min(

min(s(

quot(0, s(

quot(s(

log(s(0)) -> 0

log(s(s(

R

↳Dependency Pair Analysis

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

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

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

LOG(s(s(X))) -> LOG(s(quot(X, s(s(0)))))

LOG(s(s(X))) -> QUOT(X, s(s(0)))

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

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

min(X, 0) ->X

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

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

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

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost 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:

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

↳AFS

min(X, 0) ->X

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

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

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

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

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

↳AFS

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

min(X, 0) ->X

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

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

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

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

The following dependency pair can be strictly oriented:

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

The following usable rules for innermost can be oriented:

min(X, 0) ->X

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 5

↳Dependency Graph

→DP Problem 3

↳AFS

min(X, 0) ->X

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

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

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

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳Argument Filtering and Ordering

**LOG(s(s( X))) -> LOG(s(quot(X, s(s(0)))))**

min(X, 0) ->X

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

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

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

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

The following dependency pair can be strictly oriented:

LOG(s(s(X))) -> LOG(s(quot(X, s(s(0)))))

The following usable rules for innermost can be oriented:

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

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

min(X, 0) ->X

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 6

↳Dependency Graph

min(X, 0) ->X

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

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

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

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes