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i(i(

i(e) -> e

R

↳Dependency Pair Analysis

:'(:(x,y),z) -> :'(x, :(z, i(y)))

:'(:(x,y),z) -> :'(z, i(y))

:'(:(x,y),z) -> I(y)

:'(e,x) -> I(x)

:'(x, :(y, i(x))) -> I(y)

:'(x, :(y, :(i(x),z))) -> :'(i(z),y)

:'(x, :(y, :(i(x),z))) -> I(z)

:'(i(x), :(y,x)) -> I(y)

:'(i(x), :(y, :(x,z))) -> :'(i(z),y)

:'(i(x), :(y, :(x,z))) -> I(z)

I(:(x,y)) -> :'(y,x)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

**:'(i( x), :(y, :(x, z))) -> I(z)**

:(x,x) -> e

:(x, e) ->x

:(:(x,y),z) -> :(x, :(z, i(y)))

:(e,x) -> i(x)

:(x, :(y, i(x))) -> i(y)

:(x, :(y, :(i(x),z))) -> :(i(z),y)

:(i(x), :(y,x)) -> i(y)

:(i(x), :(y, :(x,z))) -> :(i(z),y)

i(:(x,y)) -> :(y,x)

i(i(x)) ->x

i(e) -> e

The following dependency pairs can be strictly oriented:

:'(i(x), :(y, :(x,z))) -> I(z)

:'(i(x), :(y, :(x,z))) -> :'(i(z),y)

:'(i(x), :(y,x)) -> I(y)

:'(x, :(y, :(i(x),z))) -> I(z)

:'(x, :(y, :(i(x),z))) -> :'(i(z),y)

:'(x, :(y, i(x))) -> I(y)

I(:(x,y)) -> :'(y,x)

:'(:(x,y),z) -> I(y)

:'(:(x,y),z) -> :'(z, i(y))

The following rules can be oriented:

i(:(x,y)) -> :(y,x)

i(i(x)) ->x

i(e) -> e

:(x,x) -> e

:(x, e) ->x

:(:(x,y),z) -> :(x, :(z, i(y)))

:(e,x) -> i(x)

:(x, :(y, i(x))) -> i(y)

:(x, :(y, :(i(x),z))) -> :(i(z),y)

:(i(x), :(y,x)) -> i(y)

:(i(x), :(y, :(x,z))) -> :(i(z),y)

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(:(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(I(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(i(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(e)= 0 _{ }^{ }_{ }^{ }POL(:'(x)_{1}, x_{2})= x _{1}+ x_{2}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

:'(x,_{1}x) -> :'(_{2}x,_{1}x)_{2}

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Dependency Graph

**:'(e, x) -> I(x)**

:(x,x) -> e

:(x, e) ->x

:(:(x,y),z) -> :(x, :(z, i(y)))

:(e,x) -> i(x)

:(x, :(y, i(x))) -> i(y)

:(x, :(y, :(i(x),z))) -> :(i(z),y)

:(i(x), :(y,x)) -> i(y)

:(i(x), :(y, :(x,z))) -> :(i(z),y)

i(:(x,y)) -> :(y,x)

i(i(x)) ->x

i(e) -> e

Using the Dependency Graph the DP problem was split into 1 DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳DGraph

...

→DP Problem 3

↳Argument Filtering and Ordering

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

:(x,x) -> e

:(x, e) ->x

:(:(x,y),z) -> :(x, :(z, i(y)))

:(e,x) -> i(x)

:(x, :(y, i(x))) -> i(y)

:(x, :(y, :(i(x),z))) -> :(i(z),y)

:(i(x), :(y,x)) -> i(y)

:(i(x), :(y, :(x,z))) -> :(i(z),y)

i(:(x,y)) -> :(y,x)

i(i(x)) ->x

i(e) -> e

The following dependency pair can be strictly oriented:

:'(:(x,y),z) -> :'(x, :(z, i(y)))

The following rules can be oriented:

:(x,x) -> e

:(x, e) ->x

:(:(x,y),z) -> :(x, :(z, i(y)))

:(e,x) -> i(x)

:(x, :(y, i(x))) -> i(y)

:(x, :(y, :(i(x),z))) -> :(i(z),y)

:(i(x), :(y,x)) -> i(y)

:(i(x), :(y, :(x,z))) -> :(i(z),y)

i(:(x,y)) -> :(y,x)

i(i(x)) ->x

i(e) -> e

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(:(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(i(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(e)= 0 _{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳DGraph

...

→DP Problem 4

↳Dependency Graph

:(x,x) -> e

:(x, e) ->x

:(:(x,y),z) -> :(x, :(z, i(y)))

:(e,x) -> i(x)

:(x, :(y, i(x))) -> i(y)

:(x, :(y, :(i(x),z))) -> :(i(z),y)

:(i(x), :(y,x)) -> i(y)

:(i(x), :(y, :(x,z))) -> :(i(z),y)

i(:(x,y)) -> :(y,x)

i(i(x)) ->x

i(e) -> e

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:01 minutes