We consider the following Problem:
Strict Trs:
{ 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(isNePal(__(I, __(P, I)))) -> mark(tt())
, mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
, mark(nil()) -> active(nil())
, mark(and(X1, X2)) -> active(and(mark(X1), X2))
, 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)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ 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(isNePal(__(I, __(P, I)))) -> mark(tt())
, mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
, mark(nil()) -> active(nil())
, mark(and(X1, X2)) -> active(and(mark(X1), X2))
, 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)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ __(mark(X1), X2) -> __(X1, X2)
, __(active(X1), X2) -> __(X1, X2)
, __(X1, active(X2)) -> __(X1, X2)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(__) = {1, 2}, Uargs(mark) = {1},
Uargs(and) = {1}, Uargs(isNePal) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[0 1] [1]
__(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
mark(x1) = [1 0] x1 + [1]
[1 0] [1]
nil() = [0]
[0]
and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
isNePal(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ 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(isNePal(__(I, __(P, I)))) -> mark(tt())
, mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
, mark(nil()) -> active(nil())
, mark(and(X1, X2)) -> active(and(mark(X1), X2))
, mark(tt()) -> active(tt())
, mark(isNePal(X)) -> active(isNePal(mark(X)))
, __(X1, mark(X2)) -> __(X1, X2)}
Weak Trs:
{ __(mark(X1), X2) -> __(X1, X2)
, __(active(X1), X2) -> __(X1, X2)
, __(X1, active(X2)) -> __(X1, X2)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {__(X1, mark(X2)) -> __(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(__) = {1, 2}, Uargs(mark) = {1},
Uargs(and) = {1}, Uargs(isNePal) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [1]
[0 0] [1]
__(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
nil() = [0]
[0]
and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
tt() = [0]
[0]
isNePal(x1) = [1 0] x1 + [0]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ 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(isNePal(__(I, __(P, I)))) -> mark(tt())
, mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
, mark(nil()) -> active(nil())
, mark(and(X1, X2)) -> active(and(mark(X1), X2))
, mark(tt()) -> active(tt())
, mark(isNePal(X)) -> active(isNePal(mark(X)))}
Weak Trs:
{ __(X1, mark(X2)) -> __(X1, X2)
, __(mark(X1), X2) -> __(X1, X2)
, __(active(X1), X2) -> __(X1, X2)
, __(X1, active(X2)) -> __(X1, X2)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(__) = {1, 2}, Uargs(mark) = {1},
Uargs(and) = {1}, Uargs(isNePal) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 2] x1 + [1]
[0 0] [1]
__(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [1 0] x1 + [1]
[0 0] [1]
nil() = [0]
[0]
and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [3]
tt() = [0]
[0]
isNePal(x1) = [1 0] x1 + [1]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ 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(and(X1, X2)) -> active(and(mark(X1), X2))
, mark(tt()) -> active(tt())
, mark(isNePal(X)) -> active(isNePal(mark(X)))}
Weak Trs:
{ active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(X1, mark(X2)) -> __(X1, X2)
, __(mark(X1), X2) -> __(X1, X2)
, __(active(X1), X2) -> __(X1, X2)
, __(X1, active(X2)) -> __(X1, X2)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(__) = {1, 2}, Uargs(mark) = {1},
Uargs(and) = {1}, Uargs(isNePal) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 1] x1 + [1]
[0 0] [0]
__(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [1 0] x1 + [1]
[0 0] [0]
nil() = [2]
[2]
and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
isNePal(x1) = [1 0] x1 + [0]
[0 0] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
, mark(nil()) -> active(nil())
, mark(and(X1, X2)) -> active(and(mark(X1), X2))
, mark(tt()) -> active(tt())
, mark(isNePal(X)) -> active(isNePal(mark(X)))}
Weak Trs:
{ active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(X1, mark(X2)) -> __(X1, X2)
, __(mark(X1), X2) -> __(X1, X2)
, __(active(X1), X2) -> __(X1, X2)
, __(X1, active(X2)) -> __(X1, X2)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(tt()) -> active(tt())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(__) = {1, 2}, Uargs(mark) = {1},
Uargs(and) = {1}, Uargs(isNePal) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 0] x1 + [0]
[0 1] [3]
__(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[1 1] [1 1] [2]
mark(x1) = [1 0] x1 + [1]
[1 1] [0]
nil() = [0]
[1]
and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[1 1] [1 1] [0]
tt() = [3]
[0]
isNePal(x1) = [1 0] x1 + [3]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
, mark(nil()) -> active(nil())
, mark(and(X1, X2)) -> active(and(mark(X1), X2))
, mark(isNePal(X)) -> active(isNePal(mark(X)))}
Weak Trs:
{ mark(tt()) -> active(tt())
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(X1, mark(X2)) -> __(X1, X2)
, __(mark(X1), X2) -> __(X1, X2)
, __(active(X1), X2) -> __(X1, X2)
, __(X1, active(X2)) -> __(X1, X2)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {mark(nil()) -> active(nil())}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(__) = {1, 2}, Uargs(mark) = {1},
Uargs(and) = {1}, Uargs(isNePal) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 1] x1 + [1]
[0 0] [1]
__(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
mark(x1) = [1 0] x1 + [3]
[0 0] [1]
nil() = [2]
[1]
and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [2]
[0]
isNePal(x1) = [1 0] x1 + [3]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
, mark(and(X1, X2)) -> active(and(mark(X1), X2))
, mark(isNePal(X)) -> active(isNePal(mark(X)))}
Weak Trs:
{ mark(nil()) -> active(nil())
, mark(tt()) -> active(tt())
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(X1, mark(X2)) -> __(X1, X2)
, __(mark(X1), X2) -> __(X1, X2)
, __(active(X1), X2) -> __(X1, X2)
, __(X1, active(X2)) -> __(X1, X2)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(active) = {1}, Uargs(__) = {1, 2}, Uargs(mark) = {1},
Uargs(and) = {1}, Uargs(isNePal) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
active(x1) = [1 2] x1 + [0]
[0 0] [0]
__(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [2]
mark(x1) = [1 0] x1 + [0]
[0 0] [0]
nil() = [0]
[0]
and(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
tt() = [0]
[0]
isNePal(x1) = [1 0] x1 + [0]
[0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
, mark(and(X1, X2)) -> active(and(mark(X1), X2))
, mark(isNePal(X)) -> active(isNePal(mark(X)))}
Weak Trs:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, mark(nil()) -> active(nil())
, mark(tt()) -> active(tt())
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(X1, mark(X2)) -> __(X1, X2)
, __(mark(X1), X2) -> __(X1, X2)
, __(active(X1), X2) -> __(X1, X2)
, __(X1, active(X2)) -> __(X1, X2)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
, mark(and(X1, X2)) -> active(and(mark(X1), X2))
, mark(isNePal(X)) -> active(isNePal(mark(X)))}
Weak Trs:
{ active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
, mark(nil()) -> active(nil())
, mark(tt()) -> active(tt())
, active(__(X, nil())) -> mark(X)
, active(__(nil(), X)) -> mark(X)
, active(and(tt(), X)) -> mark(X)
, active(isNePal(__(I, __(P, I)))) -> mark(tt())
, __(X1, mark(X2)) -> __(X1, X2)
, __(mark(X1), X2) -> __(X1, X2)
, __(active(X1), X2) -> __(X1, X2)
, __(X1, active(X2)) -> __(X1, X2)
, and(mark(X1), X2) -> and(X1, X2)
, and(X1, mark(X2)) -> and(X1, X2)
, and(active(X1), X2) -> and(X1, X2)
, and(X1, active(X2)) -> and(X1, X2)
, isNePal(mark(X)) -> isNePal(X)
, isNePal(active(X)) -> isNePal(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ active_0(2) -> 1
, ___0(2, 2) -> 1
, mark_0(2) -> 1
, nil_0() -> 2
, and_0(2, 2) -> 1
, tt_0() -> 2
, isNePal_0(2) -> 1}
Hurray, we answered YES(?,O(n^1))