Termination w.r.t. Q proof of /tmp/tmpgk42bG/PALINDROME_nokinds

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

0 QTRS

↳1 QTRSToCSRProof (⇔)

↳2 CSR

↳3 PoloCSRProof (⇔)

↳4 CSR

↳5 PoloCSRProof (⇔)

↳6 CSR

↳7 PoloCSRProof (⇔)

↳8 CSR

↳9 PoloCSRProof (⇔)

↳10 CSR

↳11 PoloCSRProof (⇔)

↳12 CSR

↳13 PoloCSRProof (⇔)

↳14 CSR

↳15 RisEmptyProof (⇔)

↳16 TRUE

(0) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(isNeList(V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(and(isList(V1), isList(V2)))
active(isNeList(V)) → mark(isQid(V))
active(isNeList(__(V1, V2))) → mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) → mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) → mark(isQid(V))
active(isNePal(__(I, __(P, I)))) → mark(and(isQid(I), isPal(P)))
active(isPal(V)) → mark(isNePal(V))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isList(X)) → isList(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isList(ok(X)) → ok(isList(X))
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
isPal(ok(X)) → ok(isPal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) QTRSToCSRProof (EQUIVALENT transformation)

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isList(V)) → mark(isNeList(V))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(and(isList(V1), isList(V2)))
active(isNeList(V)) → mark(isQid(V))
active(isNeList(__(V1, V2))) → mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) → mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) → mark(isQid(V))
active(isNePal(__(I, __(P, I)))) → mark(and(isQid(I), isPal(P)))
active(isPal(V)) → mark(isNePal(V))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isList(X)) → isList(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isList(ok(X)) → ok(isList(X))
isNeList(ok(X)) → ok(isNeList(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
isPal(ok(X)) → ok(isPal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top, active: active, mark: mark, ok: ok, proper: proper
The replacement map contains the following entries:

__: {1, 2}
nil: empty set
and: {1}
tt: empty set
isList: empty set
isNeList: empty set
isQid: empty set
isNePal: empty set
isPal: empty set
a: empty set
e: empty set
i: empty set
o: empty set
u: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).

(2) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → X
isList(V) → isNeList(V)
isList(nil) → tt
isList(__(V1, V2)) → and(isList(V1), isList(V2))
isNeList(V) → isQid(V)
isNeList(__(V1, V2)) → and(isList(V1), isNeList(V2))
isNeList(__(V1, V2)) → and(isNeList(V1), isList(V2))
isNePal(V) → isQid(V)
isNePal(__(I, __(P, I))) → and(isQid(I), isPal(P))
isPal(V) → isNePal(V)
isPal(nil) → tt
isQid(a) → tt
isQid(e) → tt
isQid(i) → tt
isQid(o) → tt
isQid(u) → tt

The replacement map contains the following entries:

(3) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

__(X, nil) → X
__(nil, X) → X

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(e) = 2
POL(i) = 2
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isQid(x₁)) = 0
POL(nil) = 1
POL(o) = 0
POL(tt) = 0
POL(u) = 2

(4) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
and(tt, X) → X
isList(V) → isNeList(V)
isList(nil) → tt
isList(__(V1, V2)) → and(isList(V1), isList(V2))
isNeList(V) → isQid(V)
isNeList(__(V1, V2)) → and(isList(V1), isNeList(V2))
isNeList(__(V1, V2)) → and(isNeList(V1), isList(V2))
isNePal(V) → isQid(V)
isNePal(__(I, __(P, I))) → and(isQid(I), isPal(P))
isPal(V) → isNePal(V)
isPal(nil) → tt
isQid(a) → tt
isQid(e) → tt
isQid(i) → tt
isQid(o) → tt
isQid(u) → tt

The replacement map contains the following entries:

(5) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

isQid(e) → tt
isQid(i) → tt
isQid(o) → tt

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 2·x₁ + x₂
POL(a) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 2
POL(i) = 1
POL(isList(x₁)) = 2·x₁
POL(isNeList(x₁)) = 2·x₁
POL(isNePal(x₁)) = 2·x₁
POL(isPal(x₁)) = 2·x₁
POL(isQid(x₁)) = 2·x₁
POL(nil) = 0
POL(o) = 2
POL(tt) = 0
POL(u) = 0

(6) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
and(tt, X) → X
isList(V) → isNeList(V)
isList(nil) → tt
isList(__(V1, V2)) → and(isList(V1), isList(V2))
isNeList(V) → isQid(V)
isNeList(__(V1, V2)) → and(isList(V1), isNeList(V2))
isNeList(__(V1, V2)) → and(isNeList(V1), isList(V2))
isNePal(V) → isQid(V)
isNePal(__(I, __(P, I))) → and(isQid(I), isPal(P))
isPal(V) → isNePal(V)
isPal(nil) → tt
isQid(a) → tt
isQid(u) → tt

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
and: {1}
tt: empty set
isList: empty set
isNeList: empty set
isQid: empty set
isNePal: empty set
isPal: empty set
a: empty set
u: empty set

(7) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

__(__(X, Y), Z) → __(X, __(Y, Z))

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(a) = 2
POL(and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isQid(x₁)) = 0
POL(nil) = 2
POL(tt) = 0
POL(u) = 0

(8) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

and(tt, X) → X
isList(V) → isNeList(V)
isList(nil) → tt
isList(__(V1, V2)) → and(isList(V1), isList(V2))
isNeList(V) → isQid(V)
isNeList(__(V1, V2)) → and(isList(V1), isNeList(V2))
isNeList(__(V1, V2)) → and(isNeList(V1), isList(V2))
isNePal(V) → isQid(V)
isNePal(__(I, __(P, I))) → and(isQid(I), isPal(P))
isPal(V) → isNePal(V)
isPal(nil) → tt
isQid(a) → tt
isQid(u) → tt

The replacement map contains the following entries:

__: {1, 2}
nil: empty set
and: {1}
tt: empty set
isList: empty set
isNeList: empty set
isQid: empty set
isNePal: empty set
isPal: empty set
a: empty set
u: empty set

(9) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

and(tt, X) → X
isList(nil) → tt
isList(__(V1, V2)) → and(isList(V1), isList(V2))
isNeList(__(V1, V2)) → and(isList(V1), isNeList(V2))
isNeList(__(V1, V2)) → and(isNeList(V1), isList(V2))
isNePal(V) → isQid(V)
isNePal(__(I, __(P, I))) → and(isQid(I), isPal(P))
isPal(nil) → tt
isQid(a) → tt
isQid(u) → tt

Used ordering:
Polynomial interpretation [POLO]:

POL(__(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(a) = 2
POL(and(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(isList(x₁)) = 2·x₁
POL(isNeList(x₁)) = 2·x₁
POL(isNePal(x₁)) = 2 + 2·x₁
POL(isPal(x₁)) = 2 + 2·x₁
POL(isQid(x₁)) = 2·x₁
POL(nil) = 2
POL(tt) = 1
POL(u) = 2

(10) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

isList(V) → isNeList(V)
isNeList(V) → isQid(V)
isPal(V) → isNePal(V)

The replacement map contains the following entries:

isList: empty set
isNeList: empty set
isQid: empty set
isNePal: empty set
isPal: empty set

(11) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

isNeList(V) → isQid(V)
isPal(V) → isNePal(V)

Used ordering:
Polynomial interpretation [POLO]:

POL(isList(x₁)) = 2 + 2·x₁
POL(isNeList(x₁)) = 2 + 2·x₁
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 2 + 2·x₁
POL(isQid(x₁)) = 1 + x₁

(12) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

isList(V) → isNeList(V)

The replacement map contains the following entries:

isList: empty set
isNeList: empty set

(13) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

isList(V) → isNeList(V)

Used ordering:
Polynomial interpretation [POLO]:

POL(isList(x₁)) = 2 + 2·x₁
POL(isNeList(x₁)) = 1 + x₁

(14) Obligation:

Context-sensitive rewrite system:
R is empty.

(15) RisEmptyProof (EQUIVALENT transformation)

The CSR R is empty. Hence, termination is trivially proven.

(0) Obligation:

(1) QTRSToCSRProof (EQUIVALENT transformation)

(2) Obligation:

(3) PoloCSRProof (EQUIVALENT transformation)

(4) Obligation:

(5) PoloCSRProof (EQUIVALENT transformation)

(6) Obligation:

(7) PoloCSRProof (EQUIVALENT transformation)

(8) Obligation:

(9) PoloCSRProof (EQUIVALENT transformation)

(10) Obligation:

(11) PoloCSRProof (EQUIVALENT transformation)

(12) Obligation:

(13) PoloCSRProof (EQUIVALENT transformation)

(14) Obligation:

(15) RisEmptyProof (EQUIVALENT transformation)

(16) TRUE