Termination w.r.t. Q proof of Transformed_CSR_04_Ex1_2_AEL03

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

0 QTRS

↳1 DependencyPairsProof (⇔, 42 ms)

↳2 QDP

↳3 TransformationProof (⇔, 0 ms)

↳4 QDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 QDP

↳7 QDPOrderProof (⇔, 137 ms)

↳8 QDP

↳9 QDPOrderProof (⇔, 134 ms)

↳10 QDP

↳11 QDPOrderProof (⇔, 66 ms)

↳12 QDP

↳13 DependencyGraphProof (⇔, 0 ms)

↳14 QDP

↳15 QDPOrderProof (⇔, 124 ms)

↳16 QDP

↳17 QDPOrderProof (⇔, 176 ms)

↳18 QDP

↳19 DependencyGraphProof (⇔, 0 ms)

↳20 QDP

↳21 QDPOrderProof (⇔, 110 ms)

↳22 QDP

↳23 DependencyGraphProof (⇔, 0 ms)

↳24 QDP

↳25 QDPOrderProof (⇔, 61 ms)

↳26 QDP

↳27 DependencyGraphProof (⇔, 0 ms)

↳28 AND

    ↳29 QDP

        ↳30 QDPOrderProof (⇔, 55 ms)

        ↳31 QDP

        ↳32 QDPOrderProof (⇔, 0 ms)

        ↳33 QDP

        ↳34 QDPOrderProof (⇔, 0 ms)

        ↳35 QDP

        ↳36 DependencyGraphProof (⇔, 0 ms)

        ↳37 AND

            ↳38 QDP

                ↳39 UsableRulesProof (⇔, 0 ms)

                ↳40 QDP

                ↳41 QDPSizeChangeProof (⇔, 0 ms)

                ↳42 YES

            ↳43 QDP

                ↳44 QDPOrderProof (⇔, 0 ms)

                ↳45 QDP

                ↳46 PisEmptyProof (⇔, 0 ms)

                ↳47 YES

    ↳48 QDP

        ↳49 QDPOrderProof (⇔, 75 ms)

        ↳50 QDP

        ↳51 PisEmptyProof (⇔, 0 ms)

        ↳52 YES

(0) Obligation:

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

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__PI(X) → A__2NDSPOS(mark(X), a__from(0))
A__PI(X) → MARK(X)
A__PI(X) → A__FROM(0)
A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__SQUARE(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(3) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule A__PI(X) → A__2NDSPOS(mark(X), a__from(0)) at position [1] we obtained the following new rules [LPAR04]:

A__PI(y0) → A__2NDSPOS(mark(y0), cons(mark(0), from(s(0)))) → A__PI(y0) → A__2NDSPOS(mark(y0), cons(mark(0), from(s(0))))
A__PI(y0) → A__2NDSPOS(mark(y0), from(0)) → A__PI(y0) → A__2NDSPOS(mark(y0), from(0))

(4) Obligation:

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

A__FROM(X) → MARK(X)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__PI(X) → MARK(X)
A__PI(X) → A__FROM(0)
A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__SQUARE(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PI(y0) → A__2NDSPOS(mark(y0), cons(mark(0), from(s(0))))
A__PI(y0) → A__2NDSPOS(mark(y0), from(0))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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 2 less nodes.

(6) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
A__PI(X) → MARK(X)
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__PI(X) → A__FROM(0)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MARK(pi(X)) → MARK(X)
A__PLUS(s(X), Y) → MARK(X)
MARK(square(X)) → MARK(X)
A__SQUARE(X) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(MARK(x₁)) = 4A + 0A · x₁

POL(from(x₁)) = 2A + 2A · x₁

POL(A__FROM(x₁)) = 4A + 2A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = 4A + 0A · x₁ + 3A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 4A + 0A · x₁ + 1A · x₂

POL(s(x₁)) = 0A + 0A · x₁

POL(cons(x₁, x₂)) = 0A + 1A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 4A + 0A · x₁ + 3A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 4A + 0A · x₁ + 1A · x₂

POL(pi(x₁)) = 5A + 1A · x₁

POL(A__PI(x₁)) = 4A + 1A · x₁

POL(plus(x₁, x₂)) = 3A + 5A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 3A + 5A · x₁ + 0A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = 3A + 0A · x₁ + 5A · x₂

POL(A__TIMES(x₁, x₂)) = 4A + 0A · x₁ + 5A · x₂

POL(a__times(x₁, x₂)) = 3A + 0A · x₁ + 5A · x₂

POL(square(x₁)) = 5A + 5A · x₁

POL(A__SQUARE(x₁)) = 5A + 5A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__from(x₁)) = 2A + 2A · x₁

POL(a__2ndspos(x₁, x₂)) = 4A + 0A · x₁ + 3A · x₂

POL(a__2ndsneg(x₁, x₂)) = 4A + 0A · x₁ + 3A · x₂

POL(a__pi(x₁)) = 5A + 1A · x₁

POL(a__plus(x₁, x₂)) = 3A + 5A · x₁ + 0A · x₂

POL(a__square(x₁)) = 5A + 5A · x₁

POL(nil) = 2A

POL(rnil) = 3A

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(8) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
A__PI(X) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__PI(X) → A__FROM(0)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

A__PI(X) → A__FROM(0)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(MARK(x₁)) = 3A + 0A · x₁

POL(from(x₁)) = 5A + 0A · x₁

POL(A__FROM(x₁)) = 4A + 0A · x₁

POL(mark(x₁)) = 5A + 0A · x₁

POL(2ndspos(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = 4A + 0A · x₁

POL(cons(x₁, x₂)) = 3A + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(pi(x₁)) = 5A + 0A · x₁

POL(A__PI(x₁)) = 5A + 0A · x₁

POL(plus(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 5A + -I · x₁ + 0A · x₂

POL(0) = 4A

POL(times(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(square(x₁)) = 5A + 0A · x₁

POL(A__SQUARE(x₁)) = 5A + 0A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__from(x₁)) = 5A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(a__2ndsneg(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 0A · x₁

POL(a__plus(x₁, x₂)) = 5A + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 5A + 0A · x₁

POL(nil) = 4A

POL(rnil) = 3A

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(10) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
A__PI(X) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

A__PI(X) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(MARK(x₁)) = 2A + 1A · x₁

POL(from(x₁)) = 1A + 1A · x₁

POL(A__FROM(x₁)) = 2A + 2A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 2A + 2A · x₁ + 2A · x₂

POL(s(x₁)) = 0A + 0A · x₁

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

POL(2ndsneg(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 2A + 2A · x₁ + 2A · x₂

POL(pi(x₁)) = 2A + 2A · x₁

POL(A__PI(x₁)) = 3A + 3A · x₁

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

POL(A__PLUS(x₁, x₂)) = 2A + -I · x₁ + 1A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(A__TIMES(x₁, x₂)) = 1A + 2A · x₁ + 2A · x₂

POL(a__times(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(square(x₁)) = 4A + 1A · x₁

POL(A__SQUARE(x₁)) = 3A + 2A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__from(x₁)) = 1A + 1A · x₁

POL(a__2ndspos(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(a__2ndsneg(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(a__pi(x₁)) = 2A + 2A · x₁

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

POL(a__square(x₁)) = 4A + 1A · x₁

POL(nil) = 0A

POL(rnil) = 1A

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(12) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(13) DependencyGraphProof (EQUIVALENT transformation)

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

(14) Obligation:

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

A__FROM(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X1)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__FROM(x₁)) = 1A + 0A · x₁

POL(MARK(x₁)) = 1A + 0A · x₁

POL(from(x₁)) = 2A + 0A · x₁

POL(mark(x₁)) = 1A + 0A · x₁

POL(2ndspos(x₁, x₂)) = 5A + 4A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 3A + 4A · x₁ + 0A · x₂

POL(s(x₁)) = 2A + 0A · x₁

POL(cons(x₁, x₂)) = 2A + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 5A + 4A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 3A + 4A · x₁ + 0A · x₂

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

POL(A__PLUS(x₁, x₂)) = 1A + -I · x₁ + 0A · x₂

POL(0) = 5A

POL(times(x₁, x₂)) = 2A + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 2A + 0A · x₁ + 0A · x₂

POL(square(x₁)) = 2A + 0A · x₁

POL(A__SQUARE(x₁)) = 2A + 0A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__from(x₁)) = 2A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 5A + 4A · x₁ + 0A · x₂

POL(a__2ndsneg(x₁, x₂)) = 5A + 4A · x₁ + 0A · x₂

POL(pi(x₁)) = 5A + 4A · x₁

POL(a__pi(x₁)) = 5A + 4A · x₁

POL(a__plus(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(a__square(x₁)) = 2A + 0A · x₁

POL(nil) = 4A

POL(rnil) = 5A

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(16) Obligation:

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

A__FROM(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MARK(from(X)) → MARK(X)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(cons(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__FROM(x₁)) = 4A + 3A · x₁

POL(MARK(x₁)) = 4A + 2A · x₁

POL(from(x₁)) = 5A + 1A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = 4A + 4A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 5A + 2A · x₁ + 2A · x₂

POL(s(x₁)) = 0A + 0A · x₁

POL(cons(x₁, x₂)) = 3A + 1A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 0A + 4A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 4A + 2A · x₁ + 2A · x₂

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

POL(A__PLUS(x₁, x₂)) = 4A + -I · x₁ + 2A · x₂

POL(0) = 2A

POL(times(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 4A + 2A · x₁ + 2A · x₂

POL(a__times(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(square(x₁)) = 3A + 2A · x₁

POL(A__SQUARE(x₁)) = 4A + 3A · x₁

POL(posrecip(x₁)) = 3A + 0A · x₁

POL(negrecip(x₁)) = -I + 1A · x₁

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

POL(a__from(x₁)) = 5A + 1A · x₁

POL(a__2ndspos(x₁, x₂)) = 4A + 4A · x₁ + 0A · x₂

POL(a__2ndsneg(x₁, x₂)) = 0A + 4A · x₁ + 0A · x₂

POL(pi(x₁)) = 5A + 5A · x₁

POL(a__pi(x₁)) = 5A + 5A · x₁

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

POL(a__square(x₁)) = 3A + 2A · x₁

POL(nil) = 3A

POL(rnil) = 5A

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(18) Obligation:

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

A__FROM(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

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

(20) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MARK(from(X)) → A__FROM(mark(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(MARK(x₁)) = 5A + 1A · x₁

POL(from(x₁)) = 5A + 1A · x₁

POL(A__FROM(x₁)) = 5A + 1A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 5A + -I · x₁ + 1A · x₂

POL(s(x₁)) = -I + 0A · x₁

POL(cons(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 5A + -I · x₁ + 1A · x₂

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(plus(x₁, x₂)) = 4A + 1A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 5A + 2A · x₁ + 1A · x₂

POL(0) = 1A

POL(times(x₁, x₂)) = 4A + 0A · x₁ + 1A · x₂

POL(A__TIMES(x₁, x₂)) = 5A + 1A · x₁ + 2A · x₂

POL(a__times(x₁, x₂)) = 4A + 0A · x₁ + 1A · x₂

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

POL(rcons(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(a__from(x₁)) = 5A + 1A · x₁

POL(a__2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(pi(x₁)) = 5A + 4A · x₁

POL(a__pi(x₁)) = 5A + 4A · x₁

POL(a__plus(x₁, x₂)) = 4A + 1A · x₁ + 0A · x₂

POL(square(x₁)) = 4A + 5A · x₁

POL(a__square(x₁)) = 4A + 5A · x₁

POL(nil) = 1A

POL(rnil) = 0A

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(22) Obligation:

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

A__FROM(X) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(24) Obligation:

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

A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → MARK(X1)
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(A__2NDSPOS(x₁, x₂)) = 3A + 0A · x₁ + 4A · x₂

POL(s(x₁)) = -I + 0A · x₁

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

POL(A__2NDSNEG(x₁, x₂)) = 3A + 0A · x₁ + 4A · x₂

POL(mark(x₁)) = -I + 0A · x₁

POL(MARK(x₁)) = 3A + 4A · x₁

POL(2ndspos(x₁, x₂)) = 2A + 0A · x₁ + 5A · x₂

POL(2ndsneg(x₁, x₂)) = 2A + 0A · x₁ + 5A · x₂

POL(plus(x₁, x₂)) = 0A + 1A · x₁ + 0A · x₂

POL(A__PLUS(x₁, x₂)) = 3A + -I · x₁ + 4A · x₂

POL(0) = 0A

POL(times(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(A__TIMES(x₁, x₂)) = 5A + 5A · x₁ + 5A · x₂

POL(a__times(x₁, x₂)) = 1A + 1A · x₁ + 1A · x₂

POL(posrecip(x₁)) = -I + 4A · x₁

POL(negrecip(x₁)) = -I + 4A · x₁

POL(rcons(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(from(x₁)) = -I + 0A · x₁

POL(a__from(x₁)) = -I + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 2A + 0A · x₁ + 5A · x₂

POL(a__2ndsneg(x₁, x₂)) = 2A + 0A · x₁ + 5A · x₂

POL(pi(x₁)) = 5A + 5A · x₁

POL(a__pi(x₁)) = 5A + 5A · x₁

POL(a__plus(x₁, x₂)) = 0A + 1A · x₁ + 0A · x₂

POL(square(x₁)) = 3A + 3A · x₁

POL(a__square(x₁)) = 3A + 3A · x₁

POL(nil) = 2A

POL(rnil) = 0A

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(26) Obligation:

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

A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

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

(28) Complex Obligation (AND)

(29) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MARK(negrecip(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( A__PLUS(x₁, x₂) ) = 2x₂

POL( A__TIMES(x₁, x₂) ) = 2

POL( mark(x₁) ) = x₁

POL( from(x₁) ) = max{0, -1}

POL( a__from(x₁) ) = 0

POL( 2ndspos(x₁, x₂) ) = x₂ + 1

POL( a__2ndspos(x₁, x₂) ) = x₂ + 1

POL( 2ndsneg(x₁, x₂) ) = 2x₂ + 1

POL( a__2ndsneg(x₁, x₂) ) = 2x₂ + 1

POL( pi(x₁) ) = x₁ + 2

POL( a__pi(x₁) ) = x₁ + 2

POL( plus(x₁, x₂) ) = x₂

POL( a__plus(x₁, x₂) ) = x₂

POL( 0 ) = 0

POL( times(x₁, x₂) ) = 1

POL( a__times(x₁, x₂) ) = 1

POL( s(x₁) ) = x₁

POL( square(x₁) ) = 1

POL( a__square(x₁) ) = 1

POL( posrecip(x₁) ) = x₁

POL( negrecip(x₁) ) = 2x₁ + 2

POL( nil ) = 0

POL( cons(x₁, x₂) ) = 2x₂

POL( rnil ) = 1

POL( rcons(x₁, x₂) ) = x₂

POL( MARK(x₁) ) = 2x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(31) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MARK(posrecip(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( A__PLUS(x₁, x₂) ) = x₂

POL( A__TIMES(x₁, x₂) ) = 0

POL( mark(x₁) ) = x₁

POL( from(x₁) ) = 1

POL( a__from(x₁) ) = 1

POL( 2ndspos(x₁, x₂) ) = 0

POL( a__2ndspos(x₁, x₂) ) = 0

POL( 2ndsneg(x₁, x₂) ) = 0

POL( a__2ndsneg(x₁, x₂) ) = max{0, -1}

POL( pi(x₁) ) = 0

POL( a__pi(x₁) ) = 0

POL( plus(x₁, x₂) ) = x₂

POL( a__plus(x₁, x₂) ) = x₂

POL( 0 ) = 0

POL( times(x₁, x₂) ) = 0

POL( a__times(x₁, x₂) ) = 0

POL( s(x₁) ) = x₁

POL( square(x₁) ) = 2x₁ + 1

POL( a__square(x₁) ) = 2x₁ + 1

POL( posrecip(x₁) ) = x₁ + 1

POL( negrecip(x₁) ) = max{0, -2}

POL( nil ) = 0

POL( cons(x₁, x₂) ) = 1

POL( rnil ) = 0

POL( rcons(x₁, x₂) ) = 2x₂

POL( MARK(x₁) ) = x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(33) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
MARK(s(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__PLUS(x1, x2) = A__PLUS(x2)
0 = 0
MARK(x1) = MARK(x1)
plus(x1, x2) = plus(x1, x2)
mark(x1) = x1
s(x1) = s(x1)
times(x1, x2) = times(x1, x2)
A__TIMES(x1, x2) = A__TIMES(x1, x2)
a__times(x1, x2) = a__times(x1, x2)
rcons(x1, x2) = x2
from(x1) = x1
a__from(x1) = x1
2ndspos(x1, x2) = 2ndspos(x1)
a__2ndspos(x1, x2) = a__2ndspos(x1)
2ndsneg(x1, x2) = 2ndsneg(x1)
a__2ndsneg(x1, x2) = a__2ndsneg(x1)
pi(x1) = pi(x1)
a__pi(x1) = a__pi(x1)
a__plus(x1, x2) = a__plus(x1, x2)
square(x1) = square(x1)
a__square(x1) = a__square(x1)
posrecip(x1) = posrecip
negrecip(x1) = negrecip
nil = nil
cons(x1, x2) = x1
rnil = rnil

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

[2ndspos₁, a_{_2ndspos₁}, 2ndsneg₁, a_{_2ndsneg₁}, pi₁, a_{_pi₁}] > [0, s₁, rnil]
[square₁, a_{_square₁}] > [times₂, A_{_TIMES₂}, a_{_times₂}] > [A_{_PLUS₁}, MARK₁] > [0, s₁, rnil]
[square₁, a_{_square₁}] > [times₂, A_{_TIMES₂}, a_{_times₂}] > [plus₂, a_{_plus₂}] > [0, s₁, rnil]
posrecip > [0, s₁, rnil]
negrecip > [0, s₁, rnil]
nil > [0, s₁, rnil]

Status:

A_{_PLUS₁}: [1]
0: multiset
MARK₁: [1]
plus₂: [1,2]
s₁: [1]
times₂: [1,2]
A_{_TIMES₂}: [1,2]
a_{_times₂}: [1,2]
2ndspos₁: multiset
a_{_2ndspos₁}: multiset
2ndsneg₁: multiset
a_{_2ndsneg₁}: multiset
pi₁: multiset
a_{_pi₁}: multiset
a_{_plus₂}: [1,2]
square₁: [1]
a_{_square₁}: [1]
posrecip: multiset
negrecip: []
nil: multiset
rnil: multiset

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(35) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)
MARK(rcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(36) DependencyGraphProof (EQUIVALENT transformation)

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

(37) Complex Obligation (AND)

(38) Obligation:

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

MARK(rcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(39) 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.

(40) Obligation:

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

MARK(rcons(X1, X2)) → MARK(X2)

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

(41) 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(rcons(X1, X2)) → MARK(X2)
The graph contains the following edges 1 > 1

(42) YES

(43) Obligation:

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

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__PLUS(x1, x2) = A__PLUS(x1, x2)
s(x1) = s(x1)
mark(x1) = x1
from(x1) = from
a__from(x1) = a__from
2ndspos(x1, x2) = 2ndspos(x2)
a__2ndspos(x1, x2) = a__2ndspos(x2)
2ndsneg(x1, x2) = 2ndsneg(x2)
a__2ndsneg(x1, x2) = a__2ndsneg(x2)
pi(x1) = pi(x1)
a__pi(x1) = a__pi(x1)
plus(x1, x2) = plus(x1, x2)
a__plus(x1, x2) = a__plus(x1, x2)
0 = 0
times(x1, x2) = times(x1, x2)
a__times(x1, x2) = a__times(x1, x2)
square(x1) = square(x1)
a__square(x1) = a__square(x1)
posrecip(x1) = posrecip
negrecip(x1) = negrecip
nil = nil
cons(x1, x2) = cons
rnil = rnil
rcons(x1, x2) = rcons

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

A_{_PLUS₂} > [from, a_{_from}, cons]
[2ndsneg₁, a_{_2ndsneg₁}] > [s₁, 2ndspos₁, a_{_2ndspos₁}, pi₁, a_{_pi₁}, negrecip] > rnil > [from, a_{_from}, cons]
[2ndsneg₁, a_{_2ndsneg₁}] > [s₁, 2ndspos₁, a_{_2ndspos₁}, pi₁, a_{_pi₁}, negrecip] > rcons > [from, a_{_from}, cons]
0 > rnil > [from, a_{_from}, cons]
[square₁, a_{_square₁}] > [times₂, a_{_times₂}] > [plus₂, a_{_plus₂}] > [s₁, 2ndspos₁, a_{_2ndspos₁}, pi₁, a_{_pi₁}, negrecip] > rnil > [from, a_{_from}, cons]
[square₁, a_{_square₁}] > [times₂, a_{_times₂}] > [plus₂, a_{_plus₂}] > [s₁, 2ndspos₁, a_{_2ndspos₁}, pi₁, a_{_pi₁}, negrecip] > rcons > [from, a_{_from}, cons]
posrecip > [from, a_{_from}, cons]
nil > [from, a_{_from}, cons]

Status:

A_{_PLUS₂}: [1,2]
s₁: [1]
from: []
a_{_from}: []
2ndspos₁: [1]
a_{_2ndspos₁}: [1]
2ndsneg₁: multiset
a_{_2ndsneg₁}: multiset
pi₁: [1]
a_{_pi₁}: [1]
plus₂: [1,2]
a_{_plus₂}: [1,2]
0: multiset
times₂: multiset
a_{_times₂}: multiset
square₁: [1]
a_{_square₁}: [1]
posrecip: multiset
negrecip: multiset
nil: multiset
cons: []
rnil: multiset
rcons: multiset

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(45) 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__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(46) PisEmptyProof (EQUIVALENT transformation)

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

(47) YES

(48) Obligation:

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

A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__2NDSNEG(x1, x2) = A__2NDSNEG(x1)
s(x1) = s(x1)
cons(x1, x2) = x1
A__2NDSPOS(x1, x2) = A__2NDSPOS(x1)
mark(x1) = x1
from(x1) = x1
a__from(x1) = x1
2ndspos(x1, x2) = x1
a__2ndspos(x1, x2) = x1
2ndsneg(x1, x2) = x1
a__2ndsneg(x1, x2) = x1
pi(x1) = pi(x1)
a__pi(x1) = a__pi(x1)
plus(x1, x2) = plus(x1, x2)
a__plus(x1, x2) = a__plus(x1, x2)
0 = 0
times(x1, x2) = times(x1, x2)
a__times(x1, x2) = a__times(x1, x2)
square(x1) = square(x1)
a__square(x1) = a__square(x1)
posrecip(x1) = posrecip
negrecip(x1) = negrecip(x1)
nil = nil
rnil = rnil
rcons(x1, x2) = rcons

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

[pi₁, a_{_pi₁}] > [A_{_2NDSNEG₁}, A_{_2NDSPOS₁}, 0, rnil]
[square₁, a_{_square₁}] > [times₂, a_{_times₂}] > [plus₂, a_{_plus₂}] > s₁ > negrecip₁ > [A_{_2NDSNEG₁}, A_{_2NDSPOS₁}, 0, rnil]
[square₁, a_{_square₁}] > [times₂, a_{_times₂}] > [plus₂, a_{_plus₂}] > s₁ > rcons > [A_{_2NDSNEG₁}, A_{_2NDSPOS₁}, 0, rnil]
posrecip > [A_{_2NDSNEG₁}, A_{_2NDSPOS₁}, 0, rnil]
nil > [A_{_2NDSNEG₁}, A_{_2NDSPOS₁}, 0, rnil]

Status:

A_{_2NDSNEG₁}: multiset
s₁: multiset
A_{_2NDSPOS₁}: multiset
pi₁: multiset
a_{_pi₁}: multiset
plus₂: [1,2]
a_{_plus₂}: [1,2]
0: multiset
times₂: multiset
a_{_times₂}: multiset
square₁: multiset
a_{_square₁}: multiset
posrecip: multiset
negrecip₁: multiset
nil: multiset
rnil: multiset
rcons: multiset

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__plus(0, Y) → mark(Y)
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
mark(square(X)) → a__square(mark(X))
a__square(X) → a__times(mark(X), mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(0, Z) → rnil
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(0, Z) → rnil
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(0, Y) → 0
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))

(50) 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__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(51) 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) TransformationProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyGraphProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) DependencyGraphProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Complex Obligation (AND)

(29) Obligation:

(30) QDPOrderProof (EQUIVALENT transformation)

(31) Obligation:

(32) QDPOrderProof (EQUIVALENT transformation)

(33) Obligation:

(34) QDPOrderProof (EQUIVALENT transformation)

(35) Obligation:

(36) DependencyGraphProof (EQUIVALENT transformation)

(37) Complex Obligation (AND)

(38) Obligation:

(39) UsableRulesProof (EQUIVALENT transformation)

(40) Obligation:

(41) QDPSizeChangeProof (EQUIVALENT transformation)

(42) YES

(43) Obligation:

(44) QDPOrderProof (EQUIVALENT transformation)

(45) Obligation:

(46) PisEmptyProof (EQUIVALENT transformation)

(47) YES

(48) Obligation:

(49) QDPOrderProof (EQUIVALENT transformation)

(50) Obligation:

(51) PisEmptyProof (EQUIVALENT transformation)

(52) YES