plus(0,

plus(s(

min(

min(s(

min(min(

quot(0, s(

quot(s(

R

↳Dependency Pair Analysis

PLUS(s(X),Y) -> PLUS(X,Y)

MIN(s(X), s(Y)) -> MIN(X,Y)

MIN(min(X,Y), Z) -> MIN(X, plus(Y, Z))

MIN(min(X,Y), Z) -> PLUS(Y, Z)

QUOT(s(X), s(Y)) -> QUOT(min(X,Y), s(Y))

QUOT(s(X), s(Y)) -> MIN(X,Y)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

→DP Problem 3

↳Remaining

**PLUS(s( X), Y) -> PLUS(X, Y)**

plus(0,Y) ->Y

plus(s(X),Y) -> s(plus(X,Y))

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

min(min(X,Y), Z) -> min(X, plus(Y, Z))

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

The following dependency pair can be strictly oriented:

PLUS(s(X),Y) -> PLUS(X,Y)

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

PLUS(x,_{1}x) -> PLUS(_{2}x,_{1}x)_{2}

s(x) -> s(_{1}x)_{1}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 4

↳Dependency Graph

→DP Problem 2

↳AFS

→DP Problem 3

↳Remaining

plus(0,Y) ->Y

plus(s(X),Y) -> s(plus(X,Y))

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

min(min(X,Y), Z) -> min(X, plus(Y, Z))

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

→DP Problem 3

↳Remaining

**MIN(min( X, Y), Z) -> MIN(X, plus(Y, Z))**

plus(0,Y) ->Y

plus(s(X),Y) -> s(plus(X,Y))

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

min(min(X,Y), Z) -> min(X, plus(Y, Z))

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

The following dependency pair can be strictly oriented:

MIN(min(X,Y), Z) -> MIN(X, plus(Y, Z))

The following usable rules using the Ce-refinement can be oriented:

plus(0,Y) ->Y

plus(s(X),Y) -> s(plus(X,Y))

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

MIN(x,_{1}x) -> MIN(_{2}x,_{1}x)_{2}

s(x) ->_{1}x_{1}

min(x,_{1}x) -> min(_{2}x,_{1}x)_{2}

plus(x,_{1}x) ->_{2}x_{2}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 5

↳Argument Filtering and Ordering

→DP Problem 3

↳Remaining

**MIN(s( X), s(Y)) -> MIN(X, Y)**

plus(0,Y) ->Y

plus(s(X),Y) -> s(plus(X,Y))

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

min(min(X,Y), Z) -> min(X, plus(Y, Z))

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

The following dependency pair can be strictly oriented:

MIN(s(X), s(Y)) -> MIN(X,Y)

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

MIN(x,_{1}x) -> MIN(_{2}x,_{1}x)_{2}

s(x) -> s(_{1}x)_{1}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 5

↳AFS

...

→DP Problem 6

↳Dependency Graph

→DP Problem 3

↳Remaining

plus(0,Y) ->Y

plus(s(X),Y) -> s(plus(X,Y))

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

min(min(X,Y), Z) -> min(X, plus(Y, Z))

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳Remaining Obligation(s)

The following remains to be proven:

**QUOT(s( X), s(Y)) -> QUOT(min(X, Y), s(Y))**

plus(0,Y) ->Y

plus(s(X),Y) -> s(plus(X,Y))

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

min(min(X,Y), Z) -> min(X, plus(Y, Z))

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

Duration:

0:00 minutes