We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, head(cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
, sel(0(), cons(X, XS)) -> X
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, head(cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
, sel(0(), cons(X, XS)) -> X
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, 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:
{ from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(head) = {1}, Uargs(2nd) = {},
Uargs(activate) = {}, Uargs(take) = {}, Uargs(n__take) = {2},
Uargs(sel) = {2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [2]
[0 0] [2]
cons(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
n__from(x1) = [0 0] x1 + [0]
[1 1] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [1 0] x1 + [0]
[0 0] [1]
2nd(x1) = [1 0] x1 + [0]
[0 0] [1]
activate(x1) = [1 1] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
sel(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ head(cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, sel(0(), cons(X, XS)) -> X
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X}
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
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)
, take(X1, X2) -> n__take(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(head) = {1}, Uargs(2nd) = {},
Uargs(activate) = {}, Uargs(take) = {}, Uargs(n__take) = {2},
Uargs(sel) = {2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [2]
[0 0] [2]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
n__from(x1) = [1 1] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [1 0] x1 + [1]
[0 0] [1]
2nd(x1) = [1 0] x1 + [1]
[0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [1]
take(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
sel(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ head(cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, sel(0(), cons(X, XS)) -> X
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X}
Weak Trs:
{ from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {sel(s(N), cons(X, XS)) -> sel(N, activate(XS))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(head) = {1}, Uargs(2nd) = {},
Uargs(activate) = {}, Uargs(take) = {}, Uargs(n__take) = {2},
Uargs(sel) = {2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 1] x1 + [0]
[0 0] [3]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [3]
n__from(x1) = [1 1] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [1 0] x1 + [1]
[0 0] [1]
2nd(x1) = [1 0] x1 + [1]
[0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [2]
take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
sel(x1, x2) = [0 0] x1 + [1 3] x2 + [0]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ head(cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, sel(0(), cons(X, XS)) -> X
, activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X}
Weak Trs:
{ sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {2nd(cons(X, XS)) -> head(activate(XS))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(head) = {1}, Uargs(2nd) = {},
Uargs(activate) = {}, Uargs(take) = {}, Uargs(n__take) = {2},
Uargs(sel) = {2}
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 + [1 0] x2 + [0]
[0 0] [0 0] [2]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 1] [0]
head(x1) = [1 0] x1 + [0]
[0 0] [1]
2nd(x1) = [1 0] x1 + [3]
[0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [1]
take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
0() = [0]
[3]
nil() = [0]
[0]
n__take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
sel(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ head(cons(X, XS)) -> X
, sel(0(), cons(X, XS)) -> X
, activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X}
Weak Trs:
{ 2nd(cons(X, XS)) -> head(activate(XS))
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {sel(0(), cons(X, XS)) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(head) = {1}, Uargs(2nd) = {},
Uargs(activate) = {}, Uargs(take) = {}, Uargs(n__take) = {2},
Uargs(sel) = {2}
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 + [1 0] x2 + [0]
[0 1] [0 0] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
head(x1) = [1 0] x1 + [0]
[1 0] [1]
2nd(x1) = [1 0] x1 + [0]
[1 0] [2]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
sel(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ head(cons(X, XS)) -> X
, activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X}
Weak Trs:
{ sel(0(), cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {head(cons(X, XS)) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(head) = {1}, Uargs(2nd) = {},
Uargs(activate) = {}, Uargs(take) = {}, Uargs(n__take) = {2},
Uargs(sel) = {2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 0] x1 + [0]
[1 1] [0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 1] [2]
head(x1) = [1 0] x1 + [1]
[1 1] [1]
2nd(x1) = [1 0] x1 + [1]
[1 0] [2]
activate(x1) = [1 0] x1 + [0]
[0 0] [1]
take(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
sel(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X}
Weak Trs:
{ head(cons(X, XS)) -> X
, sel(0(), cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
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(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(head) = {1}, Uargs(2nd) = {},
Uargs(activate) = {}, Uargs(take) = {}, Uargs(n__take) = {2},
Uargs(sel) = {2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 0] x1 + [2]
[0 1] [0]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [0]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [1]
[0 0] [0]
head(x1) = [1 0] x1 + [0]
[0 1] [1]
2nd(x1) = [1 0] x1 + [0]
[0 1] [1]
activate(x1) = [1 0] x1 + [1]
[0 1] [0]
take(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
sel(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)}
Weak Trs:
{ activate(X) -> X
, head(cons(X, XS)) -> X
, sel(0(), cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
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__take(X1, X2)) -> take(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(head) = {1}, Uargs(2nd) = {},
Uargs(activate) = {}, Uargs(take) = {}, Uargs(n__take) = {2},
Uargs(sel) = {2}
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 + [1 0] x2 + [0]
[0 1] [0 1] [2]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 1] [3]
head(x1) = [1 0] x1 + [0]
[0 1] [0]
2nd(x1) = [1 0] x1 + [1]
[0 1] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [0]
take(x1, x2) = [0 3] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
0() = [0]
[2]
nil() = [0]
[0]
n__take(x1, x2) = [0 3] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
sel(x1, x2) = [0 3] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {activate(n__from(X)) -> from(X)}
Weak Trs:
{ activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, head(cons(X, XS)) -> X
, sel(0(), cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {activate(n__from(X)) -> from(X)}
Weak Trs:
{ activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, head(cons(X, XS)) -> X
, sel(0(), cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs: {activate^#(n__from(X)) -> from^#(X)}
Weak DPs:
{ activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, activate^#(X) -> c_3()
, head^#(cons(X, XS)) -> c_4()
, sel^#(0(), cons(X, XS)) -> c_5()
, 2nd^#(cons(X, XS)) -> head^#(activate(XS))
, sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS))
, from^#(X) -> c_8()
, take^#(X1, X2) -> c_9()
, from^#(X) -> c_10()
, take^#(0(), XS) -> c_11()
, take^#(s(N), cons(X, XS)) -> activate^#(XS)}
We consider the following Problem:
Strict DPs: {activate^#(n__from(X)) -> from^#(X)}
Strict Trs: {activate(n__from(X)) -> from(X)}
Weak DPs:
{ activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, activate^#(X) -> c_3()
, head^#(cons(X, XS)) -> c_4()
, sel^#(0(), cons(X, XS)) -> c_5()
, 2nd^#(cons(X, XS)) -> head^#(activate(XS))
, sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS))
, from^#(X) -> c_8()
, take^#(X1, X2) -> c_9()
, from^#(X) -> c_10()
, take^#(0(), XS) -> c_11()
, take^#(s(N), cons(X, XS)) -> activate^#(XS)}
Weak Trs:
{ activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, head(cons(X, XS)) -> X
, sel(0(), cons(X, XS)) -> X
, 2nd(cons(X, XS)) -> head(activate(XS))
, sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Strict Usable Rules: {activate(n__from(X)) -> from(X)}
Weak Usable Rules:
{ activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
We consider the following Problem:
Strict DPs: {activate^#(n__from(X)) -> from^#(X)}
Strict Trs: {activate(n__from(X)) -> from(X)}
Weak DPs:
{ activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, activate^#(X) -> c_3()
, head^#(cons(X, XS)) -> c_4()
, sel^#(0(), cons(X, XS)) -> c_5()
, 2nd^#(cons(X, XS)) -> head^#(activate(XS))
, sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS))
, from^#(X) -> c_8()
, take^#(X1, X2) -> c_9()
, from^#(X) -> c_10()
, take^#(0(), XS) -> c_11()
, take^#(s(N), cons(X, XS)) -> activate^#(XS)}
Weak Trs:
{ activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
Dependency Pairs: {activate^#(n__from(X)) -> from^#(X)}
TRS Component: {activate(n__from(X)) -> from(X)}
Interpretation of constant growth:
----------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(head) = {}, Uargs(2nd) = {},
Uargs(activate) = {}, Uargs(take) = {}, Uargs(n__take) = {2},
Uargs(sel) = {}, Uargs(activate^#) = {}, Uargs(from^#) = {},
Uargs(take^#) = {}, Uargs(head^#) = {1}, Uargs(sel^#) = {2},
Uargs(2nd^#) = {}
We have the following constructor-based EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [0 0] x1 + [0]
[0 0] [1]
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
n__from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [0]
[0 1] [1]
head(x1) = [0 0] x1 + [0]
[0 0] [0]
2nd(x1) = [0 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 2] [1]
take(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__take(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
activate^#(x1) = [0 0] x1 + [1]
[0 0] [1]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
take^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
c_3() = [0]
[0]
head^#(x1) = [1 0] x1 + [1]
[0 0] [1]
c_4() = [0]
[0]
sel^#(x1, x2) = [0 1] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
c_5() = [0]
[0]
2nd^#(x1) = [1 0] x1 + [3]
[0 0] [1]
c_8() = [0]
[0]
c_9() = [0]
[0]
c_10() = [0]
[0]
c_11() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Weak DPs:
{ activate^#(n__from(X)) -> from^#(X)
, activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, activate^#(X) -> c_3()
, head^#(cons(X, XS)) -> c_4()
, sel^#(0(), cons(X, XS)) -> c_5()
, 2nd^#(cons(X, XS)) -> head^#(activate(XS))
, sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS))
, from^#(X) -> c_8()
, take^#(X1, X2) -> c_9()
, from^#(X) -> c_10()
, take^#(0(), XS) -> c_11()
, take^#(s(N), cons(X, XS)) -> activate^#(XS)}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->5:{2,12} [ subsumed ]
|
|->9:{1} [ subsumed ]
| |
| |->10:{8} [ YES(O(1),O(1)) ]
| |
| `->11:{10} [ YES(O(1),O(1)) ]
|
|->8:{3} [ YES(O(1),O(1)) ]
|
|->6:{9} [ YES(O(1),O(1)) ]
|
`->7:{11} [ YES(O(1),O(1)) ]
->2:{6} [ subsumed ]
|
`->4:{4} [ YES(O(1),O(1)) ]
->1:{7} [ subsumed ]
|
`->3:{5} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: activate^#(n__from(X)) -> from^#(X)
, 2: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 3: activate^#(X) -> c_3()
, 4: head^#(cons(X, XS)) -> c_4()
, 5: sel^#(0(), cons(X, XS)) -> c_5()
, 6: 2nd^#(cons(X, XS)) -> head^#(activate(XS))
, 7: sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS))
, 8: from^#(X) -> c_8()
, 9: take^#(X1, X2) -> c_9()
, 10: from^#(X) -> c_10()
, 11: take^#(0(), XS) -> c_11()
, 12: take^#(s(N), cons(X, XS)) -> activate^#(XS)}
* Path 5:{2,12}: subsumed
-----------------------
This path is subsumed by the proof of paths 5:{2,12}->9:{1},
5:{2,12}->8:{3},
5:{2,12}->7:{11},
5:{2,12}->6:{9}.
* Path 5:{2,12}->9:{1}: subsumed
------------------------------
This path is subsumed by the proof of paths 5:{2,12}->9:{1}->11:{10},
5:{2,12}->9:{1}->10:{8}.
* Path 5:{2,12}->9:{1}->10:{8}: YES(O(1),O(1))
--------------------------------------------
We consider the following Problem:
Weak DPs:
{ activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, take^#(s(N), cons(X, XS)) -> activate^#(XS)
, activate^#(n__from(X)) -> from^#(X)}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
-->_1 take^#(s(N), cons(X, XS)) -> activate^#(XS) :2
2: take^#(s(N), cons(X, XS)) -> activate^#(XS)
-->_1 activate^#(n__from(X)) -> from^#(X) :3
-->_1 activate^#(n__take(X1, X2)) -> take^#(X1, X2) :1
3: activate^#(n__from(X)) -> from^#(X)
together with the congruence-graph
->1:{1,2} Weak SCC
|
`->2:{3} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)
, 3: activate^#(n__from(X)) -> from^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)
, 3: activate^#(n__from(X)) -> from^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) ->
cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{2,12}->9:{1}->11:{10}: YES(O(1),O(1))
---------------------------------------------
We consider the following Problem:
Weak DPs:
{ activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, take^#(s(N), cons(X, XS)) -> activate^#(XS)
, activate^#(n__from(X)) -> from^#(X)}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
-->_1 take^#(s(N), cons(X, XS)) -> activate^#(XS) :2
2: take^#(s(N), cons(X, XS)) -> activate^#(XS)
-->_1 activate^#(n__from(X)) -> from^#(X) :3
-->_1 activate^#(n__take(X1, X2)) -> take^#(X1, X2) :1
3: activate^#(n__from(X)) -> from^#(X)
together with the congruence-graph
->1:{1,2} Weak SCC
|
`->2:{3} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)
, 3: activate^#(n__from(X)) -> from^#(X)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)
, 3: activate^#(n__from(X)) -> from^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) ->
cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{2,12}->8:{3}: YES(O(1),O(1))
------------------------------------
We consider the following Problem:
Weak DPs:
{ activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, take^#(s(N), cons(X, XS)) -> activate^#(XS)}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
-->_1 take^#(s(N), cons(X, XS)) -> activate^#(XS) :2
2: take^#(s(N), cons(X, XS)) -> activate^#(XS)
-->_1 activate^#(n__take(X1, X2)) -> take^#(X1, X2) :1
together with the congruence-graph
->1:{1,2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)}
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) ->
cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{2,12}->6:{9}: YES(O(1),O(1))
------------------------------------
We consider the following Problem:
Weak DPs:
{ activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, take^#(s(N), cons(X, XS)) -> activate^#(XS)}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
-->_1 take^#(s(N), cons(X, XS)) -> activate^#(XS) :2
2: take^#(s(N), cons(X, XS)) -> activate^#(XS)
-->_1 activate^#(n__take(X1, X2)) -> take^#(X1, X2) :1
together with the congruence-graph
->1:{1,2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)}
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) ->
cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 5:{2,12}->7:{11}: YES(O(1),O(1))
-------------------------------------
We consider the following Problem:
Weak DPs:
{ activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, take^#(s(N), cons(X, XS)) -> activate^#(XS)}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
-->_1 take^#(s(N), cons(X, XS)) -> activate^#(XS) :2
2: take^#(s(N), cons(X, XS)) -> activate^#(XS)
-->_1 activate^#(n__take(X1, X2)) -> take^#(X1, X2) :1
together with the congruence-graph
->1:{1,2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: activate^#(n__take(X1, X2)) -> take^#(X1, X2)
, 2: take^#(s(N), cons(X, XS)) -> activate^#(XS)}
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) ->
cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 2:{6}: subsumed
--------------------
This path is subsumed by the proof of paths 2:{6}->4:{4}.
* Path 2:{6}->4:{4}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Weak DPs: {2nd^#(cons(X, XS)) -> head^#(activate(XS))}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: 2nd^#(cons(X, XS)) -> head^#(activate(XS))
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: 2nd^#(cons(X, XS)) -> head^#(activate(XS))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: 2nd^#(cons(X, XS)) -> head^#(activate(XS))}
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) ->
cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
* Path 1:{7}: subsumed
--------------------
This path is subsumed by the proof of paths 1:{7}->3:{5}.
* Path 1:{7}->3:{5}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Weak DPs: {sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS))}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS))
-->_1 sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS)) :1
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: sel^#(s(N), cons(X, XS)) -> sel^#(N, activate(XS))}
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(n__take(X1, X2)) -> take(X1, X2)
, activate(X) -> X
, from(X) -> n__from(X)
, take(X1, X2) -> n__take(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, take(0(), XS) -> nil()
, take(s(N), cons(X, XS)) ->
cons(X, n__take(N, activate(XS)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
No rule is usable.
We consider the following Problem:
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
Empty rules are trivially bounded
Hurray, we answered YES(?,O(n^1))