Term Rewriting System R:
[y, x, z]
or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(x, nil) -> false
mem(x, set(y)) -> =(x, y)
mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

MEM(x, union(y, z)) -> OR(mem(x, y), mem(x, z))
MEM(x, union(y, z)) -> MEM(x, y)
MEM(x, union(y, z)) -> MEM(x, z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

MEM(x, union(y, z)) -> MEM(x, z)
MEM(x, union(y, z)) -> MEM(x, y)


Rules:


or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(x, nil) -> false
mem(x, set(y)) -> =(x, y)
mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))





The following dependency pairs can be strictly oriented:

MEM(x, union(y, z)) -> MEM(x, z)
MEM(x, union(y, z)) -> MEM(x, y)


Additionally, the following rules can be oriented:

or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(x, nil) -> false
mem(x, set(y)) -> =(x, y)
mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(MEM(x1, x2))=  x2  
  POL(mem(x1, x2))=  0  
  POL(false)=  0  
  POL(union(x1, x2))=  1 + x1 + x2  
  POL(=(x1, x2))=  0  
  POL(or(x1, x2))=  0  
  POL(true)=  0  
  POL(nil)=  0  
  POL(set(x1))=  0  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(x, nil) -> false
mem(x, set(y)) -> =(x, y)
mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))





Using the Dependency Graph resulted in no new DP problems.

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