Term Rewriting System R:

   [X, Y, N, M]
   eq(0, 0) -> true
   eq(0, s(X)) -> false
   eq(s(X), 0) -> false
   eq(s(X), s(Y)) -> eq(X, Y)
   rm(N, nil) -> nil
   rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))
   ifrm(true, N, add(M, X)) -> rm(N, X)
   ifrm(false, N, add(M, X)) -> add(M, rm(N, X))
   purge(nil) -> nil
   purge(add(N, X)) -> add(N, purge(rm(N, X)))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   EQ(s(X), s(Y)) -> EQ(X, Y)
   IFRM(true, N, add(M, X)) -> RM(N, X)
   IFRM(false, N, add(M, X)) -> RM(N, X)
   RM(N, add(M, X)) -> IFRM(eq(N, M), N, add(M, X))
   RM(N, add(M, X)) -> EQ(N, M)
   PURGE(add(N, X)) -> PURGE(rm(N, X))
   PURGE(add(N, X)) -> RM(N, X)

Furthermore, R contains three SCCs.


SCC1:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(EQ(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

IFRM(false, N, add(M, X)) -> RM(N, X)
RM(N, add(M, X)) -> IFRM(eq(N, M), N, add(M, X))
IFRM(true, N, add(M, X)) -> RM(N, X)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(add(x_1, x_2)) = 1 + x_2
POL(s(x_1)) = 0
POL(eq(x_1, x_2)) = 0
POL(RM(x_1, x_2)) = x_2
POL(true) = 0
POL(0) = 0
POL(IFRM(x_1, x_2, x_3)) = x_3
POL(false) = 0

 resulting in no subcycles.


SCC3:

PURGE(add(N, X)) -> PURGE(rm(N, X))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   ifrm(true, N, add(M, X)) -> rm(N, X)
   ifrm(false, N, add(M, X)) -> add(M, rm(N, X))
   rm(N, nil) -> nil
   rm(N, add(M, X)) -> ifrm(eq(N, M), N, add(M, X))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(add(x_1, x_2)) = 1 + x_2
POL(ifrm(x_1, x_2, x_3)) = x_3
POL(s(x_1)) = 0
POL(eq(x_1, x_2)) = 0
POL(rm(x_1, x_2)) = x_2
POL(true) = 0
POL(PURGE(x_1)) = x_1
POL(0) = 0
POL(false) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.866 seconds.