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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 QDPOrderProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 QDP
↳7 QDPOrderProof (⇔)
↳8 QDP
↳9 DependencyGraphProof (⇔)
↳10 AND
  ↳11 QDP
    ↳12 UsableRulesProof (⇔)
    ↳13 QDP
    ↳14 QDPSizeChangeProof (⇔)
    ↳15 TRUE
  ↳16 QDP
    ↳17 QDPOrderProof (⇔)
    ↳18 QDP
    ↳19 PisEmptyProof (⇔)
    ↳20 TRUE

(0) Obligation:

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

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__from(X) → from(X)
a__after(X1, X2) → after(X1, X2)

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:

A__FROM(X) → MARK(X)
A__AFTER(0, XS) → MARK(XS)
A__AFTER(s(N), cons(X, XS)) → A__AFTER(mark(N), mark(XS))
A__AFTER(s(N), cons(X, XS)) → MARK(N)
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(after(X1, X2)) → A__AFTER(mark(X1), mark(X2))
MARK(after(X1, X2)) → MARK(X1)
MARK(after(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__from(X) → from(X)
a__after(X1, X2) → after(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__FROM(x1)) = 0A + 0A·x1

POL(MARK(x1)) = 0A + 0A·x1

POL(A__AFTER(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(0) = 0A

POL(s(x1)) = -I + 0A·x1

POL(cons(x1, x2)) = 0A + 0A·x1 + 0A·x2

POL(mark(x1)) = 0A + 0A·x1

POL(from(x1)) = 1A + 1A·x1

POL(after(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(a__after(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(a__from(x1)) = 1A + 1A·x1

The following usable rules [FROCOS05] were oriented:

a__after(X1, X2) → after(X1, X2)
mark(0) → 0
a__from(X) → from(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
a__after(0, XS) → mark(XS)
mark(from(X)) → a__from(mark(X))
a__from(X) → cons(mark(X), from(s(X)))

(4) Obligation:

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

A__FROM(X) → MARK(X)
A__AFTER(0, XS) → MARK(XS)
A__AFTER(s(N), cons(X, XS)) → A__AFTER(mark(N), mark(XS))
A__AFTER(s(N), cons(X, XS)) → MARK(N)
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
MARK(after(X1, X2)) → A__AFTER(mark(X1), mark(X2))
MARK(after(X1, X2)) → MARK(X1)
MARK(after(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__from(X) → from(X)
a__after(X1, X2) → after(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

MARK(after(X1, X2)) → A__AFTER(mark(X1), mark(X2))
A__AFTER(0, XS) → MARK(XS)
MARK(after(X1, X2)) → MARK(X1)
MARK(after(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__AFTER(s(N), cons(X, XS)) → A__AFTER(mark(N), mark(XS))
A__AFTER(s(N), cons(X, XS)) → MARK(N)
A__AFTER(s(N), cons(X, XS)) → MARK(XS)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__from(X) → from(X)
a__after(X1, X2) → after(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(after(X1, X2)) → A__AFTER(mark(X1), mark(X2))
MARK(after(X1, X2)) → MARK(X1)
MARK(after(X1, X2)) → MARK(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(MARK(x1)) = -I + 0A·x1

POL(after(x1, x2)) = -I + 5A·x1 + 1A·x2

POL(A__AFTER(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(mark(x1)) = -I + 0A·x1

POL(0) = 0A

POL(cons(x1, x2)) = -I + 0A·x1 + 0A·x2

POL(s(x1)) = -I + 0A·x1

POL(a__after(x1, x2)) = -I + 5A·x1 + 1A·x2

POL(a__from(x1)) = -I + 0A·x1

POL(from(x1)) = -I + 0A·x1

The following usable rules [FROCOS05] were oriented:

a__after(X1, X2) → after(X1, X2)
mark(0) → 0
a__from(X) → from(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
a__after(0, XS) → mark(XS)
mark(from(X)) → a__from(mark(X))
a__from(X) → cons(mark(X), from(s(X)))

(8) Obligation:

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

A__AFTER(0, XS) → MARK(XS)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__AFTER(s(N), cons(X, XS)) → A__AFTER(mark(N), mark(XS))
A__AFTER(s(N), cons(X, XS)) → MARK(N)
A__AFTER(s(N), cons(X, XS)) → MARK(XS)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__from(X) → from(X)
a__after(X1, X2) → after(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__from(X) → from(X)
a__after(X1, X2) → after(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(13) Obligation:

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

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • MARK(s(X)) → MARK(X)
    The graph contains the following edges 1 > 1

  • MARK(cons(X1, X2)) → MARK(X1)
    The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

A__AFTER(s(N), cons(X, XS)) → A__AFTER(mark(N), mark(XS))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__from(X) → from(X)
a__after(X1, X2) → after(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__AFTER(s(N), cons(X, XS)) → A__AFTER(mark(N), mark(XS))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(A__AFTER(x1, x2)) = x1   
POL(a__after(x1, x2)) = x2   
POL(a__from(x1)) = 0   
POL(after(x1, x2)) = x2   
POL(cons(x1, x2)) = x2   
POL(from(x1)) = 0   
POL(mark(x1)) = x1   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
a__after(0, XS) → mark(XS)
mark(from(X)) → a__from(mark(X))
a__from(X) → cons(mark(X), from(s(X)))
a__after(X1, X2) → after(X1, X2)
mark(0) → 0
a__from(X) → from(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))

(18) Obligation:

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

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__from(X) → from(X)
a__after(X1, X2) → after(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE