Term Rewriting System R:
[X, Y]
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)


Rules:


ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)


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

ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(u22(x1))=  x1  
  POL(U21(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(ACKIN(x1, x2))=  1 + x1 + x2  
  POL(ackout(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
ACKIN(x1, x2) -> ACKIN(x1, x2)
U21(x1, x2) -> U21(x1, x2)
s(x1) -> s(x1)
ackin(x1, x2) -> x2
ackout(x1) -> ackout(x1)
u21(x1, x2) -> x1
u22(x1) -> u22(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pairs:

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)


Rules:


ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))


Strategy:

innermost




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


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


Dependency Pair:

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)


Rules:


ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)


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

resulting in one new DP problem.
Used Argument Filtering System:
ACKIN(x1, x2) -> ACKIN(x1, x2)
s(x1) -> s(x1)


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


Dependency Pair:


Rules:


ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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