plus(plus(

times(

R

↳Dependency Pair Analysis

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

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳Remaining

**PLUS(plus( X, Y), Z) -> PLUS(Y, Z)**

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

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

innermost

The following dependency pair can be strictly oriented:

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

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:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳Remaining

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Remaining Obligation(s)

The following remains to be proven:

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

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

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

innermost

Duration:

0:00 minutes