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 PisEmptyProof (⇔)
    ↳9 TRUE
  ↳10 QDP
    ↳11 QDPOrderProof (⇔)
    ↳12 QDP
    ↳13 PisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 QDP
    ↳16 QDPOrderProof (⇔)
    ↳17 QDP
    ↳18 DependencyGraphProof (⇔)
    ↳19 AND
      ↳20 QDP
        ↳21 QDPOrderProof (⇔)
        ↳22 QDP
        ↳23 PisEmptyProof (⇔)
        ↳24 TRUE
      ↳25 QDP
        ↳26 QDPOrderProof (⇔)
        ↳27 QDP
        ↳28 PisEmptyProof (⇔)
        ↳29 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)
SUM_SUB(xi, Cons_sum(psi, k, s)) → SUM_SUB(xi, s)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SUM_SUB(x1, x2)  =  SUM_SUB(x2)
Cons_usual(x1, x2, x3)  =  Cons_usual(x2, x3)
Cons_sum(x1, x2, x3)  =  Cons_sum(x2, x3)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(7) 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.

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) 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.

(11) 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)
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: Combined order from the following AFS and order.
TERM_SUB(x1, x2)  =  x2
Term_var(x1)  =  x1
Cons_sum(x1, x2, x3)  =  Cons_sum(x1, x2, x3)
Cons_usual(x1, x2, x3)  =  Cons_usual(x3)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(12) 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.

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) 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.

(16) 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)
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)
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: 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)  =  x1
Case(x1, x2, x3)  =  Case(x1, x2, x3)
Sum_sub(x1, x2)  =  Sum_sub
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)  =  x1
Cons_usual(x1, x2, x3)  =  Cons_usual(x2, x3)
Cons_sum(x1, x2, x3)  =  x3
Sum_term_var(x1)  =  Sum_term_var
Id  =  Id
Term_var(x1)  =  Term_var(x1)
Frozen(x1, x2, x3, x4)  =  Frozen(x2, x3, x4)

Lexicographic Path Order [LPO].
Precedence:
Left > [FROZEN3, Sumconstant, Case3, Sumsub, Termapp2, Termpair2, Terminr1, Termsub2, Concat2, Sumtermvar, Termvar1]
Right > [FROZEN3, Sumconstant, Case3, Sumsub, Termapp2, Termpair2, Terminr1, Termsub2, Concat2, Sumtermvar, Termvar1]
Consusual2 > [FROZEN3, Sumconstant, Case3, Sumsub, Termapp2, Termpair2, Terminr1, Termsub2, Concat2, Sumtermvar, Termvar1]
Id > [FROZEN3, Sumconstant, Case3, Sumsub, Termapp2, Termpair2, Terminr1, Termsub2, Concat2, Sumtermvar, Termvar1]
Frozen3 > [FROZEN3, Sumconstant, Case3, Sumsub, Termapp2, Termpair2, Terminr1, Termsub2, Concat2, Sumtermvar, Termvar1]


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)

(17) 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_inl(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.

(18) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(19) Complex Obligation (AND)

(20) Obligation:

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.

(21) 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: Combined order from the following AFS and order.
CONCAT(x1, x2)  =  CONCAT(x1, x2)
Cons_sum(x1, x2, x3)  =  Cons_sum(x2, x3)

Lexicographic Path Order [LPO].
Precedence:
Conssum2 > CONCAT2


The following usable rules [FROCOS05] were oriented: none

(22) 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.

(23) PisEmptyProof (EQUIVALENT transformation)

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

(24) TRUE

(25) 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.

(26) 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: Combined order from the following AFS and order.
TERM_SUB(x1, x2)  =  x1
Term_inl(x1)  =  Term_inl(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(27) 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.

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE