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

         ↳Usable Rules (Innermost)

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Neg 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

As we are in the innermost case, we can delete all 19 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

           →DP Problem 4

             ↳Size-Change Principle

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Neg POLO

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1
2	>	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳Usable Rules (Innermost)

       →DP Problem 3

         ↳Neg 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

As we are in the innermost case, we can delete all 19 non-usable-rules.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

           →DP Problem 5

             ↳Size-Change Principle

       →DP Problem 3

         ↳Neg POLO

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:

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

and get the following Size-Change Graph(s):

{1}	,	{1}
1	>	1
2	>	2

which lead(s) to this/these maximal multigraph(s):

{1}	,	{1}
1	>	1
2	>	2

D_P: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.

trivial

with Argument Filtering System:

s(x₁) -> s(x₁)

We obtain no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Negative Polynomial Order

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 Pair can be strictly oriented using the given order.

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

Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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

Used ordering:
Polynomial Order with Interpretation:

POL( MARK(x₁) ) = x₁

POL( div(x₁, x₂) ) = x₁ + 1

POL( DIV_ACTIVE(x₁, x₂) ) = 1

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

POL( s(x₁) ) = x₁

POL( minus(x₁, x₂) ) = 0

POL( 0 ) = 0

POL( if(x₁, ..., x₃) ) = x₁ + x₂ + x₃

POL( if_active(x₁, ..., x₃) ) = x₁ + x₂ + x₃

POL( true ) = 0

POL( mark(x₁) ) = x₁

POL( div_active(x₁, x₂) ) = x₁ + 1

POL( minus_active(x₁, x₂) ) = 0

POL( ge_active(x₁, x₂) ) = 0

POL( false ) = 0

POL( ge(x₁, x₂) ) = 0

This results in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Neg POLO

           →DP Problem 6

             ↳Negative Polynomial Order

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)
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 using the given order.

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

Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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

Used ordering:
Polynomial Order with Interpretation:

POL( MARK(x₁) ) = x₁

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

POL( DIV_ACTIVE(x₁, x₂) ) = 0

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

POL( s(x₁) ) = x₁

POL( div(x₁, x₂) ) = 0

POL( 0 ) = 0

POL( if_active(x₁, ..., x₃) ) = x₁ + x₂ + x₃ + 1

POL( true ) = 0

POL( mark(x₁) ) = x₁ + 1

POL( div_active(x₁, x₂) ) = 1

POL( minus(x₁, x₂) ) = 0

POL( minus_active(x₁, x₂) ) = 0

POL( ge_active(x₁, x₂) ) = 0

POL( false ) = 0

POL( ge(x₁, x₂) ) = 0

This results in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Neg POLO

           →DP Problem 6

             ↳Neg POLO

...

               →DP Problem 7

                 ↳Negative Polynomial Order

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 using the given order.

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

Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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

Used ordering:
Polynomial Order with Interpretation:

POL( MARK(x₁) ) = x₁

POL( s(x₁) ) = x₁ + 1

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

POL( DIV_ACTIVE(x₁, x₂) ) = x₁

POL( div(x₁, x₂) ) = x₁

POL( minus(x₁, x₂) ) = 0

POL( 0 ) = 0

POL( mark(x₁) ) = x₁

POL( ge_active(x₁, x₂) ) = 0

POL( ge(x₁, x₂) ) = 0

POL( true ) = 0

POL( false ) = 0

POL( if_active(x₁, ..., x₃) ) = x₂ + x₃

POL( div_active(x₁, x₂) ) = x₁

POL( minus_active(x₁, x₂) ) = 0

POL( if(x₁, ..., x₃) ) = x₂ + x₃

This results in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Neg POLO

           →DP Problem 6

             ↳Neg POLO

...

               →DP Problem 8

                 ↳Negative Polynomial Order

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 using the given order.

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

Moreover, the following usable rules (regarding the implicit AFS) are oriented.

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

Used ordering:
Polynomial Order with Interpretation:

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

POL( MARK(x₁) ) = x₁

POL( DIV_ACTIVE(x₁, x₂) ) = 1

POL( s(x₁) ) = 0

POL( 0 ) = 0

POL( div(x₁, x₂) ) = 1

POL( ge_active(x₁, x₂) ) = 0

POL( ge(x₁, x₂) ) = 0

POL( true ) = 0

POL( false ) = 0

POL( if_active(x₁, ..., x₃) ) = 1

POL( mark(x₁) ) = 1

POL( div_active(x₁, x₂) ) = 1

POL( minus_active(x₁, x₂) ) = 0

POL( if(x₁, ..., x₃) ) = 0

POL( minus(x₁, x₂) ) = 0

This results in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳UsableRules

       →DP Problem 2

         ↳UsableRules

       →DP Problem 3

         ↳Neg POLO

           →DP Problem 6

             ↳Neg 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:
0:14 minutes