Term Rewriting System R:
[x]
half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

HALF(s(s(x))) -> HALF(x)
LASTBIT(s(s(x))) -> LASTBIT(x)
CONV(s(x)) -> CONV(half(s(x)))
CONV(s(x)) -> HALF(s(x))
CONV(s(x)) -> LASTBIT(s(x))

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar


Dependency Pair:

HALF(s(s(x))) -> HALF(x)


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




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

HALF(s(s(x))) -> HALF(x)
one new Dependency Pair is created:

HALF(s(s(s(s(x''))))) -> HALF(s(s(x'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 4
             ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar


Dependency Pair:

HALF(s(s(s(s(x''))))) -> HALF(s(s(x'')))


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




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

HALF(s(s(s(s(x''))))) -> HALF(s(s(x'')))
one new Dependency Pair is created:

HALF(s(s(s(s(s(s(x''''))))))) -> HALF(s(s(s(s(x'''')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 4
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Polynomial Ordering
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar


Dependency Pair:

HALF(s(s(s(s(s(s(x''''))))))) -> HALF(s(s(s(s(x'''')))))


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

HALF(s(s(s(s(s(s(x''''))))))) -> HALF(s(s(s(s(x'''')))))


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(HALF(x1))=  1 + x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 4
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Dependency Graph
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar


Dependency Pair:


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳Nar


Dependency Pair:

LASTBIT(s(s(x))) -> LASTBIT(x)


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




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

LASTBIT(s(s(x))) -> LASTBIT(x)
one new Dependency Pair is created:

LASTBIT(s(s(s(s(x''))))) -> LASTBIT(s(s(x'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 7
             ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳Nar


Dependency Pair:

LASTBIT(s(s(s(s(x''))))) -> LASTBIT(s(s(x'')))


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




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

LASTBIT(s(s(s(s(x''))))) -> LASTBIT(s(s(x'')))
one new Dependency Pair is created:

LASTBIT(s(s(s(s(s(s(x''''))))))) -> LASTBIT(s(s(s(s(x'''')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 7
             ↳FwdInst
             ...
               →DP Problem 8
                 ↳Polynomial Ordering
       →DP Problem 3
         ↳Nar


Dependency Pair:

LASTBIT(s(s(s(s(s(s(x''''))))))) -> LASTBIT(s(s(s(s(x'''')))))


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

LASTBIT(s(s(s(s(s(s(x''''))))))) -> LASTBIT(s(s(s(s(x'''')))))


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(LASTBIT(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 7
             ↳FwdInst
             ...
               →DP Problem 9
                 ↳Dependency Graph
       →DP Problem 3
         ↳Nar


Dependency Pair:


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Narrowing Transformation


Dependency Pair:

CONV(s(x)) -> CONV(half(s(x)))


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




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

CONV(s(x)) -> CONV(half(s(x)))
two new Dependency Pairs are created:

CONV(s(0)) -> CONV(0)
CONV(s(s(x''))) -> CONV(s(half(x'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar
           →DP Problem 10
             ↳Narrowing Transformation


Dependency Pair:

CONV(s(s(x''))) -> CONV(s(half(x'')))


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




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

CONV(s(s(x''))) -> CONV(s(half(x'')))
three new Dependency Pairs are created:

CONV(s(s(0))) -> CONV(s(0))
CONV(s(s(s(0)))) -> CONV(s(0))
CONV(s(s(s(s(x'))))) -> CONV(s(s(half(x'))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar
           →DP Problem 10
             ↳Nar
             ...
               →DP Problem 11
                 ↳Polynomial Ordering


Dependency Pair:

CONV(s(s(s(s(x'))))) -> CONV(s(s(half(x'))))


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

CONV(s(s(s(s(x'))))) -> CONV(s(s(half(x'))))


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

half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(s(x1))=  1 + x1  
  POL(CONV(x1))=  1 + x1  
  POL(half(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Nar
           →DP Problem 10
             ↳Nar
             ...
               →DP Problem 12
                 ↳Dependency Graph


Dependency Pair:


Rules:


half(0) -> 0
half(s(0)) -> 0
half(s(s(x))) -> s(half(x))
lastbit(0) -> 0
lastbit(s(0)) -> s(0)
lastbit(s(s(x))) -> lastbit(x)
conv(0) -> cons(nil, 0)
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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