Term Rewriting System R:
[YS, X, XS, X1, X2, Y, L]
aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

AAPP(nil, YS) -> MARK(YS)
AAPP(cons(X, XS), YS) -> MARK(X)
AFROM(X) -> MARK(X)
AZWADR(cons(X, XS), cons(Y, YS)) -> AAPP(mark(Y), cons(mark(X), nil))
AZWADR(cons(X, XS), cons(Y, YS)) -> MARK(Y)
AZWADR(cons(X, XS), cons(Y, YS)) -> MARK(X)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))
MARK(app(X1, X2)) -> MARK(X1)
MARK(app(X1, X2)) -> MARK(X2)
MARK(from(X)) -> AFROM(mark(X))
MARK(from(X)) -> MARK(X)
MARK(zWadr(X1, X2)) -> AZWADR(mark(X1), mark(X2))
MARK(zWadr(X1, X2)) -> MARK(X1)
MARK(zWadr(X1, X2)) -> MARK(X2)
MARK(prefix(X)) -> APREFIX(mark(X))
MARK(prefix(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

AZWADR(cons(X, XS), cons(Y, YS)) -> MARK(X)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(prefix(X)) -> MARK(X)
MARK(zWadr(X1, X2)) -> MARK(X2)
MARK(zWadr(X1, X2)) -> MARK(X1)
AZWADR(cons(X, XS), cons(Y, YS)) -> MARK(Y)
AZWADR(cons(X, XS), cons(Y, YS)) -> AAPP(mark(Y), cons(mark(X), nil))
MARK(zWadr(X1, X2)) -> AZWADR(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
AFROM(X) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))
MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
AAPP(cons(X, XS), YS) -> MARK(X)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))
AAPP(nil, YS) -> MARK(YS)


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

MARK(prefix(X)) -> MARK(X)


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

aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(A__APP(x1, x2))=  x1 + x2  
  POL(MARK(x1))=  x1  
  POL(A__FROM(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(a__from(x1))=  x1  
  POL(A__ZWADR(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  x1  
  POL(a__prefix(x1))=  1 + x1  
  POL(a__app(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(s(x1))=  x1  
  POL(a__zWadr(x1, x2))=  x1 + x2  
  POL(prefix(x1))=  1 + x1  
  POL(app(x1, x2))=  x1 + x2  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



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


Dependency Pairs:

AZWADR(cons(X, XS), cons(Y, YS)) -> MARK(X)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(zWadr(X1, X2)) -> MARK(X2)
MARK(zWadr(X1, X2)) -> MARK(X1)
AZWADR(cons(X, XS), cons(Y, YS)) -> MARK(Y)
AZWADR(cons(X, XS), cons(Y, YS)) -> AAPP(mark(Y), cons(mark(X), nil))
MARK(zWadr(X1, X2)) -> AZWADR(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
AFROM(X) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))
MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
AAPP(cons(X, XS), YS) -> MARK(X)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))
AAPP(nil, YS) -> MARK(YS)


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

MARK(zWadr(X1, X2)) -> MARK(X2)
MARK(zWadr(X1, X2)) -> MARK(X1)
MARK(zWadr(X1, X2)) -> AZWADR(mark(X1), mark(X2))


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

aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(A__APP(x1, x2))=  x1 + x2  
  POL(MARK(x1))=  x1  
  POL(A__FROM(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(a__from(x1))=  x1  
  POL(A__ZWADR(x1, x2))=  x1 + x2  
  POL(cons(x1, x2))=  x1  
  POL(a__prefix(x1))=  0  
  POL(a__app(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(s(x1))=  x1  
  POL(a__zWadr(x1, x2))=  1 + x1 + x2  
  POL(prefix(x1))=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(zWadr(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



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


Dependency Pairs:

AZWADR(cons(X, XS), cons(Y, YS)) -> MARK(X)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
AZWADR(cons(X, XS), cons(Y, YS)) -> MARK(Y)
AZWADR(cons(X, XS), cons(Y, YS)) -> AAPP(mark(Y), cons(mark(X), nil))
MARK(from(X)) -> MARK(X)
AFROM(X) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))
MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
AAPP(cons(X, XS), YS) -> MARK(X)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))
AAPP(nil, YS) -> MARK(YS)


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


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


Dependency Pairs:

AAPP(cons(X, XS), YS) -> MARK(X)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(from(X)) -> MARK(X)
AFROM(X) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))
MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
AAPP(nil, YS) -> MARK(YS)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

MARK(s(X)) -> MARK(X)


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

aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(A__APP(x1, x2))=  x1 + x2  
  POL(MARK(x1))=  x1  
  POL(A__FROM(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(a__from(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(a__prefix(x1))=  0  
  POL(a__app(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(s(x1))=  1 + x1  
  POL(a__zWadr(x1, x2))=  x1 + x2  
  POL(prefix(x1))=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 5
                 ↳Polynomial Ordering


Dependency Pairs:

AAPP(cons(X, XS), YS) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(from(X)) -> MARK(X)
AFROM(X) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))
MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
AAPP(nil, YS) -> MARK(YS)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

MARK(from(X)) -> MARK(X)
MARK(from(X)) -> AFROM(mark(X))


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

aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  1 + x1  
  POL(A__APP(x1, x2))=  x1 + x2  
  POL(MARK(x1))=  x1  
  POL(A__FROM(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(a__from(x1))=  1 + x1  
  POL(cons(x1, x2))=  x1  
  POL(a__prefix(x1))=  0  
  POL(a__app(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(s(x1))=  0  
  POL(a__zWadr(x1, x2))=  x1 + x2  
  POL(prefix(x1))=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pairs:

AAPP(cons(X, XS), YS) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
AFROM(X) -> MARK(X)
MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
AAPP(nil, YS) -> MARK(YS)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 7
                 ↳Polynomial Ordering


Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
AAPP(nil, YS) -> MARK(YS)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))
AAPP(cons(X, XS), YS) -> MARK(X)


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

AAPP(nil, YS) -> MARK(YS)


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

aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(A__APP(x1, x2))=  x1 + x2  
  POL(MARK(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(a__from(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(a__prefix(x1))=  1  
  POL(a__app(x1, x2))=  x1 + x2  
  POL(nil)=  1  
  POL(s(x1))=  0  
  POL(a__zWadr(x1, x2))=  x1 + x2  
  POL(app(x1, x2))=  x1 + x2  
  POL(prefix(x1))=  1  
  POL(zWadr(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 8
                 ↳Polynomial Ordering


Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))
AAPP(cons(X, XS), YS) -> MARK(X)


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

MARK(app(X1, X2)) -> MARK(X2)
MARK(app(X1, X2)) -> MARK(X1)
MARK(app(X1, X2)) -> AAPP(mark(X1), mark(X2))


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

aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(from(x1))=  x1  
  POL(A__APP(x1, x2))=  x1  
  POL(MARK(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(a__from(x1))=  x1  
  POL(cons(x1, x2))=  x1  
  POL(a__prefix(x1))=  0  
  POL(a__app(x1, x2))=  1 + x1 + x2  
  POL(nil)=  0  
  POL(s(x1))=  0  
  POL(a__zWadr(x1, x2))=  1 + x1 + x2  
  POL(app(x1, x2))=  1 + x1 + x2  
  POL(prefix(x1))=  0  
  POL(zWadr(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 9
                 ↳Dependency Graph


Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
AAPP(cons(X, XS), YS) -> MARK(X)


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 10
                 ↳Polynomial Ordering


Dependency Pair:

MARK(cons(X1, X2)) -> MARK(X1)


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

MARK(cons(X1, X2)) -> MARK(X1)


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(MARK(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 11
                 ↳Dependency Graph


Dependency Pair:


Rules:


aapp(nil, YS) -> mark(YS)
aapp(cons(X, XS), YS) -> cons(mark(X), app(XS, YS))
aapp(X1, X2) -> app(X1, X2)
afrom(X) -> cons(mark(X), from(s(X)))
afrom(X) -> from(X)
azWadr(nil, YS) -> nil
azWadr(XS, nil) -> nil
azWadr(cons(X, XS), cons(Y, YS)) -> cons(aapp(mark(Y), cons(mark(X), nil)), zWadr(XS, YS))
azWadr(X1, X2) -> zWadr(X1, X2)
aprefix(L) -> cons(nil, zWadr(L, prefix(L)))
aprefix(X) -> prefix(X)
mark(app(X1, X2)) -> aapp(mark(X1), mark(X2))
mark(from(X)) -> afrom(mark(X))
mark(zWadr(X1, X2)) -> azWadr(mark(X1), mark(X2))
mark(prefix(X)) -> aprefix(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:09 minutes