We consider the following Problem:
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(0(), cons(X, Z)) -> X
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(n__first(X1, X2)) -> first(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)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(0(), cons(X, Z)) -> X
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(n__first(X1, X2)) -> first(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)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {2},
Uargs(activate) = {}, 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]
first(x1, x2) = [0 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
activate(x1) = [1 1] x1 + [0]
[0 0] [0]
sel(x1, x2) = [0 0] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ sel(0(), cons(X, Z)) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, activate(X) -> X}
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))}
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)
, first(X1, X2) -> n__first(X1, X2)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {2},
Uargs(activate) = {}, 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]
[1 0] [2]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [0 0] x1 + [1]
[1 1] [0]
first(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
0() = [3]
[1]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [0]
[1 0] [1]
sel(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[1 1] [1 0] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ sel(0(), cons(X, Z)) -> X
, activate(n__from(X)) -> from(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, activate(X) -> X}
Weak Trs:
{ from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))}
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(first) = {}, Uargs(n__first) = {2},
Uargs(activate) = {}, 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]
[1 0] [2]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
n__from(x1) = [1 0] x1 + [0]
[0 0] [1]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [1]
activate(x1) = [1 0] x1 + [2]
[0 1] [0]
sel(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [1 0] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ sel(0(), cons(X, Z)) -> X
, activate(n__from(X)) -> from(X)
, activate(n__first(X1, X2)) -> first(X1, X2)}
Weak Trs:
{ activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))}
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__from(X)) -> from(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {2},
Uargs(activate) = {}, Uargs(sel) = {2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
from(x1) = [1 0] x1 + [1]
[0 1] [1]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [1]
n__from(x1) = [1 0] x1 + [1]
[0 1] [1]
s(x1) = [0 0] x1 + [0]
[0 1] [3]
first(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [0]
sel(x1, x2) = [0 1] x1 + [1 0] x2 + [1]
[0 1] [1 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ sel(0(), cons(X, Z)) -> X
, activate(n__first(X1, X2)) -> first(X1, X2)}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))}
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, Z)) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {2},
Uargs(activate) = {}, 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] [1]
cons(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
n__from(x1) = [1 0] x1 + [0]
[0 1] [1]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
first(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
0() = [0]
[0]
nil() = [0]
[0]
n__first(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [0]
[0 1] [0]
sel(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {activate(n__first(X1, X2)) -> first(X1, X2)}
Weak Trs:
{ sel(0(), cons(X, Z)) -> X
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {activate(n__first(X1, X2)) -> first(X1, X2)}
Weak Trs:
{ sel(0(), cons(X, Z)) -> X
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs: {activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
Weak DPs:
{ sel^#(0(), cons(X, Z)) -> c_2()
, activate^#(n__from(X)) -> from^#(X)
, activate^#(X) -> c_4()
, from^#(X) -> c_5()
, first^#(X1, X2) -> c_6()
, from^#(X) -> c_7()
, first^#(0(), Z) -> c_8()
, first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z))}
We consider the following Problem:
Strict DPs: {activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
Strict Trs: {activate(n__first(X1, X2)) -> first(X1, X2)}
Weak DPs:
{ sel^#(0(), cons(X, Z)) -> c_2()
, activate^#(n__from(X)) -> from^#(X)
, activate^#(X) -> c_4()
, from^#(X) -> c_5()
, first^#(X1, X2) -> c_6()
, from^#(X) -> c_7()
, first^#(0(), Z) -> c_8()
, first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z))}
Weak Trs:
{ sel(0(), cons(X, Z)) -> X
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))}
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__first(X1, X2)) -> first(X1, X2)}
Weak Usable Rules:
{ activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
We consider the following Problem:
Strict DPs: {activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
Strict Trs: {activate(n__first(X1, X2)) -> first(X1, X2)}
Weak DPs:
{ sel^#(0(), cons(X, Z)) -> c_2()
, activate^#(n__from(X)) -> from^#(X)
, activate^#(X) -> c_4()
, from^#(X) -> c_5()
, first^#(X1, X2) -> c_6()
, from^#(X) -> c_7()
, first^#(0(), Z) -> c_8()
, first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z))}
Weak Trs:
{ activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
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__first(X1, X2)) -> first(X1, X2)}
Interpretation of constant growth:
----------------------------------
The following argument positions are usable:
Uargs(from) = {}, Uargs(cons) = {2}, Uargs(n__from) = {},
Uargs(s) = {}, Uargs(first) = {}, Uargs(n__first) = {2},
Uargs(activate) = {}, Uargs(sel) = {}, Uargs(activate^#) = {},
Uargs(first^#) = {}, Uargs(sel^#) = {2}, Uargs(from^#) = {}
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] [0]
cons(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__from(x1) = [0 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [2]
[0 0] [0]
first(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [0]
0() = [0]
[0]
nil() = [0]
[0]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [0]
activate(x1) = [1 0] x1 + [3]
[0 2] [0]
sel(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
activate^#(x1) = [0 0] x1 + [0]
[0 0] [1]
first^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
sel^#(x1, x2) = [3 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [1]
c_2() = [0]
[0]
from^#(x1) = [0 0] x1 + [0]
[0 0] [0]
c_4() = [0]
[0]
c_5() = [0]
[0]
c_6() = [0]
[0]
c_7() = [0]
[0]
c_8() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict DPs: {activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
Weak DPs:
{ sel^#(0(), cons(X, Z)) -> c_2()
, activate^#(n__from(X)) -> from^#(X)
, activate^#(X) -> c_4()
, from^#(X) -> c_5()
, first^#(X1, X2) -> c_6()
, from^#(X) -> c_7()
, first^#(0(), Z) -> c_8()
, first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z))}
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->3:{1,9} [ YES(O(1),O(1)) ]
|
|->6:{3} [ subsumed ]
| |
| |->7:{5} [ YES(O(1),O(1)) ]
| |
| `->8:{7} [ YES(O(1),O(1)) ]
|
|->9:{4} [ YES(O(1),O(1)) ]
|
|->4:{6} [ YES(O(1),O(1)) ]
|
`->5:{8} [ YES(O(1),O(1)) ]
->1:{10} [ subsumed ]
|
`->2:{2} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{1: activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
WeakDPs DPs:
{ 2: sel^#(0(), cons(X, Z)) -> c_2()
, 3: activate^#(n__from(X)) -> from^#(X)
, 4: activate^#(X) -> c_4()
, 5: from^#(X) -> c_5()
, 6: first^#(X1, X2) -> c_6()
, 7: from^#(X) -> c_7()
, 8: first^#(0(), Z) -> c_8()
, 9: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 10: sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z))}
* Path 3:{1,9}: YES(O(1),O(1))
----------------------------
We consider the following Problem:
Strict DPs: {activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
together with the congruence-graph
->1:{1} Noncyclic, trivial, SCC
Here dependency-pairs are as follows:
Strict DPs:
{1: activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
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 3:{1,9}->6:{3}: subsumed
-----------------------------
This path is subsumed by the proof of paths 3:{1,9}->6:{3}->8:{7},
3:{1,9}->6:{3}->7:{5}.
* Path 3:{1,9}->6:{3}->7:{5}: YES(O(1),O(1))
------------------------------------------
We consider the following Problem:
Weak DPs:
{ first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, activate^#(n__first(X1, X2)) -> first^#(X1, X2)
, activate^#(n__from(X)) -> from^#(X)}
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
-->_1 activate^#(n__first(X1, X2)) -> first^#(X1, X2) :2
-->_1 activate^#(n__from(X)) -> from^#(X) :3
2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
-->_1 first^#(s(X), cons(Y, Z)) -> activate^#(Z) :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: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
, 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: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
, 3: activate^#(n__from(X)) -> from^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
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 3:{1,9}->6:{3}->8:{7}: YES(O(1),O(1))
------------------------------------------
We consider the following Problem:
Weak DPs:
{ first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, activate^#(n__first(X1, X2)) -> first^#(X1, X2)
, activate^#(n__from(X)) -> from^#(X)}
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
-->_1 activate^#(n__first(X1, X2)) -> first^#(X1, X2) :2
-->_1 activate^#(n__from(X)) -> from^#(X) :3
2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
-->_1 first^#(s(X), cons(Y, Z)) -> activate^#(Z) :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: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
, 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: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
, 3: activate^#(n__from(X)) -> from^#(X)}
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
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 3:{1,9}->9:{4}: YES(O(1),O(1))
-----------------------------------
We consider the following Problem:
Weak DPs:
{ first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
-->_1 activate^#(n__first(X1, X2)) -> first^#(X1, X2) :2
2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
-->_1 first^#(s(X), cons(Y, Z)) -> activate^#(Z) :1
together with the congruence-graph
->1:{1,2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
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 3:{1,9}->4:{6}: YES(O(1),O(1))
-----------------------------------
We consider the following Problem:
Weak DPs:
{ first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
-->_1 activate^#(n__first(X1, X2)) -> first^#(X1, X2) :2
2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
-->_1 first^#(s(X), cons(Y, Z)) -> activate^#(Z) :1
together with the congruence-graph
->1:{1,2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
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 3:{1,9}->5:{8}: YES(O(1),O(1))
-----------------------------------
We consider the following Problem:
Weak DPs:
{ first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
-->_1 activate^#(n__first(X1, X2)) -> first^#(X1, X2) :2
2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)
-->_1 first^#(s(X), cons(Y, Z)) -> activate^#(Z) :1
together with the congruence-graph
->1:{1,2} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: first^#(s(X), cons(Y, Z)) -> activate^#(Z)
, 2: activate^#(n__first(X1, X2)) -> first^#(X1, X2)}
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
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:{10}: subsumed
---------------------
This path is subsumed by the proof of paths 1:{10}->2:{2}.
* Path 1:{10}->2:{2}: YES(O(1),O(1))
----------------------------------
We consider the following Problem:
Weak DPs: {sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z))}
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z))
-->_1 sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z)) :1
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: sel^#(s(X), cons(Y, Z)) -> sel^#(X, activate(Z))}
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X
, from(X) -> n__from(X)
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> cons(X, n__from(s(X)))
, first(0(), Z) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))}
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))