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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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, s(x), s(y)) -> GCD(minus(x, y), s(y))
IF_GCD(true, s(x), s(y)) -> MINUS(x, y)
IF_GCD(false, s(x), s(y)) -> GCD(minus(y, x), s(x))
IF_GCD(false, s(x), s(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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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

                 ↳Argument Filtering and 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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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 AFS 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:

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

     ↳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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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

                 ↳Argument Filtering and 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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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 AFS 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(x₁, x₂) -> MINUS(x₁, x₂)
s(x₁) -> s(x₁)

     ↳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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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, s(x), s(y)) -> GCD(minus(y, x), s(x))
IF_GCD(true, s(x), s(y)) -> GCD(minus(x, y), s(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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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, 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(x), s(y)) -> GCD(minus(y, x), 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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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))

two new Dependency Pairs are created:

IF_GCD(true, s(x''), s(0)) -> GCD(x'', s(0))
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 11

                 ↳Forward Instantiation Transformation

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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

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

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

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 15

                 ↳Argument Filtering and Ordering

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{true, s} > GCD > IF_GCD > 0

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

IF_GCD(x₁, x₂, x₃) -> IF_GCD(x₁, x₂, x₃)
GCD(x₁, x₂) -> GCD(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 20

                 ↳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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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 12

                 ↳Narrowing Transformation

Dependency Pairs:

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

two new Dependency Pairs are created:

IF_GCD(false, s(0), s(y')) -> GCD(y', s(0))
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 14

                 ↳Narrowing Transformation

Dependency Pairs:

IF_GCD(false, s(s(y'')), s(s(x''))) -> GCD(minus(x'', y''), s(s(y'')))
GCD(s(s(y'')), s(s(x''))) -> IF_GCD(le(x'', y''), s(s(y'')), s(s(x'')))
IF_GCD(true, s(s(x'')), s(s(y''))) -> GCD(minus(x'', y''), 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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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 16

                 ↳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'))))
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'')))

Rules:

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

two new Dependency Pairs are created:

IF_GCD(true, s(s(x''')), s(s(0))) -> GCD(x''', s(s(0)))
IF_GCD(true, s(s(s(x'))), s(s(s(y')))) -> GCD(minus(x', y'), s(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 17

                 ↳Argument Filtering and Ordering

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{GCD, 0, IF_GCD} > s

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

GCD(x₁, x₂) -> GCD(x₁, x₂)
IF_GCD(x₁, x₂, x₃) -> IF_GCD(x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 18

                 ↳Narrowing Transformation

Dependency Pairs:

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

two new Dependency Pairs are created:

IF_GCD(false, s(s(0)), s(s(x'''))) -> GCD(x''', s(s(0)))
IF_GCD(false, s(s(s(y'))), s(s(s(x')))) -> GCD(minus(x', y'), s(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 19

                 ↳Narrowing Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

GCD(s(s(s(y''))), s(s(s(0)))) -> IF_GCD(true, s(s(s(y''))), s(s(s(0))))
GCD(s(s(s(0))), s(s(s(s(x''))))) -> IF_GCD(false, s(s(s(0))), s(s(s(s(x'')))))
GCD(s(s(s(s(y'')))), s(s(s(s(x''))))) -> IF_GCD(le(x'', y''), s(s(s(s(y'')))), s(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 22

                 ↳Narrowing Transformation

Dependency Pairs:

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

two new Dependency Pairs are created:

IF_GCD(true, s(s(s(x''))), s(s(s(0)))) -> GCD(x'', s(s(s(0))))
IF_GCD(true, s(s(s(s(x'')))), s(s(s(s(y''))))) -> GCD(minus(x'', y''), s(s(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 23

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

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 25

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

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 27

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

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 29

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

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 32

                 ↳Argument Filtering and Ordering

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{GCD, 0, s, IF_GCD} > true

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

IF_GCD(x₁, x₂, x₃) -> IF_GCD(x₁, x₂, x₃)
GCD(x₁, x₂) -> GCD(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 40

                 ↳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(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, s(x), s(y)) -> gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(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 24

                 ↳Narrowing Transformation

Dependency Pairs:

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

two new Dependency Pairs are created:

IF_GCD(false, s(s(s(0))), s(s(s(x'')))) -> GCD(x'', s(s(s(0))))
IF_GCD(false, s(s(s(s(y'')))), s(s(s(s(x''))))) -> GCD(minus(x'', y''), s(s(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 26

                 ↳Narrowing Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

GCD(s(s(s(s(y''')))), s(s(s(s(0))))) -> IF_GCD(true, s(s(s(s(y''')))), s(s(s(s(0)))))
GCD(s(s(s(s(0)))), s(s(s(s(s(x')))))) -> IF_GCD(false, s(s(s(s(0)))), s(s(s(s(s(x'))))))
GCD(s(s(s(s(s(y'))))), s(s(s(s(s(x')))))) -> IF_GCD(le(x', y'), s(s(s(s(s(y'))))), s(s(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 28

                 ↳Narrowing Transformation

Dependency Pairs:

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

two new Dependency Pairs are created:

IF_GCD(true, s(s(s(s(x''')))), s(s(s(s(0))))) -> GCD(x''', s(s(s(s(0)))))
IF_GCD(true, s(s(s(s(s(x'))))), s(s(s(s(s(y')))))) -> GCD(minus(x', y'), s(s(s(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 30

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

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 33

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

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 35

                 ↳Argument Filtering and Ordering

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{GCD, 0, s, IF_GCD} > true

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

IF_GCD(x₁, x₂, x₃) -> IF_GCD(x₁, x₂, x₃)
GCD(x₁, x₂) -> GCD(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 31

                 ↳Narrowing Transformation

Dependency Pairs:

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

two new Dependency Pairs are created:

IF_GCD(false, s(s(s(s(0)))), s(s(s(s(x'''))))) -> GCD(x''', s(s(s(s(0)))))
IF_GCD(false, s(s(s(s(s(y'))))), s(s(s(s(s(x')))))) -> GCD(minus(x', y'), s(s(s(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 34

                 ↳Narrowing Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

GCD(s(s(s(s(s(y''))))), s(s(s(s(s(0)))))) -> IF_GCD(true, s(s(s(s(s(y''))))), s(s(s(s(s(0))))))
GCD(s(s(s(s(s(0))))), s(s(s(s(s(s(x''))))))) -> IF_GCD(false, s(s(s(s(s(0))))), s(s(s(s(s(s(x'')))))))
GCD(s(s(s(s(s(s(y'')))))), s(s(s(s(s(s(x''))))))) -> IF_GCD(le(x'', y''), s(s(s(s(s(s(y'')))))), s(s(s(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 36

                 ↳Narrowing Transformation

Dependency Pairs:

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

two new Dependency Pairs are created:

IF_GCD(true, s(s(s(s(s(x''))))), s(s(s(s(s(0)))))) -> GCD(x'', s(s(s(s(s(0))))))
IF_GCD(true, s(s(s(s(s(s(x'')))))), s(s(s(s(s(s(y''))))))) -> GCD(minus(x'', y''), s(s(s(s(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 37

                 ↳Argument Filtering and Ordering

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

{GCD, 0, s, IF_GCD} > true

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

IF_GCD(x₁, x₂, x₃) -> IF_GCD(x₁, x₂, x₃)
GCD(x₁, x₂) -> GCD(x₁)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 10

             ↳Nar

...

               →DP Problem 38

                 ↳Narrowing Transformation

Dependency Pairs:

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

two new Dependency Pairs are created:

IF_GCD(false, s(s(s(s(s(0))))), s(s(s(s(s(x'')))))) -> GCD(x'', s(s(s(s(s(0))))))
IF_GCD(false, s(s(s(s(s(s(y'')))))), s(s(s(s(s(s(x''))))))) -> GCD(minus(x'', y''), s(s(s(s(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 39

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:48 minutes