+(a, b) -> +(b, a)

+(a, +(b,

+(+(

f(a,

f(b,

f(+(

R

↳Dependency Pair Analysis

+'(a, b) -> +'(b, a)

+'(a, +(b,z)) -> +'(b, +(a,z))

+'(a, +(b,z)) -> +'(a,z)

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

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

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Forward Instantiation Transformation

→DP Problem 2

↳FwdInst

→DP Problem 3

↳FwdInst

**+'(a, +(b, z)) -> +'(a, z)**

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

f(a,y) -> a

f(b,y) -> b

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

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

+'(a, +(b,z)) -> +'(a,z)

+'(a, +(b, +(b,z''))) -> +'(a, +(b,z''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 4

↳Polynomial Ordering

→DP Problem 2

↳FwdInst

→DP Problem 3

↳FwdInst

**+'(a, +(b, +(b, z''))) -> +'(a, +(b, z''))**

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

f(a,y) -> a

f(b,y) -> b

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

innermost

The following dependency pair can be strictly oriented:

+'(a, +(b, +(b,z''))) -> +'(a, +(b,z''))

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(b)= 0 _{ }^{ }_{ }^{ }POL(a)= 0 _{ }^{ }_{ }^{ }POL(+(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(+'(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 4

↳Polo

...

→DP Problem 5

↳Dependency Graph

→DP Problem 2

↳FwdInst

→DP Problem 3

↳FwdInst

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

f(a,y) -> a

f(b,y) -> b

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Forward Instantiation Transformation

→DP Problem 3

↳FwdInst

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

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

f(a,y) -> a

f(b,y) -> b

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

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

→DP Problem 6

↳Polynomial Ordering

→DP Problem 3

↳FwdInst

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

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

f(a,y) -> a

f(b,y) -> b

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

innermost

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(b)= 0 _{ }^{ }_{ }^{ }POL(a)= 0 _{ }^{ }_{ }^{ }POL(+(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(+'(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

→DP Problem 6

↳Polo

...

→DP Problem 7

↳Dependency Graph

→DP Problem 3

↳FwdInst

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

f(a,y) -> a

f(b,y) -> b

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

→DP Problem 3

↳Forward Instantiation Transformation

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

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

f(a,y) -> a

f(b,y) -> b

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

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

→DP Problem 3

↳FwdInst

→DP Problem 8

↳Polynomial Ordering

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

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

f(a,y) -> a

f(b,y) -> b

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

innermost

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(b)= 0 _{ }^{ }_{ }^{ }POL(a)= 0 _{ }^{ }_{ }^{ }POL(+(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(F(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

→DP Problem 3

↳FwdInst

→DP Problem 8

↳Polo

...

→DP Problem 9

↳Dependency Graph

+(a, b) -> +(b, a)

+(a, +(b,z)) -> +(b, +(a,z))

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

f(a,y) -> a

f(b,y) -> b

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

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes