*(

*(+(

*(*(

+(+(

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↳Dependency Pair Analysis

*'(x, +(y,z)) -> +'(*(x,y), *(x,z))

*'(x, +(y,z)) -> *'(x,y)

*'(x, +(y,z)) -> *'(x,z)

*'(+(y,z),x) -> +'(*(x,y), *(x,z))

*'(+(y,z),x) -> *'(x,y)

*'(+(y,z),x) -> *'(x,z)

*'(*(x,y),z) -> *'(x, *(y,z))

*'(*(x,y),z) -> *'(y,z)

+'(+(x,y),z) -> +'(x, +(y,z))

+'(+(x,y),z) -> +'(y,z)

Furthermore,

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↳DPs

→DP Problem 1

↳Polynomial Ordering

→DP Problem 2

↳Nar

**+'(+( x, y), z) -> +'(y, z)**

*(x, +(y,z)) -> +(*(x,y), *(x,z))

*(+(y,z),x) -> +(*(x,y), *(x,z))

*(*(x,y),z) -> *(x, *(y,z))

+(+(x,y),z) -> +(x, +(y,z))

innermost

The following dependency pair can be strictly oriented:

+'(+(x,y),z) -> +'(y,z)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(+(x)_{1}, x_{2})= 1 + x _{2}_{ }^{ }_{ }^{ }POL(+'(x)_{1}, x_{2})= x _{1}_{ }^{ }

resulting in one new DP problem.

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↳DPs

→DP Problem 1

↳Polo

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳Nar

*(x, +(y,z)) -> +(*(x,y), *(x,z))

*(+(y,z),x) -> +(*(x,y), *(x,z))

*(*(x,y),z) -> *(x, *(y,z))

+(+(x,y),z) -> +(x, +(y,z))

innermost

Using the Dependency Graph resulted in no new DP problems.

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↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Narrowing Transformation

***'(*( x, y), z) -> *'(y, z)**

*(x, +(y,z)) -> +(*(x,y), *(x,z))

*(+(y,z),x) -> +(*(x,y), *(x,z))

*(*(x,y),z) -> *(x, *(y,z))

+(+(x,y),z) -> +(x, +(y,z))

innermost

On this DP problem, a Narrowing SCC transformation can be performed.

As a result of transforming the rule

two new Dependency Pairs are created:

*'(*(x,y),z) -> *'(x, *(y,z))

*'(*(x,y0), +(y'',z'')) -> *'(x, +(*(y0,y''), *(y0,z'')))

*'(*(x, *(x'',y'')),z'') -> *'(x, *(x'', *(y'',z'')))

The transformation is resulting in one new DP problem:

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↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Nar

→DP Problem 4

↳Remaining Obligation(s)

The following remains to be proven:

***'(*( x, *(x'', y'')), z'') -> *'(x, *(x'', *(y'', z'')))**

*(x, +(y,z)) -> +(*(x,y), *(x,z))

*(+(y,z),x) -> +(*(x,y), *(x,z))

*(*(x,y),z) -> *(x, *(y,z))

+(+(x,y),z) -> +(x, +(y,z))

innermost

Duration:

0:00 minutes