Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following 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)))))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following 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)))))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 4 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) → TE(msubst(te(a), 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(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(t), sortSu(u)))
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) → 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(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(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(circ(sortSu(s), sortSu(t)))

The TRS R consists of the following 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)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.