(x, y)
DIV_ACTIVE(s(x), s(y)) -> IF_ACTIVE(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
DIV_ACTIVE(s(x), s(y)) -> GE_ACTIVE(x, y)
IF_ACTIVE(true, x, y) -> MARK(x)
IF_ACTIVE(false, x, y) -> MARK(y)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

Dependency Pair:

MINUS_ACTIVE(s(x), s(y)) -> MINUS_ACTIVE(x, y)

Rules:

minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Strategy:

innermost

The following dependency pair can be strictly oriented:

MINUS_ACTIVE(s(x), s(y)) -> MINUS_ACTIVE(x, y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(s(x₁)) = 1 + x₁

POL(MINUS_ACTIVE(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 4

             ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

Dependency Pair:

Rules:

minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

Dependency Pair:

GE_ACTIVE(s(x), s(y)) -> GE_ACTIVE(x, y)

Rules:

minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Strategy:

innermost

The following dependency pair can be strictly oriented:

GE_ACTIVE(s(x), s(y)) -> GE_ACTIVE(x, y)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(GE_ACTIVE(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

         ↳Polo

Dependency Pair:

Rules:

minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polynomial Ordering

Dependency Pairs:

MARK(if(x, y, z)) -> MARK(x)
IF_ACTIVE(false, x, y) -> MARK(y)
MARK(if(x, y, z)) -> IF_ACTIVE(mark(x), y, z)
MARK(div(x, y)) -> MARK(x)
IF_ACTIVE(true, x, y) -> MARK(x)
DIV_ACTIVE(s(x), s(y)) -> IF_ACTIVE(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
MARK(div(x, y)) -> DIV_ACTIVE(mark(x), y)
MARK(s(x)) -> MARK(x)

Rules:

minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(if(x, y, z)) -> MARK(x)
MARK(if(x, y, z)) -> IF_ACTIVE(mark(x), y, z)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ge_active(x₁, x₂)) = 0

POL(MARK(x₁)) = x₁

POL(div_active(x₁, x₂)) = 0

POL(false) = 0

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = 0

POL(minus_active(x₁, x₂)) = 0

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

POL(0) = 0

POL(s(x₁)) = x₁

POL(ge(x₁, x₂)) = 0

POL(DIV_ACTIVE(x₁, x₂)) = 0

POL(div(x₁, x₂)) = x₁

POL(if_active(x₁, x₂, x₃)) = 0

POL(IF_ACTIVE(x₁, x₂, x₃)) = x₂ + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 6

             ↳Polynomial Ordering

Dependency Pairs:

IF_ACTIVE(false, x, y) -> MARK(y)
MARK(div(x, y)) -> MARK(x)
IF_ACTIVE(true, x, y) -> MARK(x)
DIV_ACTIVE(s(x), s(y)) -> IF_ACTIVE(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
MARK(div(x, y)) -> DIV_ACTIVE(mark(x), y)
MARK(s(x)) -> MARK(x)

Rules:

minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(div(x, y)) -> MARK(x)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ge_active(x₁, x₂)) = 0

POL(MARK(x₁)) = x₁

POL(false) = 0

POL(div_active(x₁, x₂)) = 0

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = 0

POL(minus_active(x₁, x₂)) = 0

POL(if(x₁, x₂, x₃)) = 0

POL(0) = 0

POL(s(x₁)) = x₁

POL(ge(x₁, x₂)) = 0

POL(DIV_ACTIVE(x₁, x₂)) = 1

POL(div(x₁, x₂)) = 1 + x₁

POL(if_active(x₁, x₂, x₃)) = 0

POL(IF_ACTIVE(x₁, x₂, x₃)) = x₂ + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 6

             ↳Polo

...

               →DP Problem 7

                 ↳Polynomial Ordering

Dependency Pairs:

IF_ACTIVE(false, x, y) -> MARK(y)
IF_ACTIVE(true, x, y) -> MARK(x)
DIV_ACTIVE(s(x), s(y)) -> IF_ACTIVE(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
MARK(div(x, y)) -> DIV_ACTIVE(mark(x), y)
MARK(s(x)) -> MARK(x)

Rules:

minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(s(x)) -> MARK(x)

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

ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)
minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ge_active(x₁, x₂)) = 0

POL(MARK(x₁)) = x₁

POL(false) = 0

POL(div_active(x₁, x₂)) = x₁

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = x₁

POL(minus_active(x₁, x₂)) = 0

POL(if(x₁, x₂, x₃)) = x₂ + x₃

POL(0) = 0

POL(s(x₁)) = 1 + x₁

POL(ge(x₁, x₂)) = 0

POL(DIV_ACTIVE(x₁, x₂)) = x₁

POL(div(x₁, x₂)) = x₁

POL(if_active(x₁, x₂, x₃)) = x₂ + x₃

POL(IF_ACTIVE(x₁, x₂, x₃)) = x₂ + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 6

             ↳Polo

...

               →DP Problem 8

                 ↳Polynomial Ordering

Dependency Pairs:

Rules:

minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

IF_ACTIVE(false, x, y) -> MARK(y)
IF_ACTIVE(true, x, y) -> MARK(x)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ge_active(x₁, x₂)) = 0

POL(MARK(x₁)) = x₁

POL(false) = 0

POL(div_active(x₁, x₂)) = 0

POL(minus(x₁, x₂)) = 0

POL(true) = 0

POL(mark(x₁)) = 0

POL(minus_active(x₁, x₂)) = 0

POL(if(x₁, x₂, x₃)) = 0

POL(0) = 0

POL(s(x₁)) = 0

POL(ge(x₁, x₂)) = 0

POL(DIV_ACTIVE(x₁, x₂)) = 1

POL(div(x₁, x₂)) = 1

POL(if_active(x₁, x₂, x₃)) = 0

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

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 6

             ↳Polo

...

               →DP Problem 9

                 ↳Dependency Graph

Dependency Pairs:

DIV_ACTIVE(s(x), s(y)) -> IF_ACTIVE(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
MARK(div(x, y)) -> DIV_ACTIVE(mark(x), y)

Rules:

minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)
mark(0) -> 0
mark(s(x)) -> s(mark(x))
mark(minus(x, y)) -> minus_active(x, y)
mark(ge(x, y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
mark(if(x, y, z)) -> if_active(mark(x), y, z)
ge_active(x, 0) -> true
ge_active(0, s(y)) -> false
ge_active(s(x), s(y)) -> ge_active(x, y)
ge_active(x, y) -> ge(x, y)
div_active(0, s(y)) -> 0
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
div_active(x, y) -> div(x, y)
if_active(true, x, y) -> mark(x)
if_active(false, x, y) -> mark(y)
if_active(x, y, z) -> if(x, y, z)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes

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

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

POL(ge_active(x₁, x₂))	= 0
POL(MARK(x₁))	= x₁
POL(div_active(x₁, x₂))	= 0
POL(false)	= 0
POL(minus(x₁, x₂))	= 0
POL(true)	= 0
POL(mark(x₁))	= 0
POL(minus_active(x₁, x₂))	= 0
POL(if(x₁, x₂, x₃))	= 1 + x₁ + x₂ + x₃
POL(0)	= 0
POL(s(x₁))	= x₁
POL(ge(x₁, x₂))	= 0
POL(DIV_ACTIVE(x₁, x₂))	= 0
POL(div(x₁, x₂))	= x₁
POL(if_active(x₁, x₂, x₃))	= 0
POL(IF_ACTIVE(x₁, x₂, x₃))	= x₂ + x₃