Term Rewriting System R:
[X, Y, Z, X1, X2, X3, X4]
plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))
PLUS(s(X), plus(Y, Z)) -> PLUS(s(s(Y)), Z)
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X3, plus(X2, X4))
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X2, X4)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X3, plus(X2, X4))
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X), plus(Y, Z)) -> PLUS(s(s(Y)), Z)
PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))


Rules:


plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X3, plus(X2, X4))
PLUS(s(X), plus(Y, Z)) -> PLUS(s(s(Y)), Z)


The following usable rules for innermost can be oriented:

plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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


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


Dependency Pairs:

PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))


Rules:


plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))


The following usable rules for innermost can be oriented:

plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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


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


Dependency Pair:


Rules:


plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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