Termination w.r.t. Q proof of /tmp/tmplzKzTs/lse.xml

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

    ↳5 QDP

        ↳6 QDPOrderProof (⇔)

        ↳7 QDP

        ↳8 QDPOrderProof (⇔)

        ↳9 QDP

        ↳10 PisEmptyProof (⇔)

        ↳11 TRUE

    ↳12 QDP

        ↳13 QDPOrderProof (⇔)

        ↳14 QDP

        ↳15 QDPOrderProof (⇔)

        ↳16 QDP

        ↳17 PisEmptyProof (⇔)

        ↳18 TRUE

    ↳19 QDP

        ↳20 QDPOrderProof (⇔)

        ↳21 QDP

        ↳22 DependencyGraphProof (⇔)

        ↳23 AND

            ↳24 QDP

                ↳25 QDPOrderProof (⇔)

                ↳26 QDP

                ↳27 QDPOrderProof (⇔)

                ↳28 QDP

                ↳29 QDPOrderProof (⇔)

                ↳30 QDP

                ↳31 PisEmptyProof (⇔)

                ↳32 TRUE

            ↳33 QDP

                ↳34 QDPOrderProof (⇔)

                ↳35 QDP

                ↳36 QDPOrderProof (⇔)

                ↳37 QDP

                ↳38 QDPOrderProof (⇔)

                ↳39 QDP

                ↳40 QDPOrderProof (⇔)

                ↳41 QDP

                ↳42 QDPOrderProof (⇔)

                ↳43 QDP

                ↳44 QDPOrderProof (⇔)

                ↳45 QDP

                ↳46 QDPOrderProof (⇔)

                ↳47 QDP

                ↳48 QDPOrderProof (⇔)

                ↳49 QDP

                ↳50 DependencyGraphProof (⇔)

                ↳51 TRUE

(0) Obligation:

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.

(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:

TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)
TERM_SUB(Case(m, xi, n), s) → SUM_SUB(xi, 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) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, s)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
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))
TERM_SUB(Term_sub(m, s), t) → CONCAT(s, t)
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(Concat(s, t), u) → CONCAT(t, u)
CONCAT(Cons_usual(x, m, s), t) → TERM_SUB(m, t)
CONCAT(Cons_usual(x, m, s), t) → CONCAT(s, t)
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.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 1 less node.

(4) Complex Obligation (AND)

(5) Obligation:

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

SUM_SUB(xi, Cons_usual(y, m, s)) → SUM_SUB(xi, s)
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.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
SUM_SUB(x0, x1, x2) = SUM_SUB(x2)

Tags:
SUM_SUB has argument tags [2,3,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Cons_sum(x₁, x₂, x₃)) = x₃
POL(Cons_usual(x₁, x₂, x₃)) = 1 + x₃
POL(SUM_SUB(x₁, x₂)) = x₁

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
SUM_SUB(x0, x1, x2) = SUM_SUB(x0, x2)

Tags:
SUM_SUB has argument tags [3,2,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Cons_sum(x₁, x₂, x₃)) = 1 + x₃
POL(SUM_SUB(x₁, x₂)) = 0

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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

TERM_SUB(Term_var(x), Cons_sum(xi, k, s)) → TERM_SUB(Term_var(x), s)
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.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x0, x2)

Tags:
TERM_SUB has argument tags [3,0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Cons_sum(x₁, x₂, x₃)) = 1 + x₃
POL(Cons_usual(x₁, x₂, x₃)) = x₃
POL(TERM_SUB(x₁, x₂)) = 0
POL(Term_var(x₁)) = x₁

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x2)

Tags:
TERM_SUB has argument tags [0,3,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Cons_usual(x₁, x₂, x₃)) = 1 + x₃
POL(TERM_SUB(x₁, x₂)) = 1
POL(Term_var(x₁)) = x₁

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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.

(17) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(18) TRUE

(19) Obligation:

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

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TERM_SUB(Term_sub(m, s), t) → CONCAT(s, t)
CONCAT(Cons_usual(x, m, s), t) → TERM_SUB(m, t)
CONCAT(Cons_usual(x, m, s), t) → CONCAT(s, t)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
FROZEN(x0, x1, x2, x3, x4) = FROZEN(x0, x1, x2, x3)
TERM_SUB(x0, x1, x2) = TERM_SUB(x0)
CONCAT(x0, x1, x2) = CONCAT(x0, x1)

Tags:
FROZEN has argument tags [0,8,0,8,7] and root tag 1
TERM_SUB has argument tags [0,2,0] and root tag 1
CONCAT has argument tags [1,0,5] and root tag 3

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(CONCAT(x₁, x₂)) = 0
POL(Case(x₁, x₂, x₃)) = x₁ + x₃
POL(Concat(x₁, x₂)) = x₁ + x₂
POL(Cons_sum(x₁, x₂, x₃)) = x₃
POL(Cons_usual(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(FROZEN(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₃
POL(Frozen(x₁, x₂, x₃, x₄)) = 0
POL(Id) = 1
POL(Left) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = 0
POL(Sum_sub(x₁, x₂)) = 0
POL(Sum_term_var(x₁)) = 0
POL(TERM_SUB(x₁, x₂)) = 1 + x₁
POL(Term_app(x₁, x₂)) = x₁ + x₂
POL(Term_inl(x₁)) = x₁
POL(Term_inr(x₁)) = x₁
POL(Term_pair(x₁, x₂)) = x₁ + x₂
POL(Term_sub(x₁, x₂)) = x₁ + x₂
POL(Term_var(x₁)) = 1

The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

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

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

(22) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(23) Complex Obligation (AND)

(24) Obligation:

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

CONCAT(Concat(s, t), u) → CONCAT(t, u)
CONCAT(Concat(s, t), u) → CONCAT(s, Concat(t, u))
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.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
CONCAT(x0, x1, x2) = CONCAT(x1)

Tags:
CONCAT has argument tags [3,0,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(CONCAT(x₁, x₂)) = 1 + x₂
POL(Case(x₁, x₂, x₃)) = 0
POL(Concat(x₁, x₂)) = x₁ + x₂
POL(Cons_sum(x₁, x₂, x₃)) = 1 + x₃
POL(Cons_usual(x₁, x₂, x₃)) = 1
POL(Frozen(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(Id) = 0
POL(Left) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = 0
POL(Sum_sub(x₁, x₂)) = 0
POL(Sum_term_var(x₁)) = 1 + x₁
POL(Term_app(x₁, x₂)) = 1
POL(Term_inl(x₁)) = 1
POL(Term_inr(x₁)) = 1
POL(Term_pair(x₁, x₂)) = 1
POL(Term_sub(x₁, x₂)) = 0
POL(Term_var(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented:

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

(26) Obligation:

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

CONCAT(Concat(s, t), u) → CONCAT(t, u)
CONCAT(Concat(s, t), u) → CONCAT(s, Concat(t, u))

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.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONCAT(Concat(s, t), u) → CONCAT(t, u)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
CONCAT(x0, x1, x2) = CONCAT(x0)

Tags:
CONCAT has argument tags [0,2,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(CONCAT(x₁, x₂)) = x₁ + x₂
POL(Case(x₁, x₂, x₃)) = 0
POL(Concat(x₁, x₂)) = 1 + x₁ + x₂
POL(Cons_sum(x₁, x₂, x₃)) = x₁ + x₂
POL(Cons_usual(x₁, x₂, x₃)) = 1 + x₁
POL(Frozen(x₁, x₂, x₃, x₄)) = x₁ + x₂
POL(Id) = 0
POL(Left) = 1
POL(Right) = 1
POL(Sum_constant(x₁)) = 1 + x₁
POL(Sum_sub(x₁, x₂)) = x₁ + x₂
POL(Sum_term_var(x₁)) = 1
POL(Term_app(x₁, x₂)) = 1
POL(Term_inl(x₁)) = 1
POL(Term_inr(x₁)) = 0
POL(Term_pair(x₁, x₂)) = 1
POL(Term_sub(x₁, x₂)) = 0
POL(Term_var(x₁)) = 1 + x₁

The following usable rules [FROCOS05] were oriented:

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

(28) Obligation:

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

CONCAT(Concat(s, t), u) → CONCAT(s, Concat(t, u))

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.

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONCAT(Concat(s, t), u) → CONCAT(s, Concat(t, u))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
CONCAT(x0, x1, x2) = CONCAT(x1)

Tags:
CONCAT has argument tags [3,3,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(CONCAT(x₁, x₂)) = 1
POL(Case(x₁, x₂, x₃)) = 0
POL(Concat(x₁, x₂)) = 1 + x₁
POL(Cons_sum(x₁, x₂, x₃)) = 0
POL(Cons_usual(x₁, x₂, x₃)) = 0
POL(Frozen(x₁, x₂, x₃, x₄)) = x₄
POL(Id) = 0
POL(Left) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = 0
POL(Sum_sub(x₁, x₂)) = x₁
POL(Sum_term_var(x₁)) = 1
POL(Term_app(x₁, x₂)) = 1
POL(Term_inl(x₁)) = 0
POL(Term_inr(x₁)) = 0
POL(Term_pair(x₁, x₂)) = 0
POL(Term_sub(x₁, x₂)) = 0
POL(Term_var(x₁)) = 0

The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

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.

(31) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(32) TRUE

(33) Obligation:

Q DP problem:
The TRS P 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)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_sub(m, s), t) → TERM_SUB(m, Concat(s, t))
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, 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.

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TERM_SUB(Term_app(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_app(m, n), s) → TERM_SUB(n, s)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x0, x2)
FROZEN(x0, x1, x2, x3, x4) = FROZEN(x0, x1, x2)

Tags:
TERM_SUB has argument tags [4,4,5] and root tag 0
FROZEN has argument tags [5,4,4,3,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Case(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(Concat(x₁, x₂)) = x₂
POL(Cons_sum(x₁, x₂, x₃)) = x₃
POL(Cons_usual(x₁, x₂, x₃)) = x₃
POL(FROZEN(x₁, x₂, x₃, x₄)) = x₁ + x₃ + x₄
POL(Frozen(x₁, x₂, x₃, x₄)) = 0
POL(Id) = 1
POL(Left) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = 0
POL(Sum_sub(x₁, x₂)) = 1 + x₂
POL(Sum_term_var(x₁)) = 0
POL(TERM_SUB(x₁, x₂)) = x₁ + x₂
POL(Term_app(x₁, x₂)) = 1 + x₁ + x₂
POL(Term_inl(x₁)) = x₁
POL(Term_inr(x₁)) = x₁
POL(Term_pair(x₁, x₂)) = x₁ + x₂
POL(Term_sub(x₁, x₂)) = x₁
POL(Term_var(x₁)) = 0

The following usable rules [FROCOS05] were oriented:

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

(35) Obligation:

Q DP problem:
The TRS P 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)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_sub(m, s), t) → TERM_SUB(m, Concat(s, t))
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, 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.

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TERM_SUB(Term_sub(m, s), t) → TERM_SUB(m, Concat(s, t))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x0)
FROZEN(x0, x1, x2, x3, x4) = FROZEN(x0)

Tags:
TERM_SUB has argument tags [0,0,0] and root tag 1
FROZEN has argument tags [0,0,0,0,0] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Case(x₁, x₂, x₃)) = x₁ + x₃
POL(Concat(x₁, x₂)) = x₂
POL(Cons_sum(x₁, x₂, x₃)) = 0
POL(Cons_usual(x₁, x₂, x₃)) = 1 + x₃
POL(FROZEN(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₃
POL(Frozen(x₁, x₂, x₃, x₄)) = x₄
POL(Id) = 0
POL(Left) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = x₁
POL(Sum_sub(x₁, x₂)) = 0
POL(Sum_term_var(x₁)) = x₁
POL(TERM_SUB(x₁, x₂)) = 1 + x₁
POL(Term_app(x₁, x₂)) = 1
POL(Term_inl(x₁)) = x₁
POL(Term_inr(x₁)) = x₁
POL(Term_pair(x₁, x₂)) = x₁ + x₂
POL(Term_sub(x₁, x₂)) = 1 + x₁
POL(Term_var(x₁)) = 1

The following usable rules [FROCOS05] were oriented: none

(37) Obligation:

Q DP problem:
The TRS P 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)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
TERM_SUB(Term_inr(m), s) → TERM_SUB(m, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, 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.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x0, x1, x2)
FROZEN(x0, x1, x2, x3, x4) = FROZEN(x0, x1)

Tags:
TERM_SUB has argument tags [0,0,0] and root tag 1
FROZEN has argument tags [0,0,0,2,4] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Case(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(Cons_sum(x₁, x₂, x₃)) = 0
POL(Cons_usual(x₁, x₂, x₃)) = 0
POL(FROZEN(x₁, x₂, x₃, x₄)) = x₁ + x₃ + x₄
POL(Id) = 0
POL(Left) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = 0
POL(Sum_sub(x₁, x₂)) = 0
POL(Sum_term_var(x₁)) = x₁
POL(TERM_SUB(x₁, x₂)) = x₁ + x₂
POL(Term_inl(x₁)) = x₁
POL(Term_inr(x₁)) = 1 + x₁
POL(Term_pair(x₁, x₂)) = x₁ + x₂

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

Q DP problem:
The TRS P 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)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
TERM_SUB(Term_inl(m), s) → TERM_SUB(m, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, 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.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

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.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x1)
FROZEN(x0, x1, x2, x3, x4) = FROZEN(x0, x1)

Tags:
TERM_SUB has argument tags [0,4,4] and root tag 0
FROZEN has argument tags [4,2,0,0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Case(x₁, x₂, x₃)) = x₁ + x₃
POL(Cons_sum(x₁, x₂, x₃)) = 0
POL(Cons_usual(x₁, x₂, x₃)) = 0
POL(FROZEN(x₁, x₂, x₃, x₄)) = x₁ + x₃
POL(Id) = 0
POL(Left) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = 1
POL(Sum_sub(x₁, x₂)) = 1 + x₁
POL(Sum_term_var(x₁)) = x₁
POL(TERM_SUB(x₁, x₂)) = x₁ + x₂
POL(Term_inl(x₁)) = 1 + x₁
POL(Term_pair(x₁, x₂)) = x₁ + x₂

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

Q DP problem:
The TRS P 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)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, 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.

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROZEN(m, Sum_constant(Left), n, s) → TERM_SUB(m, s)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x1)
FROZEN(x0, x1, x2, x3, x4) = FROZEN(x0, x1, x2, x3)

Tags:
TERM_SUB has argument tags [0,4,4] and root tag 0
FROZEN has argument tags [2,4,4,4,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Case(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(Cons_sum(x₁, x₂, x₃)) = 0
POL(Cons_usual(x₁, x₂, x₃)) = 0
POL(FROZEN(x₁, x₂, x₃, x₄)) = x₁ + x₂
POL(Id) = 0
POL(Left) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = 1
POL(Sum_sub(x₁, x₂)) = 1 + x₁
POL(Sum_term_var(x₁)) = 0
POL(TERM_SUB(x₁, x₂)) = 1 + x₁ + x₂
POL(Term_pair(x₁, x₂)) = x₁ + x₂

The following usable rules [FROCOS05] were oriented:

Sum_sub(xi, Id) → Sum_term_var(xi)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_constant(k)
Sum_sub(xi, Cons_usual(y, m, s)) → Sum_sub(xi, s)
Sum_sub(xi, Cons_sum(psi, k, s)) → Sum_sub(xi, s)

(43) Obligation:

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

TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, 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.

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(m, s)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x0, x1, x2)
FROZEN(x0, x1, x2, x3, x4) = FROZEN(x0, x1, x2, x4)

Tags:
TERM_SUB has argument tags [0,4,4] and root tag 1
FROZEN has argument tags [4,6,4,4,5] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Case(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(Cons_sum(x₁, x₂, x₃)) = 0
POL(Cons_usual(x₁, x₂, x₃)) = 0
POL(FROZEN(x₁, x₂, x₃, x₄)) = x₁ + x₃ + x₄
POL(Id) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = 1
POL(Sum_sub(x₁, x₂)) = 1
POL(Sum_term_var(x₁)) = 0
POL(TERM_SUB(x₁, x₂)) = x₁ + x₂
POL(Term_pair(x₁, x₂)) = x₁ + x₂

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

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

TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, 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.

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TERM_SUB(Term_pair(m, n), s) → TERM_SUB(m, s)
TERM_SUB(Term_pair(m, n), s) → TERM_SUB(n, s)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x1)
FROZEN(x0, x1, x2, x3, x4) = FROZEN(x1, x3)

Tags:
TERM_SUB has argument tags [2,0,1] and root tag 0
FROZEN has argument tags [1,0,4,0,5] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Case(x₁, x₂, x₃)) = x₁ + x₃
POL(Cons_sum(x₁, x₂, x₃)) = 0
POL(Cons_usual(x₁, x₂, x₃)) = 0
POL(FROZEN(x₁, x₂, x₃, x₄)) = x₁ + x₄
POL(Id) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = 0
POL(Sum_sub(x₁, x₂)) = 0
POL(Sum_term_var(x₁)) = x₁
POL(TERM_SUB(x₁, x₂)) = x₁
POL(Term_pair(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [FROCOS05] were oriented: none

(47) Obligation:

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

TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)
FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, 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.

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TERM_SUB(Case(m, xi, n), s) → FROZEN(m, Sum_sub(xi, s), n, s)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TERM_SUB(x0, x1, x2) = TERM_SUB(x0, x2)
FROZEN(x0, x1, x2, x3, x4) = FROZEN(x0)

Tags:
TERM_SUB has argument tags [0,0,1] and root tag 0
FROZEN has argument tags [1,7,3,3,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(Case(x₁, x₂, x₃)) = 1 + x₃
POL(Cons_sum(x₁, x₂, x₃)) = 0
POL(Cons_usual(x₁, x₂, x₃)) = 0
POL(FROZEN(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(Id) = 0
POL(Right) = 0
POL(Sum_constant(x₁)) = x₁
POL(Sum_sub(x₁, x₂)) = 1 + x₁
POL(Sum_term_var(x₁)) = 0
POL(TERM_SUB(x₁, x₂)) = x₁ + x₂

The following usable rules [FROCOS05] were oriented: none

(49) Obligation:

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

FROZEN(m, Sum_constant(Right), n, s) → TERM_SUB(n, s)
FROZEN(m, Sum_term_var(xi), n, s) → TERM_SUB(n, 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.

(50) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) QDPOrderProof (EQUIVALENT transformation)

(9) Obligation:

(10) PisEmptyProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) PisEmptyProof (EQUIVALENT transformation)

(18) TRUE

(19) Obligation:

(20) QDPOrderProof (EQUIVALENT transformation)

(21) Obligation:

(22) DependencyGraphProof (EQUIVALENT transformation)

(23) Complex Obligation (AND)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) QDPOrderProof (EQUIVALENT transformation)

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) PisEmptyProof (EQUIVALENT transformation)

(32) TRUE

(33) Obligation:

(34) QDPOrderProof (EQUIVALENT transformation)

(35) Obligation:

(36) QDPOrderProof (EQUIVALENT transformation)

(37) Obligation:

(38) QDPOrderProof (EQUIVALENT transformation)

(39) Obligation:

(40) QDPOrderProof (EQUIVALENT transformation)

(41) Obligation:

(42) QDPOrderProof (EQUIVALENT transformation)

(43) Obligation:

(44) QDPOrderProof (EQUIVALENT transformation)

(45) Obligation:

(46) QDPOrderProof (EQUIVALENT transformation)

(47) Obligation:

(48) QDPOrderProof (EQUIVALENT transformation)

(49) Obligation:

(50) DependencyGraphProof (EQUIVALENT transformation)

(51) TRUE