Term Rewriting System R:
[x, y, z]
a(lambda(x), y) -> lambda(a(x, 1))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
a(x, y) -> x
a(x, y) -> y
lambda(x) -> x

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

A(lambda(x), y) -> LAMBDA(a(x, 1))
A(lambda(x), y) -> A(x, 1)
A(lambda(x), y) -> LAMBDA(a(x, a(y, t)))
A(lambda(x), y) -> A(x, a(y, t))
A(lambda(x), y) -> A(y, t)
A(a(x, y), z) -> A(x, a(y, z))
A(a(x, y), z) -> A(y, z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

A(a(x, y), z) -> A(y, z)
A(a(x, y), z) -> A(x, a(y, z))
A(lambda(x), y) -> A(y, t)
A(lambda(x), y) -> A(x, a(y, t))
A(lambda(x), y) -> A(x, 1)


Rules:


a(lambda(x), y) -> lambda(a(x, 1))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
a(x, y) -> x
a(x, y) -> y
lambda(x) -> x





The following dependency pairs can be strictly oriented:

A(lambda(x), y) -> A(y, t)
A(lambda(x), y) -> A(x, a(y, t))
A(lambda(x), y) -> A(x, 1)


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

a(lambda(x), y) -> lambda(a(x, 1))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
a(x, y) -> x
a(x, y) -> y
lambda(x) -> x


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(t)=  0  
  POL(1)=  0  
  POL(lambda(x1))=  1 + x1  
  POL(a(x1, x2))=  x1 + x2  
  POL(A(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



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


Dependency Pairs:

A(a(x, y), z) -> A(y, z)
A(a(x, y), z) -> A(x, a(y, z))


Rules:


a(lambda(x), y) -> lambda(a(x, 1))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
a(x, y) -> x
a(x, y) -> y
lambda(x) -> x





The following dependency pairs can be strictly oriented:

A(a(x, y), z) -> A(y, z)
A(a(x, y), z) -> A(x, a(y, z))


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(t)=  0  
  POL(1)=  0  
  POL(lambda(x1))=  0  
  POL(a(x1, x2))=  1 + x1 + x2  
  POL(A(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


a(lambda(x), y) -> lambda(a(x, 1))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
a(x, y) -> x
a(x, y) -> y
lambda(x) -> x





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes