We consider the following Problem:
Strict Trs:
{ 2nd(cons1(X, cons(Y, Z))) -> Y
, 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
, 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}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ 2nd(cons1(X, cons(Y, Z))) -> Y
, 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
, 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}
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)
, s(X) -> n__s(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(2nd) = {1}, Uargs(cons1) = {2}, Uargs(cons) = {},
Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
Uargs(n__s) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
2nd(x1) = [1 1] x1 + [1]
[0 0] [1]
cons1(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [0]
activate(x1) = [1 0] x1 + [0]
[0 0] [1]
from(x1) = [1 0] x1 + [1]
[0 0] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ 2nd(cons1(X, cons(Y, Z))) -> Y
, 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
, from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X}
Weak Trs:
{ from(X) -> n__from(X)
, s(X) -> n__s(X)}
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(2nd) = {1}, Uargs(cons1) = {2}, Uargs(cons) = {},
Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
Uargs(n__s) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
2nd(x1) = [1 1] x1 + [1]
[0 0] [1]
cons1(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [0]
activate(x1) = [1 0] x1 + [2]
[0 1] [0]
from(x1) = [1 0] x1 + [3]
[0 1] [1]
n__from(x1) = [1 0] x1 + [0]
[0 0] [1]
n__s(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ 2nd(cons1(X, cons(Y, Z))) -> Y
, 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
, from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))}
Weak Trs:
{ activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
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, X1)) -> 2nd(cons1(X, activate(X1)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(2nd) = {1}, Uargs(cons1) = {2}, Uargs(cons) = {},
Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
Uargs(n__s) = {}, Uargs(s) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
2nd(x1) = [1 1] x1 + [1]
[0 0] [1]
cons1(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [2]
activate(x1) = [1 0] x1 + [0]
[0 1] [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]
s(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:
{ 2nd(cons1(X, cons(Y, Z))) -> Y
, from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))}
Weak Trs:
{ 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ 2nd(cons1(X, cons(Y, Z))) -> Y
, from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))}
Weak Trs:
{ 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1()
, from^#(X) -> c_2()
, activate^#(n__from(X)) -> from^#(activate(X))
, activate^#(n__s(X)) -> s^#(activate(X))}
Weak DPs:
{ 2nd^#(cons(X, X1)) -> 2nd^#(cons1(X, activate(X1)))
, activate^#(X) -> c_6()
, from^#(X) -> c_7()
, s^#(X) -> c_8()}
We consider the following Problem:
Strict DPs:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1()
, from^#(X) -> c_2()
, activate^#(n__from(X)) -> from^#(activate(X))
, activate^#(n__s(X)) -> s^#(activate(X))}
Strict Trs:
{ 2nd(cons1(X, cons(Y, Z))) -> Y
, from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))}
Weak DPs:
{ 2nd^#(cons(X, X1)) -> 2nd^#(cons1(X, activate(X1)))
, activate^#(X) -> c_6()
, from^#(X) -> c_7()
, s^#(X) -> c_8()}
Weak Trs:
{ 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Strict Usable Rules:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))}
Weak Usable Rules:
{ activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
We consider the following Problem:
Strict DPs:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1()
, from^#(X) -> c_2()
, activate^#(n__from(X)) -> from^#(activate(X))
, activate^#(n__s(X)) -> s^#(activate(X))}
Strict Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))}
Weak DPs:
{ 2nd^#(cons(X, X1)) -> 2nd^#(cons1(X, activate(X1)))
, activate^#(X) -> c_6()
, from^#(X) -> c_7()
, s^#(X) -> c_8()}
Weak Trs:
{ activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
Dependency Pairs:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1()
, activate^#(n__from(X)) -> from^#(activate(X))
, activate^#(n__s(X)) -> s^#(activate(X))}
TRS Component:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))}
Interpretation of constant growth:
----------------------------------
The following argument positions are usable:
Uargs(2nd) = {}, Uargs(cons1) = {2}, Uargs(cons) = {},
Uargs(activate) = {}, Uargs(from) = {1}, Uargs(n__from) = {},
Uargs(n__s) = {}, Uargs(s) = {1}, Uargs(2nd^#) = {1},
Uargs(from^#) = {1}, Uargs(activate^#) = {}, Uargs(s^#) = {1}
We have the following constructor-based EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
2nd(x1) = [0 0] x1 + [0]
[0 0] [0]
cons1(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [1 0] [0]
activate(x1) = [2 0] x1 + [0]
[2 2] [0]
from(x1) = [1 0] x1 + [2]
[1 0] [3]
n__from(x1) = [1 0] x1 + [2]
[1 0] [0]
n__s(x1) = [1 0] x1 + [1]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
2nd^#(x1) = [1 2] x1 + [1]
[0 0] [1]
c_1() = [0]
[0]
from^#(x1) = [1 0] x1 + [0]
[0 0] [0]
c_2() = [0]
[0]
activate^#(x1) = [2 2] x1 + [0]
[0 0] [0]
s^#(x1) = [1 0] x1 + [0]
[0 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: {from^#(X) -> c_2()}
Weak DPs:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1()
, activate^#(n__from(X)) -> from^#(activate(X))
, activate^#(n__s(X)) -> s^#(activate(X))
, 2nd^#(cons(X, X1)) -> 2nd^#(cons1(X, activate(X1)))
, activate^#(X) -> c_6()
, from^#(X) -> c_7()
, s^#(X) -> c_8()}
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->5:{3} [ subsumed ]
|
|->8:{1} [ YES(O(1),O(1)) ]
|
`->6:{7} [ YES(O(1),O(1)) ]
->3:{4} [ subsumed ]
|
`->4:{8} [ YES(O(1),O(1)) ]
->2:{5} [ subsumed ]
|
`->7:{2} [ YES(O(1),O(1)) ]
->1:{6} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{1: from^#(X) -> c_2()}
WeakDPs DPs:
{ 2: 2nd^#(cons1(X, cons(Y, Z))) -> c_1()
, 3: activate^#(n__from(X)) -> from^#(activate(X))
, 4: activate^#(n__s(X)) -> s^#(activate(X))
, 5: 2nd^#(cons(X, X1)) -> 2nd^#(cons1(X, activate(X1)))
, 6: activate^#(X) -> c_6()
, 7: from^#(X) -> c_7()
, 8: s^#(X) -> c_8()}
* Path 5:{3}: subsumed
--------------------
This path is subsumed by the proof of paths 5:{3}->8:{1},
5:{3}->6:{7}.
* Path 5:{3}->8:{1}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Strict DPs: {from^#(X) -> c_2()}
Weak DPs: {activate^#(n__from(X)) -> from^#(activate(X))}
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: from^#(X) -> c_2()
2: activate^#(n__from(X)) -> from^#(activate(X))
-->_1 from^#(X) -> c_2() :1
together with the congruence-graph
->1:{2} Weak SCC
|
`->2:{1} Noncyclic, trivial, SCC
Here dependency-pairs are as follows:
Strict DPs:
{1: from^#(X) -> c_2()}
WeakDPs DPs:
{2: activate^#(n__from(X)) -> from^#(activate(X))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 2: activate^#(n__from(X)) -> from^#(activate(X))
, 1: from^#(X) -> c_2()}
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
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:{3}->6:{7}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Weak DPs: {activate^#(n__from(X)) -> from^#(activate(X))}
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__from(X)) -> from^#(activate(X))
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: activate^#(n__from(X)) -> from^#(activate(X))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: activate^#(n__from(X)) -> from^#(activate(X))}
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
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:{4}: subsumed
--------------------
This path is subsumed by the proof of paths 3:{4}->4:{8}.
* Path 3:{4}->4:{8}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Weak DPs: {activate^#(n__s(X)) -> s^#(activate(X))}
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: activate^#(n__s(X)) -> s^#(activate(X))
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: activate^#(n__s(X)) -> s^#(activate(X))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: activate^#(n__s(X)) -> s^#(activate(X))}
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
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:{5}: subsumed
--------------------
This path is subsumed by the proof of paths 2:{5}->7:{2}.
* Path 2:{5}->7:{2}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Weak DPs: {2nd^#(cons(X, X1)) -> 2nd^#(cons1(X, activate(X1)))}
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: 2nd^#(cons(X, X1)) -> 2nd^#(cons1(X, activate(X1)))
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: 2nd^#(cons(X, X1)) -> 2nd^#(cons1(X, activate(X1)))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: 2nd^#(cons(X, X1)) -> 2nd^#(cons1(X, activate(X1)))}
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
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:{6}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ from(X) -> cons(X, n__from(n__s(X)))
, activate(n__from(X)) -> from(activate(X))
, activate(n__s(X)) -> s(activate(X))
, activate(X) -> X
, from(X) -> n__from(X)
, s(X) -> n__s(X)}
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))