+(

+(

+(0,

+(s(

+(

f(g(f(

f(g(h(

f(h(

R

↳Dependency Pair Analysis

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

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

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

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

F(g(f(x))) -> F(h(s(0),x))

F(g(h(x,y))) -> F(h(s(x),y))

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

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

+(x, 0) ->x

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

+(0,y) ->y

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

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

f(g(f(x))) -> f(h(s(0),x))

f(g(h(x,y))) -> f(h(s(x),y))

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

innermost

The following dependency pairs can be strictly oriented:

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

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

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

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:

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 3

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

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

+(x, 0) ->x

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

+(0,y) ->y

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

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

f(g(f(x))) -> f(h(s(0),x))

f(g(h(x,y))) -> f(h(s(x),y))

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

innermost

The following dependency pair can be strictly oriented:

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

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:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 3

↳AFS

...

→DP Problem 4

↳Dependency Graph

→DP Problem 2

↳AFS

+(x, 0) ->x

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

+(0,y) ->y

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

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

f(g(f(x))) -> f(h(s(0),x))

f(g(h(x,y))) -> f(h(s(x),y))

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

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

+(x, 0) ->x

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

+(0,y) ->y

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

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

f(g(f(x))) -> f(h(s(0),x))

f(g(h(x,y))) -> f(h(s(x),y))

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

innermost

The following dependency pair can be strictly oriented:

F(h(x, h(y,z))) -> F(h(+(x,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:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 5

↳Dependency Graph

+(x, 0) ->x

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

+(0,y) ->y

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

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

f(g(f(x))) -> f(h(s(0),x))

f(g(h(x,y))) -> f(h(s(x),y))

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

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes