Term Rewriting System R:
[X]
ag(X) -> ah(X)
ag(X) -> g(X)
ac -> d
ac -> c
ah(d) -> ag(c)
ah(X) -> h(X)
mark(g(X)) -> ag(X)
mark(h(X)) -> ah(X)
mark(c) -> ac
mark(d) -> d

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

AG(X) -> AH(X)
AH(d) -> AG(c)
MARK(g(X)) -> AG(X)
MARK(h(X)) -> AH(X)
MARK(c) -> AC

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

AH(d) -> AG(c)
AG(X) -> AH(X)


Rules:


ag(X) -> ah(X)
ag(X) -> g(X)
ac -> d
ac -> c
ah(d) -> ag(c)
ah(X) -> h(X)
mark(g(X)) -> ag(X)
mark(h(X)) -> ah(X)
mark(c) -> ac
mark(d) -> d





The following dependency pair can be strictly oriented:

AH(d) -> AG(c)


Additionally, the following rules can be oriented:

ag(X) -> ah(X)
ag(X) -> g(X)
ah(d) -> ag(c)
ah(X) -> h(X)
ac -> d
ac -> c
mark(g(X)) -> ag(X)
mark(h(X)) -> ah(X)
mark(c) -> ac
mark(d) -> d


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(A__G(x1))=  x1  
  POL(a__g(x1))=  0  
  POL(c)=  0  
  POL(a__c)=  1  
  POL(g(x1))=  0  
  POL(d)=  1  
  POL(h(x1))=  0  
  POL(mark(x1))=  1  
  POL(a__h(x1))=  0  
  POL(A__H(x1))=  x1  

resulting in one new DP problem. 



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


Dependency Pair:

AG(X) -> AH(X)


Rules:


ag(X) -> ah(X)
ag(X) -> g(X)
ac -> d
ac -> c
ah(d) -> ag(c)
ah(X) -> h(X)
mark(g(X)) -> ag(X)
mark(h(X)) -> ah(X)
mark(c) -> ac
mark(d) -> d





Using the Dependency Graph resulted in no new DP problems.

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