(0) Obligation:
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.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P 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(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)))
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.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) → TE(msubst(te(a), 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)))
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)))
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(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.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
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(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(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)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation with max and min functions [POLO,MAXPOLO]:
POL(SORTSU(x1)) = x1
POL(TE(x1)) = 1 + x1
POL(circ(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = 1 + max(x1, x2)
POL(id) = 0
POL(lift) = 1
POL(msubst(x1, x2)) = x1 + x2
POL(sop(x1)) = x1
POL(sortSu(x1)) = x1
POL(subst(x1, x2)) = x1
POL(te(x1)) = x1
The following usable rules [FROCOS05] were oriented:
te(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) → te(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
te(msubst(te(a), sortSu(id))) → te(a)
te(subst(te(a), sortSu(id))) → te(a)
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(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(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(cons(te(a), sortSu(s))), sortSu(t))) → sortSu(cons(te(msubst(te(a), sortSu(t))), sortSu(circ(sortSu(s), sortSu(t)))))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SORTSU(circ(sortSu(cons(te(a), sortSu(s))), sortSu(t))) → TE(msubst(te(a), 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(circ(sortSu(s), sortSu(t))), sortSu(u))) → SORTSU(circ(sortSu(t), sortSu(u)))
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.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
(8) Complex Obligation (AND)
(9) Obligation:
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)))))
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.
(10) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
TE(msubst(te(msubst(te(a), sortSu(s))), sortSu(t))) → TE(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(TE(x1)) = x1
POL(circ(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = 0
POL(id) = 0
POL(lift) = 0
POL(msubst(x1, x2)) = 1 + x1
POL(sop(x1)) = 0
POL(sortSu(x1)) = x1
POL(subst(x1, x2)) = x1
POL(te(x1)) = 1 + x1
The following usable rules [FROCOS05] were oriented:
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(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(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(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))) → te(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
te(msubst(te(a), sortSu(id))) → te(a)
te(subst(te(a), sortSu(id))) → te(a)
(11) Obligation:
Q DP problem:
P is empty.
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.
(12) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(13) TRUE
(14) Obligation:
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(circ(sortSu(s), sortSu(t))), sortSu(u))) → SORTSU(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))
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.
(15) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) → SORTSU(circ(sortSu(t), sortSu(u)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(SORTSU(x1)) = x1
POL(circ(x1, x2)) = 1 + x1 + x2
POL(cons(x1, x2)) = 0
POL(id) = 0
POL(lift) = 0
POL(msubst(x1, x2)) = 0
POL(sop(x1)) = 0
POL(sortSu(x1)) = x1
POL(subst(x1, x2)) = 0
POL(te(x1)) = 0
The following usable rules [FROCOS05] were oriented:
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(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(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(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))) → te(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
te(msubst(te(a), sortSu(id))) → te(a)
te(subst(te(a), sortSu(id))) → te(a)
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) → SORTSU(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))
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.
(17) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
SORTSU(circ(sortSu(circ(sortSu(s), sortSu(t))), sortSu(u))) → SORTSU(circ(sortSu(s), sortSu(circ(sortSu(t), sortSu(u)))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
POL(circ(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(cons(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(msubst(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(subst(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
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(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(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(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))) → te(msubst(te(a), sortSu(circ(sortSu(s), sortSu(t)))))
te(msubst(te(a), sortSu(id))) → te(a)
te(subst(te(a), sortSu(id))) → te(a)
(18) Obligation:
Q DP problem:
P is empty.
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.
(19) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(20) TRUE