Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

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:

TERM_SUB(Term_sub(m, s), t) → CONCAT(s, t)
TERM_SUB(Term_sub(m, s), t) → TERM_SUB(m, Concat(s, t))
TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)
TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_var(x), Cons_usual(y, m, s)) → TERM_SUB(Term_var(x), s)
CONCAT(Cons_usual(x, m, s), t) → CONCAT(s, t)
CONCAT(Concat(s, t), u) → CONCAT(s, Concat(t, u))
CONCAT(Cons_usual(x, m, s), t) → TERM_SUB(m, t)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Case(m, xi, n), s) → SUM_SUB(xi, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
CONCAT(Concat(s, t), u) → CONCAT(t, u)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
FROZEN(m, Sum_constant(Left), n, s) → TERM_SUB(m, s)
SUM_SUB(xi, Cons_sum(psi, k, s)) → SUM_SUB(xi, s)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, s)
SUM_SUB(xi, Cons_usual(y, m, s)) → SUM_SUB(xi, s)
TERM_SUB(Term_var(x), Cons_sum(xi, k, s)) → TERM_SUB(Term_var(x), s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
CONCAT(Cons_sum(xi, k, s), t) → CONCAT(s, t)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

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:

TERM_SUB(Term_sub(m, s), t) → CONCAT(s, t)
TERM_SUB(Term_sub(m, s), t) → TERM_SUB(m, Concat(s, t))
TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)
TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_var(x), Cons_usual(y, m, s)) → TERM_SUB(Term_var(x), s)
CONCAT(Cons_usual(x, m, s), t) → CONCAT(s, t)
CONCAT(Concat(s, t), u) → CONCAT(s, Concat(t, u))
CONCAT(Cons_usual(x, m, s), t) → TERM_SUB(m, t)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Case(m, xi, n), s) → SUM_SUB(xi, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
CONCAT(Concat(s, t), u) → CONCAT(t, u)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
FROZEN(m, Sum_constant(Left), n, s) → TERM_SUB(m, s)
SUM_SUB(xi, Cons_sum(psi, k, s)) → SUM_SUB(xi, s)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, s)
SUM_SUB(xi, Cons_usual(y, m, s)) → SUM_SUB(xi, s)
TERM_SUB(Term_var(x), Cons_sum(xi, k, s)) → TERM_SUB(Term_var(x), s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
CONCAT(Cons_sum(xi, k, s), t) → CONCAT(s, t)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

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:

TERM_SUB(Term_sub(m, s), t) → CONCAT(s, t)
TERM_SUB(Term_var(x), Cons_usual(y, m, s)) → TERM_SUB(Term_var(x), s)
TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)
TERM_SUB(Term_sub(m, s), t) → TERM_SUB(m, Concat(s, t))
CONCAT(Concat(s, t), u) → CONCAT(s, Concat(t, u))
CONCAT(Cons_usual(x, m, s), t) → CONCAT(s, t)
CONCAT(Cons_usual(x, m, s), t) → TERM_SUB(m, t)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Case(m, xi, n), s) → SUM_SUB(xi, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
CONCAT(Concat(s, t), u) → CONCAT(t, u)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
FROZEN(m, Sum_constant(Left), n, s) → TERM_SUB(m, s)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(m, s)
SUM_SUB(xi, Cons_sum(psi, k, s)) → SUM_SUB(xi, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, s)
TERM_SUB(Term_var(x), Cons_sum(xi, k, s)) → TERM_SUB(Term_var(x), s)
SUM_SUB(xi, Cons_usual(y, m, s)) → SUM_SUB(xi, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
CONCAT(Cons_sum(xi, k, s), t) → CONCAT(s, t)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
              ↳ QDP
              ↳ QDP

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

SUM_SUB(xi, Cons_sum(psi, k, s)) → SUM_SUB(xi, s)
SUM_SUB(xi, Cons_usual(y, m, s)) → SUM_SUB(xi, s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SUM_SUB(xi, Cons_usual(y, m, s)) → SUM_SUB(xi, s)
The remaining pairs can at least be oriented weakly.

SUM_SUB(xi, Cons_sum(psi, k, s)) → SUM_SUB(xi, s)
Used ordering: Combined order from the following AFS and order.
SUM_SUB(x1, x2)  =  SUM_SUB(x1, x2)
Cons_sum(x1, x2, x3)  =  x3
Cons_usual(x1, x2, x3)  =  Cons_usual(x1, x2, x3)

Lexicographic path order with status [19].
Precedence:
trivial

Status:
Consusual3: multiset
SUMSUB2: [1,2]

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
              ↳ QDP
              ↳ QDP

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

SUM_SUB(xi, Cons_sum(psi, k, s)) → SUM_SUB(xi, s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SUM_SUB(xi, Cons_sum(psi, k, s)) → SUM_SUB(xi, s)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
SUM_SUB(x1, x2)  =  SUM_SUB(x1, x2)
Cons_sum(x1, x2, x3)  =  Cons_sum(x1, x3)

Lexicographic path order with status [19].
Precedence:
Conssum2 > SUMSUB2

Status:
Conssum2: multiset
SUMSUB2: [1,2]

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
              ↳ QDP

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

TERM_SUB(Term_var(x), Cons_usual(y, m, s)) → TERM_SUB(Term_var(x), s)
TERM_SUB(Term_var(x), Cons_sum(xi, k, s)) → TERM_SUB(Term_var(x), s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TERM_SUB(Term_var(x), Cons_sum(xi, k, s)) → TERM_SUB(Term_var(x), s)
The remaining pairs can at least be oriented weakly.

TERM_SUB(Term_var(x), Cons_usual(y, m, s)) → TERM_SUB(Term_var(x), s)
Used ordering: Combined order from the following AFS and order.
TERM_SUB(x1, x2)  =  TERM_SUB(x1, x2)
Term_var(x1)  =  Term_var
Cons_usual(x1, x2, x3)  =  x3
Cons_sum(x1, x2, x3)  =  Cons_sum(x1, x2, x3)

Lexicographic path order with status [19].
Precedence:
TERMSUB2 > Termvar
Conssum3 > Termvar

Status:
Termvar: multiset
TERMSUB2: [1,2]
Conssum3: multiset

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
              ↳ QDP

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

TERM_SUB(Term_var(x), Cons_usual(y, m, s)) → TERM_SUB(Term_var(x), s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TERM_SUB(Term_var(x), Cons_usual(y, m, s)) → TERM_SUB(Term_var(x), s)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
TERM_SUB(x1, x2)  =  x2
Term_var(x1)  =  Term_var
Cons_usual(x1, x2, x3)  =  Cons_usual(x1, x2, x3)

Lexicographic path order with status [19].
Precedence:
trivial

Status:
Consusual3: multiset
Termvar: multiset

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof

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

TERM_SUB(Term_sub(m, s), t) → CONCAT(s, t)
TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)
TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_sub(m, s), t) → TERM_SUB(m, Concat(s, t))
CONCAT(Cons_usual(x, m, s), t) → CONCAT(s, t)
CONCAT(Concat(s, t), u) → CONCAT(s, Concat(t, u))
CONCAT(Cons_usual(x, m, s), t) → TERM_SUB(m, t)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(n, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
CONCAT(Concat(s, t), u) → CONCAT(t, u)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
FROZEN(m, Sum_constant(Left), n, s) → TERM_SUB(m, s)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
CONCAT(Cons_sum(xi, k, s), t) → CONCAT(s, t)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TERM_SUB(Term_sub(m, s), t) → CONCAT(s, t)
TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)
TERM_SUB(Term_sub(m, s), t) → TERM_SUB(m, Concat(s, t))
CONCAT(Cons_usual(x, m, s), t) → CONCAT(s, t)
CONCAT(Concat(s, t), u) → CONCAT(s, Concat(t, u))
CONCAT(Cons_usual(x, m, s), t) → TERM_SUB(m, t)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(n, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
CONCAT(Concat(s, t), u) → CONCAT(t, u)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
FROZEN(m, Sum_constant(Left), n, s) → TERM_SUB(m, s)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
The remaining pairs can at least be oriented weakly.

TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
CONCAT(Cons_sum(xi, k, s), t) → CONCAT(s, t)
Used ordering: Combined order from the following AFS and order.
TERM_SUB(x1, x2)  =  TERM_SUB(x1)
Term_sub(x1, x2)  =  Term_sub(x1, x2)
CONCAT(x1, x2)  =  CONCAT(x1)
Case(x1, x2, x3)  =  Case(x1, x3)
FROZEN(x1, x2, x3, x4)  =  FROZEN(x1, x3)
Sum_sub(x1, x2)  =  x1
Term_inl(x1)  =  x1
Concat(x1, x2)  =  Concat(x1, x2)
Cons_usual(x1, x2, x3)  =  Cons_usual(x1, x2, x3)
Term_app(x1, x2)  =  Term_app(x1, x2)
Sum_constant(x1)  =  Sum_constant(x1)
Right  =  Right
Term_pair(x1, x2)  =  Term_pair(x1, x2)
Term_inr(x1)  =  x1
Left  =  Left
Sum_term_var(x1)  =  Sum_term_var(x1)
Cons_sum(x1, x2, x3)  =  x3
Frozen(x1, x2, x3, x4)  =  Frozen(x1, x3, x4)
Id  =  Id
Term_var(x1)  =  Term_var

Lexicographic path order with status [19].
Precedence:
Case2 > FROZEN2 > TERMSUB1 > CONCAT1 > Sumtermvar1
Case2 > Frozen3 > Sumtermvar1
Concat2 > Termsub2 > Sumtermvar1
Concat2 > CONCAT1 > Sumtermvar1
Consusual3 > TERMSUB1 > CONCAT1 > Sumtermvar1
Consusual3 > Termsub2 > Sumtermvar1
Termapp2 > Termsub2 > Sumtermvar1
Sumconstant1 > Sumtermvar1
Right > Sumtermvar1
Termpair2 > Sumtermvar1
Left > Termsub2 > Sumtermvar1
Id > Sumtermvar1
Termvar > Termsub2 > Sumtermvar1

Status:
Termapp2: [1,2]
CONCAT1: [1]
Case2: multiset
Termvar: multiset
Right: multiset
Concat2: [1,2]
Id: multiset
TERMSUB1: [1]
Termpair2: [1,2]
Consusual3: [3,2,1]
Sumtermvar1: multiset
Sumconstant1: multiset
Left: multiset
Frozen3: multiset
FROZEN2: multiset
Termsub2: [2,1]

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ DependencyGraphProof

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

TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
CONCAT(Cons_sum(xi, k, s), t) → CONCAT(s, t)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ AND
                        ↳ QDP
                          ↳ QDPOrderProof
                        ↳ QDP

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

CONCAT(Cons_sum(xi, k, s), t) → CONCAT(s, t)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONCAT(Cons_sum(xi, k, s), t) → CONCAT(s, t)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
CONCAT(x1, x2)  =  CONCAT(x1)
Cons_sum(x1, x2, x3)  =  Cons_sum(x1, x3)

Lexicographic path order with status [19].
Precedence:
Conssum2 > CONCAT1

Status:
Conssum2: multiset
CONCAT1: [1]

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ AND
                        ↳ QDP
                          ↳ QDPOrderProof
                            ↳ QDP
                              ↳ PisEmptyProof
                        ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ AND
                        ↳ QDP
                        ↳ QDP
                          ↳ QDPOrderProof

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

TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
The remaining pairs can at least be oriented weakly.

TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
Used ordering: Combined order from the following AFS and order.
TERM_SUB(x1, x2)  =  TERM_SUB(x1)
Term_inl(x1)  =  x1
Term_inr(x1)  =  Term_inr(x1)

Lexicographic path order with status [19].
Precedence:
trivial

Status:
Terminr1: multiset
TERMSUB1: [1]

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ AND
                        ↳ QDP
                        ↳ QDP
                          ↳ QDPOrderProof
                            ↳ QDP
                              ↳ QDPOrderProof

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

TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)

The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
TERM_SUB(x1, x2)  =  TERM_SUB(x1)
Term_inl(x1)  =  Term_inl(x1)

Lexicographic path order with status [19].
Precedence:
Terminl1 > TERMSUB1

Status:
Terminl1: multiset
TERMSUB1: [1]

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ AND
                        ↳ QDP
                        ↳ QDP
                          ↳ QDPOrderProof
                            ↳ QDP
                              ↳ QDPOrderProof
                                ↳ QDP
                                  ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

Term_sub(Case(m, xi, n), s) → Frozen(m, Sum_sub(xi, s), n, s)
Frozen(m, Sum_constant(Left), n, s) → Term_sub(m, s)
Frozen(m, Sum_constant(Right), n, s) → Term_sub(n, s)
Frozen(m, Sum_term_var(xi), n, s) → Case(Term_sub(m, s), xi, Term_sub(n, s))
Term_sub(Term_app(m, n), s) → Term_app(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_pair(m, n), s) → Term_pair(Term_sub(m, s), Term_sub(n, s))
Term_sub(Term_inl(m), s) → Term_inl(Term_sub(m, s))
Term_sub(Term_inr(m), s) → Term_inr(Term_sub(m, s))
Term_sub(Term_var(x), Id) → Term_var(x)
Term_sub(Term_var(x), Cons_usual(y, m, s)) → m
Term_sub(Term_var(x), Cons_usual(y, m, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_var(x), Cons_sum(xi, k, s)) → Term_sub(Term_var(x), s)
Term_sub(Term_sub(m, s), t) → Term_sub(m, Concat(s, t))
Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Concat(Concat(s, t), u) → Concat(s, Concat(t, u))
Concat(Cons_usual(x, m, s), t) → Cons_usual(x, Term_sub(m, t), Concat(s, t))
Concat(Cons_sum(xi, k, s), t) → Cons_sum(xi, k, Concat(s, t))
Concat(Id, s) → s

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.