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

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

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 that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

MINUS_ACTIVE(x₁, x₂) -> MINUS_ACTIVE(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 4

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

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

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳AFS

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 that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

GE_ACTIVE(x₁, x₂) -> GE_ACTIVE(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 3

         ↳AFS

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

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Argument Filtering and 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)
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)

The following usable rules for innermost 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)
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)
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)
minus_active(0, y) -> 0
minus_active(s(x), s(y)) -> minus_active(x, y)
minus_active(x, y) -> minus(x, y)

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{ge_active, ge, div, div_active, DIV_ACTIVE} > true
{ge_active, ge, div, div_active, DIV_ACTIVE} > {0, false}
{ge_active, ge, div, div_active, DIV_ACTIVE} > {if_active, if} > IF_ACTIVE > MARK
{ge_active, ge, div, div_active, DIV_ACTIVE} > s

resulting in one new DP problem.
Used Argument Filtering System:

MARK(x₁) -> MARK(x₁)
div(x₁, x₂) -> div(x₁, x₂)
DIV_ACTIVE(x₁, x₂) -> DIV_ACTIVE(x₁, x₂)
IF_ACTIVE(x₁, x₂, x₃) -> IF_ACTIVE(x₁, x₂, x₃)
s(x₁) -> s(x₁)
ge_active(x₁, x₂) -> ge_active(x₁, x₂)
minus(x₁, x₂) -> x₁
mark(x₁) -> x₁
if(x₁, x₂, x₃) -> if(x₁, x₂, x₃)
ge(x₁, x₂) -> ge(x₁, x₂)
minus_active(x₁, x₂) -> x₁
div_active(x₁, x₂) -> div_active(x₁, x₂)
if_active(x₁, x₂, x₃) -> if_active(x₁, x₂, x₃)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳AFS

           →DP Problem 6

             ↳Dependency Graph

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.

Innermost Termination of R successfully shown.
Duration:
0:08 minutes