Termination w.r.t. Q proof of /tmp/tmpXmDZ7a/Ex26_Luc03b

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 QDPOrderProof (⇔)

↳8 QDP

↳9 DependencyGraphProof (⇔)

↳10 QDP

↳11 QDPOrderProof (⇔)

↳12 QDP

↳13 PisEmptyProof (⇔)

↳14 TRUE

(0) Obligation:

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

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), 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)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, 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)
TERMS(N) → S(N)
SQR(s(X)) → S(n__add(sqr(activate(X)), dbl(activate(X))))
SQR(s(X)) → SQR(activate(X))
SQR(s(X)) → ACTIVATE(X)
SQR(s(X)) → DBL(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(X)
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
ACTIVATE(n__s(X)) → S(X)
ACTIVATE(n__dbl(X)) → DBL(X)
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), 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)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, 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 5 less nodes.

(4) Obligation:

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

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

The TRS R consists of the following rules:

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
SQR(x0, x1) = SQR(x0)
ACTIVATE(x0, x1) = ACTIVATE(x0, x1)
TERMS(x0, x1) = TERMS(x0)
DBL(x0, x1) = DBL(x0)
ADD(x0, x1, x2) = ADD(x0, x2)
FIRST(x0, x1, x2) = FIRST(x0, x1)

Tags:
SQR has argument tags [8,8] and root tag 4
ACTIVATE has argument tags [8,0] and root tag 4
TERMS has argument tags [8,6] and root tag 4
DBL has argument tags [8,0] and root tag 4
ADD has argument tags [12,8,0] and root tag 2
FIRST has argument tags [0,12,1] and root tag 2

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
SQR(x1) = SQR(x1)
s(x1) = x1
activate(x1) = x1
ACTIVATE(x1) = ACTIVATE(x1)
n__terms(x1) = x1
TERMS(x1) = TERMS(x1)
DBL(x1) = DBL(x1)
n__add(x1, x2) = n__add(x1, x2)
ADD(x1, x2) = ADD(x1, x2)
n__dbl(x1) = x1
n__first(x1, x2) = n__first(x1, x2)
FIRST(x1, x2) = FIRST(x1, x2)
cons(x1, x2) = x2
terms(x1) = x1
add(x1, x2) = add(x1, x2)
n__s(x1) = x1
dbl(x1) = x1
first(x1, x2) = first(x1, x2)
recip(x1) = recip
sqr(x1) = sqr
0 = 0
nil = nil

Lexicographic path order with status [LPO].
Quasi-Precedence:

[n_{_first₂}, first₂] > FIRST₂ > [SQR₁, ACTIVATE₁, TERMS₁, DBL₁] > nil
recip > nil
sqr > [n_{_add₂}, ADD₂, add₂] > [SQR₁, ACTIVATE₁, TERMS₁, DBL₁] > nil
sqr > 0 > nil

Status:

SQR₁: [1]
ACTIVATE₁: [1]
TERMS₁: [1]
DBL₁: [1]
n_{_add₂}: [2,1]
ADD₂: [2,1]
n_{_first₂}: [2,1]
FIRST₂: [2,1]
add₂: [2,1]
first₂: [2,1]
recip: []
sqr: []
0: []
nil: []

The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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

SQR(s(X)) → SQR(activate(X))
SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → TERMS(X)
TERMS(N) → SQR(N)
SQR(s(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)

The TRS R consists of the following rules:

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

The following pairs can be oriented strictly and are deleted.

SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → TERMS(X)
SQR(s(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
SQR(x0, x1) = SQR(x1)
ACTIVATE(x0, x1) = ACTIVATE(x1)
TERMS(x0, x1) = TERMS(x1)
DBL(x0, x1) = DBL(x0, x1)

Tags:
SQR has argument tags [0,4] and root tag 0
ACTIVATE has argument tags [4,4] and root tag 1
TERMS has argument tags [5,4] and root tag 0
DBL has argument tags [2,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
SQR(x1) = SQR
s(x1) = s(x1)
activate(x1) = activate(x1)
ACTIVATE(x1) = ACTIVATE(x1)
n__terms(x1) = x1
TERMS(x1) = TERMS(x1)
DBL(x1) = DBL
n__dbl(x1) = x1
terms(x1) = terms(x1)
n__add(x1, x2) = x2
add(x1, x2) = add(x2)
n__s(x1) = x1
dbl(x1) = dbl(x1)
n__first(x1, x2) = n__first(x1, x2)
first(x1, x2) = first(x1, x2)
cons(x1, x2) = x1
recip(x1) = recip
sqr(x1) = sqr
0 = 0
nil = nil

Lexicographic path order with status [LPO].
Quasi-Precedence:

SQR > [s₁, activate₁, add₁, dbl₁] > [terms₁, recip] > DBL
SQR > [s₁, activate₁, add₁, dbl₁] > first₂ > n_{_first₂} > DBL
SQR > [ACTIVATE₁, TERMS₁] > DBL
sqr > [s₁, activate₁, add₁, dbl₁] > [terms₁, recip] > DBL
sqr > [s₁, activate₁, add₁, dbl₁] > first₂ > n_{_first₂} > DBL
sqr > 0 > nil > DBL

Status:

SQR: []
s₁: [1]
activate₁: [1]
ACTIVATE₁: [1]
TERMS₁: [1]
DBL: []
terms₁: [1]
add₁: [1]
dbl₁: [1]
n_{_first₂}: [2,1]
first₂: [2,1]
recip: []
sqr: []
0: []
nil: []

The following usable rules [FROCOS05] were oriented: none

(8) Obligation:

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

SQR(s(X)) → SQR(activate(X))
TERMS(N) → SQR(N)

The TRS R consists of the following rules:

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

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

(9) DependencyGraphProof (EQUIVALENT transformation)

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

(10) Obligation:

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

SQR(s(X)) → SQR(activate(X))

The TRS R consists of the following rules:

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

The following pairs can be oriented strictly and are deleted.

SQR(s(X)) → SQR(activate(X))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
SQR(x0, x1) = SQR(x1)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
SQR(x1) = SQR
s(x1) = s(x1)
activate(x1) = activate(x1)
n__terms(x1) = x1
terms(x1) = terms(x1)
n__add(x1, x2) = n__add(x1, x2)
add(x1, x2) = add(x1, x2)
n__s(x1) = n__s(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) = cons(x1)
recip(x1) = recip
sqr(x1) = sqr(x1)
0 = 0
nil = nil

Lexicographic path order with status [LPO].
Quasi-Precedence:

SQR > [activate₁, terms₁, cons₁] > [recip, sqr₁]
[n_{_add₂}, add₂] > [s₁, n_{_s₁}] > [activate₁, terms₁, cons₁] > [recip, sqr₁]
[n_{_dbl₁}, dbl₁] > [s₁, n_{_s₁}] > [activate₁, terms₁, cons₁] > [recip, sqr₁]
[n_{_first₂}, first₂] > [activate₁, terms₁, cons₁] > [recip, sqr₁]
[n_{_first₂}, first₂] > nil > [recip, sqr₁]
0 > nil > [recip, sqr₁]

Status:

SQR: []
s₁: [1]
activate₁: [1]
terms₁: [1]
n_{_add₂}: [2,1]
add₂: [2,1]
n_{_s₁}: [1]
n_{_dbl₁}: [1]
dbl₁: [1]
n_{_first₂}: [2,1]
first₂: [2,1]
cons₁: [1]
recip: []
sqr₁: [1]
0: []
nil: []

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

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

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(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) DependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) PisEmptyProof (EQUIVALENT transformation)

(14) TRUE