Term Rewriting System R:
[x, u, v, z, y]
admit(x, nil) -> nil
admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ADMIT(x, .(u, .(v, .(w, z)))) -> COND(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pair:

ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)


Rules:


admit(x, nil) -> nil
admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y


Strategy:

innermost




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

ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)
one new Dependency Pair is created:

ADMIT(carry(x'', u''', v''), .(u'', .(v0, .(w, z'')))) -> ADMIT(carry(carry(x'', u''', v''), u'', v0), z'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Instantiation Transformation


Dependency Pair:

ADMIT(carry(x'', u''', v''), .(u'', .(v0, .(w, z'')))) -> ADMIT(carry(carry(x'', u''', v''), u'', v0), z'')


Rules:


admit(x, nil) -> nil
admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y


Strategy:

innermost




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

ADMIT(carry(x'', u''', v''), .(u'', .(v0, .(w, z'')))) -> ADMIT(carry(carry(x'', u''', v''), u'', v0), z'')
one new Dependency Pair is created:

ADMIT(carry(carry(x'''', u'''''', v''''), u''''', v''0), .(u''1, .(v0'', .(w, z'''')))) -> ADMIT(carry(carry(carry(x'''', u'''''', v''''), u''''', v''0), u''1, v0''), z'''')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


Dependency Pair:

ADMIT(carry(carry(x'''', u'''''', v''''), u''''', v''0), .(u''1, .(v0'', .(w, z'''')))) -> ADMIT(carry(carry(carry(x'''', u'''''', v''''), u''''', v''0), u''1, v0''), z'''')


Rules:


admit(x, nil) -> nil
admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y


Strategy:

innermost




The following dependency pair can be strictly oriented:

ADMIT(carry(carry(x'''', u'''''', v''''), u''''', v''0), .(u''1, .(v0'', .(w, z'''')))) -> ADMIT(carry(carry(carry(x'''', u'''''', v''''), u''''', v''0), u''1, v0''), z'''')


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(carry(x1, x2, x3))=  x1 + x2 + x3  
  POL(.(x1, x2))=  x1 + x2  
  POL(w)=  1  
  POL(ADMIT(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
ADMIT(x1, x2) -> ADMIT(x1, x2)
carry(x1, x2, x3) -> carry(x1, x2, x3)
.(x1, x2) -> .(x1, x2)


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


Dependency Pair:


Rules:


admit(x, nil) -> nil
admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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