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)

Termination of R to be shown.

`   R`
`     ↳Overlay and local confluence Check`

The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳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`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳Usable Rules (Innermost)`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`

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

As we are in the innermost case, we can delete all 11 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`             ...`
`               →DP Problem 4`
`                 ↳Size-Change Principle`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. LE(s(x), s(y)) -> LE(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2>2

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳Usable Rules (Innermost)`
`           →DP Problem 3`
`             ↳UsableRules`

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

As we are in the innermost case, we can delete all 11 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`             ...`
`               →DP Problem 5`
`                 ↳Size-Change Principle`
`           →DP Problem 3`
`             ↳UsableRules`

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. MINUS(s(x), s(y)) -> MINUS(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2>2

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳Usable Rules (Innermost)`

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

As we are in the innermost case, we can delete all 5 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 6`
`                 ↳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:

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 7`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 8`
`                 ↳Narrowing Transformation`

Dependency Pairs:

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

Rules:

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 9`
`                 ↳Instantiation Transformation`

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 10`
`                 ↳Rewriting Transformation`

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

IFGCD(false, s(0), s(s(x''''))) -> GCD(minus(s(x''''), 0), s(0))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 11`
`                 ↳Instantiation Transformation`

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in two new DP problems:

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 12`
`                 ↳Usable Rules (Innermost)`

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 4 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 14`
`                 ↳Modular Removal of Rules`

Dependency Pairs:

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

Rules:

minus(x, 0) -> x
minus(0, x) -> 0

Strategy:

innermost

We have the following set of usable rules:

minus(x, 0) -> x
minus(0, x) -> 0
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(0) =  0 POL(GCD(x1, x2)) =  x1 + x2 POL(minus(x1, x2)) =  x1 + x2 POL(IF_GCD(x1, x2, x3)) =  x1 + x2 + x3 POL(true) =  0 POL(s(x1)) =  1 + x1

We have the following set D of usable symbols: {0, GCD, minus, IFGCD, true, s}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

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

No Rules can be deleted.

The result of this processor delivers one new DP problem.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 15`
`                 ↳Modular Removal of Rules`

Dependency Pair:

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

Rules:

minus(x, 0) -> x
minus(0, x) -> 0

Strategy:

innermost

We have the following set of usable rules: none
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(0) =  0 POL(GCD(x1, x2)) =  x1 + x2 POL(IF_GCD(x1, x2, x3)) =  x1 + x2 + x3 POL(true) =  0 POL(s(x1)) =  x1

We have the following set D of usable symbols: {0, IFGCD, true, s}
The following Dependency Pairs can be deleted as they contain symbols in their lhs which do not occur in D:

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

No Rules can be deleted.

After the removal, there are no SCCs in the dependency graph which results in no DP problems which have to be solved.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 13`
`                 ↳Negative Polynomial Order`

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following Dependency Pairs can be strictly oriented using the given order.

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

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

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

Used ordering:
Polynomial Order with Interpretation:

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

POL( s(x1) ) = x1 + 1

POL( GCD(x1, x2) ) = x1 + x2

POL( minus(x1, x2) ) = x1

POL( 0 ) = 0

POL( le(x1, x2) ) = 0

POL( true ) = 0

POL( false ) = 0

This results in one new DP problem.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`           →DP Problem 2`
`             ↳UsableRules`
`           →DP Problem 3`
`             ↳UsableRules`
`             ...`
`               →DP Problem 16`
`                 ↳Dependency Graph`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:09 minutes