Termination Proof using AProVE

Term Rewriting System R:
[y, x]
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

LE(s(x), s(y)) -> LE(x, y)
MINUS(s(x), s(y)) -> MINUS(x, y)
GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y))
GCD(s(x), s(y)) -> LE(y, x)
IF_GCD(true, x, y) -> GCD(minus(x, y), y)
IF_GCD(true, x, y) -> MINUS(x, y)
IF_GCD(false, x, y) -> GCD(minus(y, x), x)
IF_GCD(false, x, y) -> MINUS(y, x)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

Dependency Pair:

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

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

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

one new Dependency Pair is created:

LE(s(s(x'')), s(s(y''))) -> LE(s(x''), s(y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

Dependency Pair:

LE(s(s(x'')), s(s(y''))) -> LE(s(x''), s(y''))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

LE(s(s(x'')), s(s(y''))) -> LE(s(x''), s(y''))

one new Dependency Pair is created:

LE(s(s(s(x''''))), s(s(s(y'''')))) -> LE(s(s(x'''')), s(s(y'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 5

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

Dependency Pair:

LE(s(s(s(x''''))), s(s(s(y'''')))) -> LE(s(s(x'''')), s(s(y'''')))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

LE(s(s(s(x''''))), s(s(s(y'''')))) -> LE(s(s(x'''')), s(s(y'''')))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(LE(x₁, x₂)) = 1 + x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 6

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Nar

Dependency Pair:

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

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

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

one new Dependency Pair is created:

MINUS(s(s(x'')), s(s(y''))) -> MINUS(s(x''), s(y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Nar

Dependency Pair:

MINUS(s(s(x'')), s(s(y''))) -> MINUS(s(x''), s(y''))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

MINUS(s(s(x'')), s(s(y''))) -> MINUS(s(x''), s(y''))

one new Dependency Pair is created:

MINUS(s(s(s(x''''))), s(s(s(y'''')))) -> MINUS(s(s(x'''')), s(s(y'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 8

                 ↳Polynomial Ordering

       →DP Problem 3

         ↳Nar

Dependency Pair:

MINUS(s(s(s(x''''))), s(s(s(y'''')))) -> MINUS(s(s(x'''')), s(s(y'''')))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

MINUS(s(s(s(x''''))), s(s(s(y'''')))) -> MINUS(s(s(x'''')), s(s(y'''')))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(MINUS(x₁, x₂)) = 1 + x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 9

                 ↳Dependency Graph

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Narrowing Transformation

Dependency Pairs:

IF_GCD(false, x, y) -> GCD(minus(y, x), x)
IF_GCD(true, x, y) -> GCD(minus(x, y), y)
GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y))

three new Dependency Pairs are created:

GCD(s(x'), s(0)) -> IF_GCD(true, s(x'), s(0))
GCD(s(0), s(s(x''))) -> IF_GCD(false, s(0), s(s(x'')))
GCD(s(s(y'')), s(s(x''))) -> IF_GCD(le(x'', y''), s(s(y'')), s(s(x'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Narrowing Transformation

Dependency Pairs:

GCD(s(s(y'')), s(s(x''))) -> IF_GCD(le(x'', y''), s(s(y'')), s(s(x'')))
GCD(s(0), s(s(x''))) -> IF_GCD(false, s(0), s(s(x'')))
IF_GCD(true, x, y) -> GCD(minus(x, y), y)
GCD(s(x'), s(0)) -> IF_GCD(true, s(x'), s(0))
IF_GCD(false, x, y) -> GCD(minus(y, x), x)

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

IF_GCD(true, x, y) -> GCD(minus(x, y), y)

three new Dependency Pairs are created:

IF_GCD(true, x'', 0) -> GCD(x'', 0)
IF_GCD(true, 0, y') -> GCD(0, y')
IF_GCD(true, s(x''), s(y'')) -> GCD(minus(x'', y''), s(y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pairs:

GCD(s(0), s(s(x''))) -> IF_GCD(false, s(0), s(s(x'')))
IF_GCD(true, s(x''), s(y'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(x'), s(0)) -> IF_GCD(true, s(x'), s(0))
IF_GCD(false, x, y) -> GCD(minus(y, x), x)
GCD(s(s(y'')), s(s(x''))) -> IF_GCD(le(x'', y''), s(s(y'')), s(s(x'')))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

IF_GCD(false, x, y) -> GCD(minus(y, x), x)

three new Dependency Pairs are created:

IF_GCD(false, 0, y') -> GCD(y', 0)
IF_GCD(false, x'', 0) -> GCD(0, x'')
IF_GCD(false, s(y''), s(x'')) -> GCD(minus(x'', y''), s(y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

Dependency Pairs:

GCD(s(s(y'')), s(s(x''))) -> IF_GCD(le(x'', y''), s(s(y'')), s(s(x'')))
IF_GCD(true, s(x''), s(y'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(x'), s(0)) -> IF_GCD(true, s(x'), s(0))
IF_GCD(false, s(y''), s(x'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(0), s(s(x''))) -> IF_GCD(false, s(0), s(s(x'')))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

GCD(s(s(y'')), s(s(x''))) -> IF_GCD(le(x'', y''), s(s(y'')), s(s(x'')))

three new Dependency Pairs are created:

GCD(s(s(y''')), s(s(0))) -> IF_GCD(true, s(s(y''')), s(s(0)))
GCD(s(s(0)), s(s(s(x')))) -> IF_GCD(false, s(s(0)), s(s(s(x'))))
GCD(s(s(s(y'))), s(s(s(x')))) -> IF_GCD(le(x', y'), s(s(s(y'))), s(s(s(x'))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

Dependency Pairs:

GCD(s(s(s(y'))), s(s(s(x')))) -> IF_GCD(le(x', y'), s(s(s(y'))), s(s(s(x'))))
GCD(s(s(0)), s(s(s(x')))) -> IF_GCD(false, s(s(0)), s(s(s(x'))))
GCD(s(s(y''')), s(s(0))) -> IF_GCD(true, s(s(y''')), s(s(0)))
IF_GCD(false, s(y''), s(x'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(0), s(s(x''))) -> IF_GCD(false, s(0), s(s(x'')))
GCD(s(x'), s(0)) -> IF_GCD(true, s(x'), s(0))
IF_GCD(true, s(x''), s(y'')) -> GCD(minus(x'', y''), s(y''))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

IF_GCD(true, s(x''), s(y'')) -> GCD(minus(x'', y''), s(y''))

three new Dependency Pairs are created:

IF_GCD(true, s(x'''), s(0)) -> GCD(x''', s(0))
IF_GCD(true, s(0), s(y''')) -> GCD(0, s(y'''))
IF_GCD(true, s(s(x')), s(s(y'))) -> GCD(minus(x', y'), s(s(y')))

The transformation is resulting in two new DP problems:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 14

                 ↳Polynomial Ordering

Dependency Pairs:

IF_GCD(true, s(x'''), s(0)) -> GCD(x''', s(0))
GCD(s(x'), s(0)) -> IF_GCD(true, s(x'), s(0))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

IF_GCD(true, s(x'''), s(0)) -> GCD(x''', s(0))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(0) = 0

POL(GCD(x₁, x₂)) = x₁

POL(IF_GCD(x₁, x₂, x₃)) = x₂

POL(true) = 0

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 17

                 ↳Dependency Graph

Dependency Pair:

GCD(s(x'), s(0)) -> IF_GCD(true, s(x'), s(0))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 15

                 ↳Narrowing Transformation

Dependency Pairs:

GCD(s(s(0)), s(s(s(x')))) -> IF_GCD(false, s(s(0)), s(s(s(x'))))
IF_GCD(true, s(s(x')), s(s(y'))) -> GCD(minus(x', y'), s(s(y')))
GCD(s(s(y''')), s(s(0))) -> IF_GCD(true, s(s(y''')), s(s(0)))
GCD(s(0), s(s(x''))) -> IF_GCD(false, s(0), s(s(x'')))
IF_GCD(false, s(y''), s(x'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(s(s(y'))), s(s(s(x')))) -> IF_GCD(le(x', y'), s(s(s(y'))), s(s(s(x'))))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

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

IF_GCD(false, s(y''), s(x'')) -> GCD(minus(x'', y''), s(y''))

three new Dependency Pairs are created:

IF_GCD(false, s(0), s(x''')) -> GCD(x''', s(0))
IF_GCD(false, s(y'''), s(0)) -> GCD(0, s(y'''))
IF_GCD(false, s(s(y')), s(s(x'))) -> GCD(minus(x', y'), s(s(y')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 16

                 ↳Polynomial Ordering

Dependency Pairs:

GCD(s(s(s(y'))), s(s(s(x')))) -> IF_GCD(le(x', y'), s(s(s(y'))), s(s(s(x'))))
IF_GCD(true, s(s(x')), s(s(y'))) -> GCD(minus(x', y'), s(s(y')))
GCD(s(s(y''')), s(s(0))) -> IF_GCD(true, s(s(y''')), s(s(0)))
IF_GCD(false, s(s(y')), s(s(x'))) -> GCD(minus(x', y'), s(s(y')))
GCD(s(s(0)), s(s(s(x')))) -> IF_GCD(false, s(s(0)), s(s(s(x'))))

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

GCD(s(s(s(y'))), s(s(s(x')))) -> IF_GCD(le(x', y'), s(s(s(y'))), s(s(s(x'))))
IF_GCD(true, s(s(x')), s(s(y'))) -> GCD(minus(x', y'), s(s(y')))
GCD(s(s(y''')), s(s(0))) -> IF_GCD(true, s(s(y''')), s(s(0)))
IF_GCD(false, s(s(y')), s(s(x'))) -> GCD(minus(x', y'), s(s(y')))
GCD(s(s(0)), s(s(s(x')))) -> IF_GCD(false, s(s(0)), s(s(s(x'))))

Additionally, the following usable rules for innermost can be oriented:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(0) = 0

POL(GCD(x₁, x₂)) = 1 + x₁ + x₂

POL(false) = 0

POL(minus(x₁, x₂)) = x₁

POL(true) = 0

POL(IF_GCD(x₁, x₂, x₃)) = x₂ + x₃

POL(s(x₁)) = 1 + x₁

POL(le(x₁, x₂)) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 18

                 ↳Dependency Graph

Dependency Pair:

Rules:

le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes

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

POL(0)	= 0
POL(GCD(x₁, x₂))	= x₁
POL(IF_GCD(x₁, x₂, x₃))	= x₂
POL(true)	= 0
POL(s(x₁))	= 1 + x₁

POL(0)	= 0
POL(GCD(x₁, x₂))	= 1 + x₁ + x₂
POL(false)	= 0
POL(minus(x₁, x₂))	= x₁
POL(true)	= 0
POL(IF_GCD(x₁, x₂, x₃))	= x₂ + x₃
POL(s(x₁))	= 1 + x₁
POL(le(x₁, x₂))	= 0