We consider the following Problem:
Strict Trs:
{ first(0(), X) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, from(X) -> cons(X, n__from(n__s(X)))
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> n__from(X)
, s(X) -> n__s(X)
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
Arguments of following rules are not normal-forms:
{first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
All above mentioned rules can be savely removed.
We consider the following Problem:
Strict Trs:
{ first(0(), X) -> nil()
, from(X) -> cons(X, n__from(n__s(X)))
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> n__from(X)
, s(X) -> n__s(X)
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {first(0(), X) -> nil()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
Uargs(n__from) = {}, Uargs(n__s) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
activate(x1) = [1 1] x1 + [1]
[0 0] [1]
from(x1) = [1 1] x1 + [1]
[0 0] [1]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
n__s(x1) = [0 0] x1 + [0]
[1 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> n__from(X)
, s(X) -> n__s(X)
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X}
Weak Trs: {first(0(), X) -> nil()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(n__s(X)) -> s(activate(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
Uargs(n__from) = {}, Uargs(n__s) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
activate(x1) = [1 1] x1 + [1]
[0 0] [1]
from(x1) = [1 1] x1 + [1]
[0 0] [1]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
n__s(x1) = [0 0] x1 + [0]
[1 1] [2]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> n__from(X)
, s(X) -> n__s(X)
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(n__from(X)) -> from(activate(X))
, activate(X) -> X}
Weak Trs:
{ activate(n__s(X)) -> s(activate(X))
, first(0(), X) -> nil()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {from(X) -> cons(X, n__from(n__s(X)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
Uargs(n__from) = {}, Uargs(n__s) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
activate(x1) = [1 1] x1 + [1]
[0 0] [1]
from(x1) = [1 1] x1 + [3]
[0 0] [1]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
n__s(x1) = [0 0] x1 + [0]
[1 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ first(X1, X2) -> n__first(X1, X2)
, from(X) -> n__from(X)
, s(X) -> n__s(X)
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(n__from(X)) -> from(activate(X))
, activate(X) -> X}
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__s(X)) -> s(activate(X))
, first(0(), X) -> nil()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {first(X1, X2) -> n__first(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
Uargs(n__from) = {}, Uargs(n__s) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 0] [1]
from(x1) = [1 0] x1 + [0]
[0 0] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(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:
{ from(X) -> n__from(X)
, s(X) -> n__s(X)
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(n__from(X)) -> from(activate(X))
, activate(X) -> X}
Weak Trs:
{ first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(n__s(X)))
, activate(n__s(X)) -> s(activate(X))
, first(0(), X) -> nil()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {from(X) -> n__from(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
Uargs(n__from) = {}, Uargs(n__s) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[1 1] [0 0] [1]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [1]
[1 0] [3]
from(x1) = [1 0] x1 + [2]
[1 1] [1]
n__from(x1) = [1 0] x1 + [1]
[0 0] [0]
n__s(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:
{ s(X) -> n__s(X)
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(n__from(X)) -> from(activate(X))
, activate(X) -> X}
Weak Trs:
{ from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(n__s(X)))
, activate(n__s(X)) -> s(activate(X))
, first(0(), X) -> nil()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(X) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
Uargs(n__from) = {}, Uargs(n__s) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
first(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 1] [0]
cons(x1, x2) = [0 1] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
n__first(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [1]
[0 1] [3]
from(x1) = [1 3] x1 + [0]
[0 0] [1]
n__from(x1) = [1 3] x1 + [0]
[0 0] [1]
n__s(x1) = [1 0] x1 + [0]
[0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ s(X) -> n__s(X)
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(n__from(X)) -> from(activate(X))}
Weak Trs:
{ activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(n__s(X)))
, activate(n__s(X)) -> s(activate(X))
, first(0(), X) -> nil()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ s(X) -> n__s(X)
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(n__from(X)) -> from(activate(X))}
Weak Trs:
{ activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(n__s(X)))
, activate(n__s(X)) -> s(activate(X))
, first(0(), X) -> nil()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The following argument positions are usable:
Uargs(first) = {1, 2}, Uargs(s) = {1}, Uargs(cons) = {},
Uargs(n__first) = {}, Uargs(activate) = {}, Uargs(from) = {1},
Uargs(n__from) = {}, Uargs(n__s) = {}
We have the following constructor-based EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
first(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
[0 1] [0 1] [2]
0() = [0]
[0]
nil() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 1] [0 1] [2]
activate(x1) = [1 1] x1 + [0]
[0 1] [0]
from(x1) = [1 0] x1 + [3]
[0 1] [2]
n__from(x1) = [1 0] x1 + [2]
[0 1] [2]
n__s(x1) = [1 0] x1 + [0]
[0 1] [1]
Hurray, we answered YES(?,O(n^1))