(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(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1)) = 2 + x1
POL(U12(x1)) = 1 + x1
POL(__(x1, x2)) = x1 + x2
POL(active(x1)) = x1
POL(isNePal(x1)) = 3 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
(2) 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)
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1)) = x1
POL(U12(x1)) = 2·x1
POL(__(x1, x2)) = x1 + x2
POL(active(x1)) = x1
POL(isNePal(x1)) = 2·x1
POL(mark(x1)) = x1
POL(nil) = 2
POL(tt) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
active(
x1) =
x1
__(
x1,
x2) =
__(
x1,
x2)
mark(
x1) =
x1
nil =
nil
U11(
x1) =
U11(
x1)
tt =
tt
U12(
x1) =
U12(
x1)
isNePal(
x1) =
x1
Recursive path order with status [RPO].
Quasi-Precedence:
_2 > U121
nil > U121
U111 > U121
tt > U121
Status:
_2: [1,2]
tt: multiset
U121: [1]
U111: [1]
nil: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1)) = 2 + x1
POL(U12(x1)) = 1 + x1
POL(__(x1, x2)) = 2 + x1 + x2
POL(active(x1)) = 1 + x1
POL(isNePal(x1)) = 2 + x1
POL(mark(x1)) = 1 + 2·x1
POL(nil) = 1
POL(tt) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(nil) → active(nil)
mark(U11(X)) → active(U11(mark(X)))
mark(tt) → active(tt)
mark(isNePal(X)) → active(isNePal(mark(X)))
__(mark(X1), X2) → __(X1, X2)
__(X1, mark(X2)) → __(X1, X2)
__(active(X1), X2) → __(X1, X2)
__(X1, active(X2)) → __(X1, X2)
U11(mark(X)) → U11(X)
U11(active(X)) → U11(X)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNePal(mark(X)) → isNePal(X)
isNePal(active(X)) → isNePal(X)
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(U12(X)) → active(U12(mark(X)))
Q is empty.
(9) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U12(x1)) = 2 + x1
POL(__(x1, x2)) = 2 + 2·x1 + x2
POL(active(x1)) = 1 + x1
POL(mark(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(__(X1, X2)) → active(__(mark(X1), mark(X2)))
mark(U12(X)) → active(U12(mark(X)))
(10) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(11) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(12) TRUE
(13) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(14) TRUE
(15) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(16) TRUE