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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

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), Y) -> IFMINUS(le(s(X), Y), s(X), Y)
MINUS(s(X), Y) -> LE(s(X), Y)
IFMINUS(false, s(X), Y) -> MINUS(X, Y)
QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
QUOT(s(X), s(Y)) -> MINUS(X, Y)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Remaining

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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

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

         ↳Nar

       →DP Problem 3

         ↳Remaining

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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

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

         ↳Nar

       →DP Problem 3

         ↳Remaining

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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

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: Homeomorphic Embedding Order with EMB
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

         ↳Nar

       →DP Problem 3

         ↳Remaining

Dependency Pair:

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Narrowing Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(X), Y) -> MINUS(X, Y)
MINUS(s(X), Y) -> IFMINUS(le(s(X), Y), s(X), Y)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

MINUS(s(X), Y) -> IFMINUS(le(s(X), Y), s(X), Y)

two new Dependency Pairs are created:

MINUS(s(X''), 0) -> IFMINUS(false, s(X''), 0)
MINUS(s(X''), s(Y'')) -> IFMINUS(le(X'', Y''), s(X''), s(Y''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Narrowing Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

MINUS(s(X''), s(Y'')) -> IFMINUS(le(X'', Y''), s(X''), s(Y''))
MINUS(s(X''), 0) -> IFMINUS(false, s(X''), 0)
IFMINUS(false, s(X), Y) -> MINUS(X, Y)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

MINUS(s(X''), s(Y'')) -> IFMINUS(le(X'', Y''), s(X''), s(Y''))

three new Dependency Pairs are created:

MINUS(s(0), s(Y''')) -> IFMINUS(true, s(0), s(Y'''))
MINUS(s(s(X')), s(0)) -> IFMINUS(false, s(s(X')), s(0))
MINUS(s(s(X')), s(s(Y'))) -> IFMINUS(le(X', Y'), s(s(X')), s(s(Y')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 8

                 ↳Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

MINUS(s(s(X')), s(s(Y'))) -> IFMINUS(le(X', Y'), s(s(X')), s(s(Y')))
MINUS(s(s(X')), s(0)) -> IFMINUS(false, s(s(X')), s(0))
IFMINUS(false, s(X), Y) -> MINUS(X, Y)
MINUS(s(X''), 0) -> IFMINUS(false, s(X''), 0)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(X), Y) -> MINUS(X, Y)

three new Dependency Pairs are created:

IFMINUS(false, s(X'), 0) -> MINUS(X', 0)
IFMINUS(false, s(s(X''')), s(0)) -> MINUS(s(X'''), s(0))
IFMINUS(false, s(s(X''')), s(s(Y'''))) -> MINUS(s(X'''), s(s(Y''')))

The transformation is resulting in three new DP problems:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(X''')), s(s(Y'''))) -> MINUS(s(X'''), s(s(Y''')))
MINUS(s(s(X')), s(s(Y'))) -> IFMINUS(le(X', Y'), 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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(s(X''')), s(s(Y'''))) -> MINUS(s(X'''), s(s(Y''')))

one new Dependency Pair is created:

IFMINUS(false, s(s(s(X''''))), 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

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(X''''))), s(s(Y''''))) -> MINUS(s(s(X'''')), s(s(Y'''')))
MINUS(s(s(X')), s(s(Y'))) -> IFMINUS(le(X', Y'), 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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

MINUS(s(s(X')), s(s(Y'))) -> IFMINUS(le(X', Y'), s(s(X')), s(s(Y')))

three new Dependency Pairs are created:

MINUS(s(s(0)), s(s(Y''))) -> IFMINUS(true, s(s(0)), s(s(Y'')))
MINUS(s(s(s(X''))), s(s(0))) -> IFMINUS(false, s(s(s(X''))), s(s(0)))
MINUS(s(s(s(X''))), s(s(s(Y'')))) -> IFMINUS(le(X'', Y''), s(s(s(X''))), s(s(s(Y''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 18

                 ↳Narrowing Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

MINUS(s(s(s(X''))), s(s(s(Y'')))) -> IFMINUS(le(X'', Y''), s(s(s(X''))), s(s(s(Y''))))
MINUS(s(s(s(X''))), s(s(0))) -> IFMINUS(false, s(s(s(X''))), s(s(0)))
IFMINUS(false, s(s(s(X''''))), 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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

MINUS(s(s(s(X''))), s(s(s(Y'')))) -> IFMINUS(le(X'', Y''), s(s(s(X''))), s(s(s(Y''))))

three new Dependency Pairs are created:

MINUS(s(s(s(0))), s(s(s(Y''')))) -> IFMINUS(true, s(s(s(0))), s(s(s(Y'''))))
MINUS(s(s(s(s(X')))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(X')))), s(s(s(0))))
MINUS(s(s(s(s(X')))), s(s(s(s(Y'))))) -> IFMINUS(le(X', Y'), s(s(s(s(X')))), s(s(s(s(Y')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 19

                 ↳Narrowing Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

MINUS(s(s(s(s(X')))), s(s(s(s(Y'))))) -> IFMINUS(le(X', Y'), s(s(s(s(X')))), s(s(s(s(Y')))))
MINUS(s(s(s(s(X')))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(X')))), s(s(s(0))))
IFMINUS(false, s(s(s(X''''))), s(s(Y''''))) -> MINUS(s(s(X'''')), s(s(Y'''')))
MINUS(s(s(s(X''))), s(s(0))) -> IFMINUS(false, s(s(s(X''))), s(s(0)))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

MINUS(s(s(s(s(X')))), s(s(s(s(Y'))))) -> IFMINUS(le(X', Y'), s(s(s(s(X')))), s(s(s(s(Y')))))

three new Dependency Pairs are created:

MINUS(s(s(s(s(0)))), s(s(s(s(Y''))))) -> IFMINUS(true, s(s(s(s(0)))), s(s(s(s(Y'')))))
MINUS(s(s(s(s(s(X''))))), s(s(s(s(0))))) -> IFMINUS(false, s(s(s(s(s(X''))))), s(s(s(s(0)))))
MINUS(s(s(s(s(s(X''))))), s(s(s(s(s(Y'')))))) -> IFMINUS(le(X'', Y''), s(s(s(s(s(X''))))), s(s(s(s(s(Y''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 20

                 ↳Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

MINUS(s(s(s(s(s(X''))))), s(s(s(s(s(Y'')))))) -> IFMINUS(le(X'', Y''), s(s(s(s(s(X''))))), s(s(s(s(s(Y''))))))
MINUS(s(s(s(s(s(X''))))), s(s(s(s(0))))) -> IFMINUS(false, s(s(s(s(s(X''))))), s(s(s(s(0)))))
MINUS(s(s(s(X''))), s(s(0))) -> IFMINUS(false, s(s(s(X''))), s(s(0)))
IFMINUS(false, s(s(s(X''''))), s(s(Y''''))) -> MINUS(s(s(X'''')), s(s(Y'''')))
MINUS(s(s(s(s(X')))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(X')))), s(s(s(0))))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(s(s(X''''))), s(s(Y''''))) -> MINUS(s(s(X'''')), s(s(Y'''')))

four new Dependency Pairs are created:

IFMINUS(false, s(s(s(X'''''))), s(s(0))) -> MINUS(s(s(X''''')), s(s(0)))
IFMINUS(false, s(s(s(s(X''')))), s(s(s(0)))) -> MINUS(s(s(s(X'''))), s(s(s(0))))
IFMINUS(false, s(s(s(s(s(X'''''))))), s(s(s(s(0))))) -> MINUS(s(s(s(s(X''''')))), s(s(s(s(0)))))
IFMINUS(false, s(s(s(s(s(X'''''))))), s(s(s(s(s(Y''''')))))) -> MINUS(s(s(s(s(X''''')))), s(s(s(s(s(Y'''''))))))

The transformation is resulting in four new DP problems:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 21

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(s(s(X'''''))))), s(s(s(s(s(Y''''')))))) -> MINUS(s(s(s(s(X''''')))), s(s(s(s(s(Y'''''))))))
MINUS(s(s(s(s(s(X''))))), s(s(s(s(s(Y'')))))) -> IFMINUS(le(X'', Y''), s(s(s(s(s(X''))))), 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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(s(s(s(s(X'''''))))), s(s(s(s(s(Y''''')))))) -> MINUS(s(s(s(s(X''''')))), s(s(s(s(s(Y'''''))))))

one new Dependency Pair is created:

IFMINUS(false, s(s(s(s(s(s(X'''')))))), s(s(s(s(s(Y'''''')))))) -> MINUS(s(s(s(s(s(X''''))))), s(s(s(s(s(Y''''''))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
IFMINUS(false, s(s(s(s(s(s(X'''')))))), s(s(s(s(s(Y'''''')))))) -> MINUS(s(s(s(s(s(X''''))))), s(s(s(s(s(Y''''''))))))
MINUS(s(s(s(s(s(X''))))), s(s(s(s(s(Y'')))))) -> IFMINUS(le(X'', Y''), s(s(s(s(s(X''))))), 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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:
innermost
Dependency Pair:
QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 22

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(s(s(X'''''))))), s(s(s(s(0))))) -> MINUS(s(s(s(s(X''''')))), s(s(s(s(0)))))
MINUS(s(s(s(s(s(X''))))), s(s(s(s(0))))) -> IFMINUS(false, s(s(s(s(s(X''))))), s(s(s(s(0)))))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(s(s(s(s(X'''''))))), s(s(s(s(0))))) -> MINUS(s(s(s(s(X''''')))), s(s(s(s(0)))))

one new Dependency Pair is created:

IFMINUS(false, s(s(s(s(s(s(X'''')))))), s(s(s(s(0))))) -> MINUS(s(s(s(s(s(X''''))))), s(s(s(s(0)))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 26

                 ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(s(s(s(X'''')))))), s(s(s(s(0))))) -> MINUS(s(s(s(s(s(X''''))))), s(s(s(s(0)))))
MINUS(s(s(s(s(s(X''))))), s(s(s(s(0))))) -> IFMINUS(false, s(s(s(s(s(X''))))), s(s(s(s(0)))))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IFMINUS(false, s(s(s(s(s(s(X'''')))))), s(s(s(s(0))))) -> MINUS(s(s(s(s(s(X''''))))), s(s(s(s(0)))))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

IFMINUS(x₁, x₂, x₃) -> x₂
s(x₁) -> s(x₁)
MINUS(x₁, x₂) -> x₁

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 34

                 ↳Dependency Graph

       →DP Problem 3

         ↳Remaining

Dependency Pair:

MINUS(s(s(s(s(s(X''))))), s(s(s(s(0))))) -> IFMINUS(false, s(s(s(s(s(X''))))), s(s(s(s(0)))))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 23

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(X'''''))), s(s(0))) -> MINUS(s(s(X''''')), s(s(0)))
MINUS(s(s(s(X''))), s(s(0))) -> IFMINUS(false, s(s(s(X''))), s(s(0)))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(s(s(X'''''))), s(s(0))) -> MINUS(s(s(X''''')), s(s(0)))

one new Dependency Pair is created:

IFMINUS(false, s(s(s(s(X'''')))), s(s(0))) -> MINUS(s(s(s(X''''))), s(s(0)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 27

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(s(X'''')))), s(s(0))) -> MINUS(s(s(s(X''''))), s(s(0)))
MINUS(s(s(s(X''))), s(s(0))) -> IFMINUS(false, s(s(s(X''))), s(s(0)))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

MINUS(s(s(s(X''))), s(s(0))) -> IFMINUS(false, s(s(s(X''))), s(s(0)))

one new Dependency Pair is created:

MINUS(s(s(s(s(X'''''')))), s(s(0))) -> IFMINUS(false, s(s(s(s(X'''''')))), s(s(0)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 29

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

MINUS(s(s(s(s(X'''''')))), s(s(0))) -> IFMINUS(false, s(s(s(s(X'''''')))), s(s(0)))
IFMINUS(false, s(s(s(s(X'''')))), s(s(0))) -> MINUS(s(s(s(X''''))), s(s(0)))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(s(s(s(X'''')))), s(s(0))) -> MINUS(s(s(s(X''''))), s(s(0)))

one new Dependency Pair is created:

IFMINUS(false, s(s(s(s(s(X''''''''))))), s(s(0))) -> MINUS(s(s(s(s(X'''''''')))), s(s(0)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 31

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(s(s(X''''''''))))), s(s(0))) -> MINUS(s(s(s(s(X'''''''')))), s(s(0)))
MINUS(s(s(s(s(X'''''')))), s(s(0))) -> IFMINUS(false, s(s(s(s(X'''''')))), s(s(0)))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

MINUS(s(s(s(s(X'''''')))), s(s(0))) -> IFMINUS(false, s(s(s(s(X'''''')))), s(s(0)))

one new Dependency Pair is created:

MINUS(s(s(s(s(s(X''''''''''))))), s(s(0))) -> IFMINUS(false, s(s(s(s(s(X''''''''''))))), s(s(0)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 33

                 ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

MINUS(s(s(s(s(s(X''''''''''))))), s(s(0))) -> IFMINUS(false, s(s(s(s(s(X''''''''''))))), s(s(0)))
IFMINUS(false, s(s(s(s(s(X''''''''))))), s(s(0))) -> MINUS(s(s(s(s(X'''''''')))), s(s(0)))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IFMINUS(false, s(s(s(s(s(X''''''''))))), s(s(0))) -> MINUS(s(s(s(s(X'''''''')))), s(s(0)))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

IFMINUS(x₁, x₂, x₃) -> x₂
s(x₁) -> s(x₁)
MINUS(x₁, x₂) -> x₁

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 36

                 ↳Dependency Graph

       →DP Problem 3

         ↳Remaining

Dependency Pair:

MINUS(s(s(s(s(s(X''''''''''))))), s(s(0))) -> IFMINUS(false, s(s(s(s(s(X''''''''''))))), s(s(0)))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 24

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(s(X''')))), s(s(s(0)))) -> MINUS(s(s(s(X'''))), s(s(s(0))))
MINUS(s(s(s(s(X')))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(X')))), s(s(s(0))))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(s(s(s(X''')))), s(s(s(0)))) -> MINUS(s(s(s(X'''))), s(s(s(0))))

one new Dependency Pair is created:

IFMINUS(false, s(s(s(s(s(X''''))))), s(s(s(0)))) -> MINUS(s(s(s(s(X'''')))), s(s(s(0))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 28

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(s(s(X''''))))), s(s(s(0)))) -> MINUS(s(s(s(s(X'''')))), s(s(s(0))))
MINUS(s(s(s(s(X')))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(X')))), s(s(s(0))))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

MINUS(s(s(s(s(X')))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(X')))), s(s(s(0))))

one new Dependency Pair is created:

MINUS(s(s(s(s(s(X''''''))))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(s(X''''''))))), s(s(s(0))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 30

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

MINUS(s(s(s(s(s(X''''''))))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(s(X''''''))))), s(s(s(0))))
IFMINUS(false, s(s(s(s(s(X''''))))), s(s(s(0)))) -> MINUS(s(s(s(s(X'''')))), s(s(s(0))))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(s(s(s(s(X''''))))), s(s(s(0)))) -> MINUS(s(s(s(s(X'''')))), s(s(s(0))))

one new Dependency Pair is created:

IFMINUS(false, s(s(s(s(s(s(X'''''''')))))), s(s(s(0)))) -> MINUS(s(s(s(s(s(X''''''''))))), s(s(s(0))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 32

                 ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(s(s(s(X'''''''')))))), s(s(s(0)))) -> MINUS(s(s(s(s(s(X''''''''))))), s(s(s(0))))
MINUS(s(s(s(s(s(X''''''))))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(s(X''''''))))), s(s(s(0))))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IFMINUS(false, s(s(s(s(s(s(X'''''''')))))), s(s(s(0)))) -> MINUS(s(s(s(s(s(X''''''''))))), s(s(s(0))))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

MINUS(x₁, x₂) -> x₁
s(x₁) -> s(x₁)
IFMINUS(x₁, x₂, x₃) -> x₂

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 35

                 ↳Dependency Graph

       →DP Problem 3

         ↳Remaining

Dependency Pair:

MINUS(s(s(s(s(s(X''''''))))), s(s(s(0)))) -> IFMINUS(false, s(s(s(s(s(X''''''))))), s(s(s(0))))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(X''')), s(0)) -> MINUS(s(X'''), s(0))
MINUS(s(s(X')), s(0)) -> IFMINUS(false, s(s(X')), s(0))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(s(X''')), s(0)) -> MINUS(s(X'''), s(0))

one new Dependency Pair is created:

IFMINUS(false, s(s(s(X''''))), s(0)) -> MINUS(s(s(X'''')), s(0))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 13

                 ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(s(X''''))), s(0)) -> MINUS(s(s(X'''')), s(0))
MINUS(s(s(X')), s(0)) -> IFMINUS(false, s(s(X')), s(0))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IFMINUS(false, s(s(s(X''''))), s(0)) -> MINUS(s(s(X'''')), s(0))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

MINUS(x₁, x₂) -> x₁
s(x₁) -> s(x₁)
IFMINUS(x₁, x₂, x₃) -> x₂

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 16

                 ↳Dependency Graph

       →DP Problem 3

         ↳Remaining

Dependency Pair:

MINUS(s(s(X')), s(0)) -> IFMINUS(false, s(s(X')), s(0))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 11

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(X'), 0) -> MINUS(X', 0)
MINUS(s(X''), 0) -> IFMINUS(false, s(X''), 0)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

IFMINUS(false, s(X'), 0) -> MINUS(X', 0)

one new Dependency Pair is created:

IFMINUS(false, s(s(X'''')), 0) -> MINUS(s(X''''), 0)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 14

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

IFMINUS(false, s(s(X'''')), 0) -> MINUS(s(X''''), 0)
MINUS(s(X''), 0) -> IFMINUS(false, s(X''), 0)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

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

MINUS(s(X''), 0) -> IFMINUS(false, s(X''), 0)

one new Dependency Pair is created:

MINUS(s(s(X'''''')), 0) -> IFMINUS(false, s(s(X'''''')), 0)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 15

                 ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Remaining

Dependency Pairs:

MINUS(s(s(X'''''')), 0) -> IFMINUS(false, s(s(X'''''')), 0)
IFMINUS(false, s(s(X'''')), 0) -> MINUS(s(X''''), 0)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IFMINUS(false, s(s(X'''')), 0) -> MINUS(s(X''''), 0)

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

IFMINUS(x₁, x₂, x₃) -> x₂
s(x₁) -> s(x₁)
MINUS(x₁, x₂) -> x₁

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 7

             ↳Nar

...

               →DP Problem 17

                 ↳Dependency Graph

       →DP Problem 3

         ↳Remaining

Dependency Pair:

MINUS(s(s(X'''''')), 0) -> IFMINUS(false, s(s(X'''''')), 0)

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
IFMINUS(false, s(s(s(s(s(s(X'''')))))), s(s(s(s(s(Y'''''')))))) -> MINUS(s(s(s(s(s(X''''))))), s(s(s(s(s(Y''''''))))))
MINUS(s(s(s(s(s(X''))))), s(s(s(s(s(Y'')))))) -> IFMINUS(le(X'', Y''), s(s(s(s(s(X''))))), 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(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:
innermost
Dependency Pair:
QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))

Rules:

le(0, Y) -> true
le(s(X), 0) -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0, Y) -> 0
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Strategy:
innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes