Term Rewriting System R:
[y, x]
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

LE(s(x), s(y)) -> LE(x, y)
MINUS(s(x), s(y)) -> MINUS(x, y)
GCD(s(x), s(y)) -> IFGCD(le(y, x), s(x), s(y))
GCD(s(x), s(y)) -> LE(y, x)
IFGCD(true, x, y) -> GCD(minus(x, y), y)
IFGCD(true, x, y) -> MINUS(x, y)
IFGCD(false, x, y) -> GCD(minus(y, x), x)
IFGCD(false, x, y) -> MINUS(y, x)

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Inst


Dependency Pair:

LE(s(x), s(y)) -> LE(x, y)


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)





The following dependency pair can be strictly oriented:

LE(s(x), s(y)) -> LE(x, y)


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(LE(x1, x2))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 4
             ↳Dependency Graph
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Inst


Dependency Pair:


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering
       →DP Problem 3
         ↳Inst


Dependency Pair:

MINUS(s(x), s(y)) -> MINUS(x, y)


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)





The following dependency pair can be strictly oriented:

MINUS(s(x), s(y)) -> MINUS(x, y)


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(MINUS(x1, x2))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 5
             ↳Dependency Graph
       →DP Problem 3
         ↳Inst


Dependency Pair:


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Instantiation Transformation


Dependency Pairs:

IFGCD(false, x, y) -> GCD(minus(y, x), x)
IFGCD(true, x, y) -> GCD(minus(x, y), y)
GCD(s(x), s(y)) -> IFGCD(le(y, x), s(x), s(y))


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)





On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

IFGCD(true, x, y) -> GCD(minus(x, y), y)
one new Dependency Pair is created:

IFGCD(true, s(x0'), s(y'')) -> GCD(minus(s(x0'), s(y'')), s(y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Inst
           →DP Problem 6
             ↳Instantiation Transformation


Dependency Pairs:

IFGCD(true, s(x0'), s(y'')) -> GCD(minus(s(x0'), s(y'')), s(y''))
GCD(s(x), s(y)) -> IFGCD(le(y, x), s(x), s(y))
IFGCD(false, x, y) -> GCD(minus(y, x), x)


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)





On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

IFGCD(false, x, y) -> GCD(minus(y, x), x)
one new Dependency Pair is created:

IFGCD(false, s(x0'), s(y'')) -> GCD(minus(s(y''), s(x0')), s(x0'))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Inst
           →DP Problem 6
             ↳Inst
             ...
               →DP Problem 7
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

IFGCD(false, s(x0'), s(y'')) -> GCD(minus(s(y''), s(x0')), s(x0'))
GCD(s(x), s(y)) -> IFGCD(le(y, x), s(x), s(y))
IFGCD(true, s(x0'), s(y'')) -> GCD(minus(s(x0'), s(y'')), s(y''))


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0) -> x
minus(0, x) -> 0
minus(s(x), s(y)) -> minus(x, y)
gcd(0, y) -> y
gcd(s(x), 0) -> s(x)
gcd(s(x), s(y)) -> ifgcd(le(y, x), s(x), s(y))
ifgcd(true, x, y) -> gcd(minus(x, y), y)
ifgcd(false, x, y) -> gcd(minus(y, x), x)




Termination of R could not be shown.
Duration:
0:00 minutes