Termination w.r.t. Q proof of Transformed_CSR_04_Ex26_Luc03b

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

0 QTRS

↳1 DependencyPairsProof (⇔, 0 ms)

↳2 QDP

↳3 DependencyGraphProof (⇔, 0 ms)

↳4 QDP

↳5 QDPOrderProof (⇔, 140 ms)

↳6 QDP

↳7 QDPOrderProof (⇔, 81 ms)

↳8 QDP

↳9 QDPOrderProof (⇔, 103 ms)

↳10 QDP

↳11 DependencyGraphProof (⇔, 0 ms)

↳12 TRUE

(0) Obligation:

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

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
add(X1, X2) → n__add(X1, X2)
sqr(X) → n__sqr(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → 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:

TERMS(N) → SQR(N)
SQR(s(X)) → S(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
SQR(s(X)) → ACTIVATE(X)
DBL(s(X)) → S(n__s(n__dbl(activate(X))))
DBL(s(X)) → ACTIVATE(X)
ADD(s(X), Y) → S(n__add(activate(X), Y))
ADD(s(X), Y) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ACTIVATE(n__terms(X)) → TERMS(activate(X))
ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__sqr(X)) → SQR(activate(X))
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
ACTIVATE(n__dbl(X)) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
add(X1, X2) → n__add(X1, X2)
sqr(X) → n__sqr(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(4) Obligation:

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

SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → TERMS(activate(X))
TERMS(N) → SQR(N)
ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__sqr(X)) → SQR(activate(X))
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
add(X1, X2) → n__add(X1, X2)
sqr(X) → n__sqr(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

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

(5) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)

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

POL(SQR(x₁)) = 0A + 1A · x₁

POL(s(x₁)) = -I + 0A · x₁

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__terms(x₁)) = -I + 5A · x₁

POL(TERMS(x₁)) = 0A + 5A · x₁

POL(activate(x₁)) = -I + 0A · x₁

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

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

POL(n__sqr(x₁)) = -I + 3A · x₁

POL(n__dbl(x₁)) = -I + 1A · x₁

POL(DBL(x₁)) = 0A + 0A · x₁

POL(n__first(x₁, x₂)) = 1A + 2A · x₁ + 0A · x₂

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

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

POL(terms(x₁)) = -I + 5A · x₁

POL(n__s(x₁)) = -I + 0A · x₁

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

POL(sqr(x₁)) = -I + 3A · x₁

POL(dbl(x₁)) = -I + 1A · x₁

POL(first(x₁, x₂)) = 1A + 2A · x₁ + 0A · x₂

POL(0) = 0A

POL(nil) = 1A

POL(recip(x₁)) = -I + 4A · x₁

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

activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
s(X) → n__s(X)
terms(X) → n__terms(X)
add(0, X) → X
add(X1, X2) → n__add(X1, X2)
sqr(0) → 0
sqr(X) → n__sqr(X)
dbl(0) → 0
dbl(X) → n__dbl(X)
first(0, X) → nil
first(X1, X2) → n__first(X1, X2)
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(s(X), Y) → s(n__add(activate(X), Y))
terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))

(6) Obligation:

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

SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → TERMS(activate(X))
TERMS(N) → SQR(N)
ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__sqr(X)) → SQR(activate(X))
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
add(X1, X2) → n__add(X1, X2)
sqr(X) → n__sqr(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → 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.

ACTIVATE(n__terms(X)) → TERMS(activate(X))
TERMS(N) → SQR(N)
ACTIVATE(n__terms(X)) → ACTIVATE(X)

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

POL(SQR(x₁)) = 2A + 1A · x₁

POL(s(x₁)) = -I + 0A · x₁

POL(ACTIVATE(x₁)) = 2A + 0A · x₁

POL(n__terms(x₁)) = 5A + 5A · x₁

POL(TERMS(x₁)) = 3A + 4A · x₁

POL(activate(x₁)) = -I + 0A · x₁

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

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

POL(n__sqr(x₁)) = -I + 1A · x₁

POL(n__dbl(x₁)) = -I + 1A · x₁

POL(DBL(x₁)) = 2A + 0A · x₁

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

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

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

POL(terms(x₁)) = 5A + 5A · x₁

POL(n__s(x₁)) = -I + 0A · x₁

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

POL(sqr(x₁)) = -I + 1A · x₁

POL(dbl(x₁)) = -I + 1A · x₁

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

POL(0) = 3A

POL(nil) = 1A

POL(recip(x₁)) = -I + 3A · x₁

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

activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
s(X) → n__s(X)
terms(X) → n__terms(X)
add(0, X) → X
add(X1, X2) → n__add(X1, X2)
sqr(0) → 0
sqr(X) → n__sqr(X)
dbl(0) → 0
dbl(X) → n__dbl(X)
first(0, X) → nil
first(X1, X2) → n__first(X1, X2)
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(s(X), Y) → s(n__add(activate(X), Y))
terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))

(8) Obligation:

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

SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__sqr(X)) → SQR(activate(X))
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
add(X1, X2) → n__add(X1, X2)
sqr(X) → n__sqr(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → 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.

SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__sqr(X)) → SQR(activate(X))
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SQR(x1) = SQR(x1)
s(x1) = s(x1)
ACTIVATE(x1) = ACTIVATE(x1)
n__add(x1, x2) = n__add(x1, x2)
ADD(x1, x2) = x1
activate(x1) = x1
n__sqr(x1) = n__sqr(x1)
n__dbl(x1) = n__dbl(x1)
DBL(x1) = DBL(x1)
n__first(x1, x2) = n__first(x1, x2)
FIRST(x1, x2) = FIRST(x1, x2)
cons(x1, x2) = x2
n__terms(x1) = n__terms
terms(x1) = terms
n__s(x1) = n__s(x1)
add(x1, x2) = add(x1, x2)
sqr(x1) = sqr(x1)
dbl(x1) = dbl(x1)
first(x1, x2) = first(x1, x2)
0 = 0
nil = nil
recip(x1) = recip

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

[n_{_sqr₁}, n_{_dbl₁}, sqr₁, dbl₁] > SQR₁ > [s₁, ACTIVATE₁, DBL₁, FIRST₂, n_{_s₁}, recip]
[n_{_sqr₁}, n_{_dbl₁}, sqr₁, dbl₁] > [n_{_add₂}, add₂] > [s₁, ACTIVATE₁, DBL₁, FIRST₂, n_{_s₁}, recip]
[n_{_sqr₁}, n_{_dbl₁}, sqr₁, dbl₁] > 0 > nil > [s₁, ACTIVATE₁, DBL₁, FIRST₂, n_{_s₁}, recip]
[n_{_first₂}, first₂] > nil > [s₁, ACTIVATE₁, DBL₁, FIRST₂, n_{_s₁}, recip]
[n_{_terms}, terms] > [s₁, ACTIVATE₁, DBL₁, FIRST₂, n_{_s₁}, recip]

Status:

SQR₁: [1]
s₁: [1]
ACTIVATE₁: [1]
n_{_add₂}: multiset
n_{_sqr₁}: [1]
n_{_dbl₁}: [1]
DBL₁: [1]
n_{_first₂}: multiset
FIRST₂: [2,1]
n_{_terms}: multiset
terms: multiset
n_{_s₁}: [1]
add₂: multiset
sqr₁: [1]
dbl₁: [1]
first₂: multiset
0: multiset
nil: multiset
recip: multiset

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

activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
s(X) → n__s(X)
terms(X) → n__terms(X)
add(0, X) → X
add(X1, X2) → n__add(X1, X2)
sqr(0) → 0
sqr(X) → n__sqr(X)
dbl(0) → 0
dbl(X) → n__dbl(X)
first(0, X) → nil
first(X1, X2) → n__first(X1, X2)
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(s(X), Y) → s(n__add(activate(X), Y))
terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))

(10) Obligation:

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

ADD(s(X), Y) → ACTIVATE(X)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
add(X1, X2) → n__add(X1, X2)
sqr(X) → n__sqr(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) DependencyGraphProof (EQUIVALENT transformation)

(12) TRUE