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

Innermost Termination of R to be shown.

`   R`
`     ↳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)) -> IFGCD(le(y, x), s(x), s(y))
GCD(s(x), s(y)) -> LE(y, x)
IFGCD(true, x, y) -> GCD(minus(x, y), y)
IFGCD(true, x, y) -> MINUS(x, y)
IFGCD(false, x, y) -> GCD(minus(y, x), x)
IFGCD(false, x, y) -> MINUS(y, x)

Furthermore, R contains three SCCs.

`   R`
`     ↳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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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:

`   R`
`     ↳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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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:

`   R`
`     ↳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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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 w.r.t. to the implicit AFS 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.

`   R`
`     ↳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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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:

`   R`
`     ↳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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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:

`   R`
`     ↳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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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 w.r.t. to the implicit AFS 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.

`   R`
`     ↳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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Narrowing Transformation`

Dependency Pairs:

IFGCD(false, x, y) -> GCD(minus(y, x), x)
IFGCD(true, x, y) -> GCD(minus(x, y), y)
GCD(s(x), s(y)) -> IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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)) -> IFGCD(le(y, x), s(x), s(y))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳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''))) -> IFGCD(le(x'', y''), s(s(y'')), s(s(x'')))
GCD(s(0), s(s(x''))) -> IFGCD(false, s(0), s(s(x'')))
IFGCD(true, x, y) -> GCD(minus(x, y), y)
GCD(s(x'), s(0)) -> IFGCD(true, s(x'), s(0))
IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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

IFGCD(true, x, y) -> GCD(minus(x, y), y)
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳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''))) -> IFGCD(false, s(0), s(s(x'')))
IFGCD(true, s(x''), s(y'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(x'), s(0)) -> IFGCD(true, s(x'), s(0))
IFGCD(false, x, y) -> GCD(minus(y, x), x)
GCD(s(s(y'')), s(s(x''))) -> IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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

IFGCD(false, x, y) -> GCD(minus(y, x), x)
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳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''))) -> IFGCD(le(x'', y''), s(s(y'')), s(s(x'')))
IFGCD(true, s(x''), s(y'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(x'), s(0)) -> IFGCD(true, s(x'), s(0))
IFGCD(false, s(y''), s(x'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(0), s(s(x''))) -> IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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''))) -> IFGCD(le(x'', y''), s(s(y'')), s(s(x'')))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳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')))) -> IFGCD(le(x', y'), s(s(s(y'))), s(s(s(x'))))
GCD(s(s(0)), s(s(s(x')))) -> IFGCD(false, s(s(0)), s(s(s(x'))))
GCD(s(s(y''')), s(s(0))) -> IFGCD(true, s(s(y''')), s(s(0)))
IFGCD(false, s(y''), s(x'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(0), s(s(x''))) -> IFGCD(false, s(0), s(s(x'')))
GCD(s(x'), s(0)) -> IFGCD(true, s(x'), s(0))
IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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

IFGCD(true, s(x''), s(y'')) -> GCD(minus(x'', y''), s(y''))
three new Dependency Pairs are created:

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

The transformation is resulting in two new DP problems:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 10`
`             ↳Nar`
`             ...`
`               →DP Problem 14`
`                 ↳Polynomial Ordering`

Dependency Pairs:

IFGCD(true, s(x'''), s(0)) -> GCD(x''', s(0))
GCD(s(x'), s(0)) -> IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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(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.

`   R`
`     ↳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)) -> IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳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')))) -> IFGCD(false, s(s(0)), s(s(s(x'))))
IFGCD(true, s(s(x')), s(s(y'))) -> GCD(minus(x', y'), s(s(y')))
GCD(s(s(y''')), s(s(0))) -> IFGCD(true, s(s(y''')), s(s(0)))
GCD(s(0), s(s(x''))) -> IFGCD(false, s(0), s(s(x'')))
IFGCD(false, s(y''), s(x'')) -> GCD(minus(x'', y''), s(y''))
GCD(s(s(s(y'))), s(s(s(x')))) -> IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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

IFGCD(false, s(y''), s(x'')) -> GCD(minus(x'', y''), s(y''))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳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')))) -> IFGCD(le(x', y'), s(s(s(y'))), s(s(s(x'))))
IFGCD(true, s(s(x')), s(s(y'))) -> GCD(minus(x', y'), s(s(y')))
GCD(s(s(y''')), s(s(0))) -> IFGCD(true, s(s(y''')), s(s(0)))
IFGCD(false, s(s(y')), s(s(x'))) -> GCD(minus(x', y'), s(s(y')))
GCD(s(s(0)), s(s(s(x')))) -> IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

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

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)) =  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.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`       →DP Problem 3`
`         ↳Nar`
`           →DP Problem 10`
`             ↳Nar`
`             ...`
`               →DP Problem 18`
`                 ↳Dependency Graph`

Dependency Pairs:

GCD(s(s(s(y'))), s(s(s(x')))) -> IFGCD(le(x', y'), s(s(s(y'))), s(s(s(x'))))
GCD(s(s(y''')), s(s(0))) -> IFGCD(true, s(s(y''')), s(s(0)))
GCD(s(s(0)), s(s(s(x')))) -> IFGCD(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)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(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