Term Rewriting System R:
[x, y, z]
app(app(*, x), app(app(+, y), z)) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(+, y), z)), x) -> app(app(+, app(app(*, x), y)), app(app(*, x), z))
app(app(*, app(app(*, x), y)), z) -> app(app(*, x), app(app(*, y), z))
app(app(+, app(app(+, x), y)), z) -> app(app(+, x), app(app(+, y), z))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Size-Change Principle
       →DP Problem 2
         ↳Polo


Dependency Pairs:

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


Rules:


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





The original DP problem is in applicative form. Its DPs and usable rules are the following.

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


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


It is proper and hence, it can be A-transformed which results in the DP problem

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


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


We number the DPs as follows:
  1. +'(+(x, y), z) -> +'(y, z)
  2. +'(+(x, y), z) -> +'(x, +(y, z))
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
2=2
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1
{1, 2} , {1, 2}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial


We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Polynomial Ordering


Dependency Pairs:

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


Rules:


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





The original DP problem is in applicative form. Its DPs and usable rules are the following.

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


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


It is proper and hence, it can be A-transformed which results in the DP problem

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


*(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))


The following dependency pairs can be strictly oriented:

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

This corresponds to the following dependency pairs in applicative form:

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


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

*(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))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(*'(x1, x2))=  1 + x1 + x1·x2  
  POL(*(x1, x2))=  x1 + x1·x2 + x2  
  POL(+(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Polo
           →DP Problem 3
             ↳Size-Change Principle


Dependency Pairs:

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


Rules:


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





The original DP problem is in applicative form. Its DPs and usable rules are the following.

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


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


It is proper and hence, it can be A-transformed which results in the DP problem

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


*(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))


We number the DPs as follows:
  1. *'(*(x, y), z) -> *'(y, z)
  2. *'(*(x, y), z) -> *'(x, *(y, z))
  3. *'(x, +(y, z)) -> *'(x, z)
  4. *'(x, +(y, z)) -> *'(x, y)
and get the following Size-Change Graph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1
2=2
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1
{1, 2, 3, 4} , {1, 2, 3, 4}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1
{1, 2, 3, 4} , {1, 2, 3, 4}
1=1
2>2
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1
2=2
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
+(x1, x2) -> +(x1, x2)

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:06 minutes