Term Rewriting System R:
[a, s, t, u]
sortSu(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> sortSu(cons(te(msubst(te(a), sortSu(t))), sortSu(circ(sortSu(s), sortSu(t)))))
sortSu(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> sortSu(cons(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
sortSu(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> sortSu(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t)))))
sortSu(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> sortSu(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))
sortSu(circ(sortSu(s), sortSu(id))) -> sortSu(s)
sortSu(circ(sortSu(id), sortSu(s))) -> sortSu(s)
sortSu(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> sortSu(circ(sortSu(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t))))), sortSu(u)))
te(subst(te(a), sortSu(id))) -> te(a)
te(msubst(te(a), sortSu(id))) -> te(a)
te(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> te(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> SORTSU(cons(te(msubst(te(a), sortSu(t))), sortSu(circ(sortSu(s), sortSu(t)))))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> TE(msubst(te(a), sortSu(t)))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> SORTSU(cons(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> SORTSU(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t)))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(t), sortSu(u)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(circ(sortSu(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t))))), sortSu(u)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t)))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> TE(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> SORTSU(circ(sortSu(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t))))), sortSu(u)))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(t), sortSu(u)))
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> SORTSU(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> SORTSU(circ(sortSu(s), sortSu(t)))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> SORTSU(circ(sortSu(s), sortSu(t)))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> TE(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> TE(msubst(te(a), sortSu(t)))


Rules:


sortSu(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) -> sortSu(cons(te(msubst(te(a), sortSu(t))), sortSu(circ(sortSu(s), sortSu(t)))))
sortSu(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(te(a), sortSu(t))))) -> sortSu(cons(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
sortSu(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(cons(sop(lift), sortSu(t))))) -> sortSu(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t)))))
sortSu(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) -> sortSu(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))
sortSu(circ(sortSu(s), sortSu(id))) -> sortSu(s)
sortSu(circ(sortSu(id), sortSu(s))) -> sortSu(s)
sortSu(circ(sortSu(cons(sop(lift), sortSu(s))), sortSu(circ(sortSu(cons(sop(lift), sortSu(t))), sortSu(u))))) -> sortSu(circ(sortSu(cons(sop(lift), sortSu(circ(sortSu(s), sortSu(t))))), sortSu(u)))
te(subst(te(a), sortSu(id))) -> te(a)
te(msubst(te(a), sortSu(id))) -> te(a)
te(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) -> te(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))




Termination of R could not be shown.
Duration:
0:03 minutes