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