Term Rewriting System R:
[x, y]
not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ODD(s(x)) -> NOT(odd(x))
ODD(s(x)) -> ODD(x)
+'(x, s(y)) -> +'(x, y)
+'(s(x), y) -> +'(x, y)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

ODD(s(x)) -> ODD(x)


Rules:


not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))





The following dependency pair can be strictly oriented:

ODD(s(x)) -> ODD(x)


Additionally, the following rules can be oriented:

not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(odd(x1))=  0  
  POL(ODD(x1))=  x1  
  POL(false)=  0  
  POL(true)=  0  
  POL(s(x1))=  1 + x1  
  POL(not(x1))=  0  
  POL(+(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pairs:

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


Rules:


not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))





The following dependency pair can be strictly oriented:

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


Additionally, the following rules can be oriented:

not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(odd(x1))=  0  
  POL(false)=  0  
  POL(true)=  0  
  POL(s(x1))=  1 + x1  
  POL(not(x1))=  0  
  POL(+(x1, x2))=  x1 + x2  
  POL(+'(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 4
             ↳Polynomial Ordering


Dependency Pair:

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


Rules:


not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))





The following dependency pair can be strictly oriented:

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


Additionally, the following rules can be oriented:

not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(odd(x1))=  0  
  POL(false)=  0  
  POL(true)=  0  
  POL(s(x1))=  1 + x1  
  POL(not(x1))=  0  
  POL(+(x1, x2))=  x1 + x2  
  POL(+'(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 4
             ↳Polo
             ...
               →DP Problem 5
                 ↳Dependency Graph


Dependency Pair:


Rules:


not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))





Using the Dependency Graph resulted in no new DP problems.

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