Termination w.r.t. Q proof of /tmp/tmpsQmth1/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 QDPOrderProof (⇔)

        ↳23 QDP

        ↳24 PisEmptyProof (⇔)

        ↳25 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_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)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
SUM_SUB(x1, x2) = SUM_SUB(x2)
Cons_usual(x1, x2, x3) = x3
Cons_sum(x1, x2, x3) = Cons_sum(x3)

Recursive path order with status [RPO].
Quasi-Precedence:

[SUM_SUB₁, Cons_sum₁]

Status:

SUM_SUB₁: [1]
Cons_sum₁: multiset

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_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.

(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_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(x0, x2)

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

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
SUM_SUB(x1, x2) = SUM_SUB(x2)
Cons_usual(x1, x2, x3) = Cons_usual(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:

[SUM_SUB₁, Cons_usual₃]

Status:

SUM_SUB₁: multiset
Cons_usual₃: multiset

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_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(x1, x2)

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

Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
TERM_SUB(x1, x2) = x2
Term_var(x1) = Term_var(x1)
Cons_sum(x1, x2, x3) = x3
Cons_usual(x1, x2, x3) = Cons_usual(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

Term_var₁: multiset
Cons_usual₃: multiset

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_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.

(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_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, x1)

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

Comparison: DMS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
TERM_SUB(x1, x2) = x2
Term_var(x1) = Term_var
Cons_sum(x1, x2, x3) = Cons_sum(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

Term_var: multiset
Cons_sum₃: [2,1,3]

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.

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_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 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)
TERM_SUB(x0, x1, x2) = TERM_SUB(x0, x1)
CONCAT(x0, x1, x2) = CONCAT(x1)

Tags:
FROZEN has argument tags [12,4,7,0,2] and root tag 1
TERM_SUB has argument tags [8,12,3] and root tag 2
CONCAT has argument tags [14,1,12] and root tag 3

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
FROZEN(x1, x2, x3, x4) = FROZEN(x1, x2, x3)
Sum_constant(x1) = Sum_constant
Left = Left
TERM_SUB(x1, x2) = TERM_SUB(x1)
Case(x1, x2, x3) = Case(x1, x2, x3)
Sum_sub(x1, x2) = x1
Right = Right
Term_app(x1, x2) = Term_app(x1, x2)
Term_pair(x1, x2) = Term_pair(x1, x2)
Term_inl(x1) = x1
Term_inr(x1) = Term_inr(x1)
Term_sub(x1, x2) = Term_sub(x1, x2)
Concat(x1, x2) = Concat(x1, x2)
CONCAT(x1, x2) = CONCAT
Cons_usual(x1, x2, x3) = Cons_usual(x1, x2, x3)
Cons_sum(x1, x2, x3) = Cons_sum(x1, x3)
Sum_term_var(x1) = x1
Id = Id
Frozen(x1, x2, x3, x4) = Frozen(x2, x3, x4)
Term_var(x1) = Term_var

Recursive path order with status [RPO].
Quasi-Precedence:

Left > [TERM_SUB₁, CONCAT, Cons_usual₃] > [FROZEN₃, Case₃] > [Sum_constant, Term_app₂, Cons_sum₂]
Left > [TERM_SUB₁, CONCAT, Cons_usual₃] > Concat₂ > [Right, Term_sub₂] > [Sum_constant, Term_app₂, Cons_sum₂]
Term_pair₂ > [Right, Term_sub₂] > [Sum_constant, Term_app₂, Cons_sum₂]
Term_inr₁ > [Right, Term_sub₂] > [Sum_constant, Term_app₂, Cons_sum₂]
Id > [Sum_constant, Term_app₂, Cons_sum₂]
Frozen₃ > [FROZEN₃, Case₃] > [Sum_constant, Term_app₂, Cons_sum₂]
Frozen₃ > [Right, Term_sub₂] > [Sum_constant, Term_app₂, Cons_sum₂]
Term_var > [Right, Term_sub₂] > [Sum_constant, Term_app₂, Cons_sum₂]

Status:

FROZEN₃: multiset
Sum_constant: multiset
Left: multiset
TERM_SUB₁: [1]
Case₃: multiset
Right: multiset
Term_app₂: multiset
Term_pair₂: [1,2]
Term_inr₁: [1]
Term_sub₂: multiset
Concat₂: multiset
CONCAT: []
Cons_usual₃: multiset
Cons_sum₂: [2,1]
Id: multiset
Frozen₃: multiset
Term_var: multiset

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)

(21) Obligation:

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.

(22) 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)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
TERM_SUB(x1, x2) = TERM_SUB
Term_inl(x1) = Term_inl(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

trivial

Status:

TERM_SUB: multiset
Term_inl₁: [1]

The following usable rules [FROCOS05] were oriented: none

(23) 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.

(24) PisEmptyProof (EQUIVALENT transformation)

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

(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) QDPOrderProof (EQUIVALENT transformation)

(23) Obligation:

(24) PisEmptyProof (EQUIVALENT transformation)

(25) TRUE