We consider the following Problem:
Strict Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, 0()) -> N
, plus(N, s(M)) -> U11(tt(), M, N)
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, 0()) -> N
, plus(N, s(M)) -> U11(tt(), M, N)
, 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: {plus(N, 0()) -> N}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {}, Uargs(U12) = {2, 3}, Uargs(activate) = {},
Uargs(s) = {1}, Uargs(plus) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 1] [1 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [1 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [1]
0() = [2]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, s(M)) -> U11(tt(), M, N)
, activate(X) -> X}
Weak Trs: {plus(N, 0()) -> N}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {plus(N, s(M)) -> U11(tt(), M, N)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {}, Uargs(U12) = {2, 3}, Uargs(activate) = {},
Uargs(s) = {1}, Uargs(plus) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 1] [0 0] [0 0] [1]
tt() = [0]
[0]
U12(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[1 0] [1 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [3]
[0 0] [1]
plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [1]
0() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, activate(X) -> X}
Weak Trs:
{ plus(N, s(M)) -> U11(tt(), M, N)
, plus(N, 0()) -> N}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{U12(tt(), M, N) -> s(plus(activate(N), activate(M)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(U11) = {}, Uargs(U12) = {2, 3}, Uargs(activate) = {},
Uargs(s) = {1}, Uargs(plus) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
[0 0] [0 0] [1 1] [0]
tt() = [3]
[1]
U12(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [3]
[1 1] [1 0] [0 0] [1]
activate(x1) = [1 0] x1 + [0]
[0 0] [0]
s(x1) = [1 0] x1 + [1]
[0 0] [1]
plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[1 1] [0 0] [0]
0() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))
, activate(X) -> X}
Weak Trs:
{ U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, s(M)) -> U11(tt(), M, N)
, plus(N, 0()) -> N}
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(U11) = {}, Uargs(U12) = {2, 3}, Uargs(activate) = {},
Uargs(s) = {1}, Uargs(plus) = {1, 2}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
U11(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [0]
[0 0] [0 0] [0 1] [1]
tt() = [1]
[3]
U12(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [3]
[1 1] [1 0] [0 1] [1]
activate(x1) = [1 0] x1 + [2]
[0 1] [0]
s(x1) = [1 0] x1 + [0]
[0 1] [3]
plus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [1]
0() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))}
Weak Trs:
{ activate(X) -> X
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, s(M)) -> U11(tt(), M, N)
, plus(N, 0()) -> N}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))}
Weak Trs:
{ activate(X) -> X
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, s(M)) -> U11(tt(), M, N)
, plus(N, 0()) -> N}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs:
{U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
Weak DPs:
{ activate^#(X) -> c_2()
, U12^#(tt(), M, N) -> plus^#(activate(N), activate(M))
, plus^#(N, s(M)) -> U11^#(tt(), M, N)
, plus^#(N, 0()) -> c_5()}
We consider the following Problem:
Strict DPs:
{U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
Strict Trs:
{U11(tt(), M, N) -> U12(tt(), activate(M), activate(N))}
Weak DPs:
{ activate^#(X) -> c_2()
, U12^#(tt(), M, N) -> plus^#(activate(N), activate(M))
, plus^#(N, s(M)) -> U11^#(tt(), M, N)
, plus^#(N, 0()) -> c_5()}
Weak Trs:
{ activate(X) -> X
, U12(tt(), M, N) -> s(plus(activate(N), activate(M)))
, plus(N, s(M)) -> U11(tt(), M, N)
, plus(N, 0()) -> N}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Weak Usable Rules: {activate(X) -> X}
We consider the following Problem:
Strict DPs:
{U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
Weak DPs:
{ activate^#(X) -> c_2()
, U12^#(tt(), M, N) -> plus^#(activate(N), activate(M))
, plus^#(N, s(M)) -> U11^#(tt(), M, N)
, plus^#(N, 0()) -> c_5()}
Weak Trs: {activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs:
{U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
Weak DPs:
{ activate^#(X) -> c_2()
, U12^#(tt(), M, N) -> plus^#(activate(N), activate(M))
, plus^#(N, s(M)) -> U11^#(tt(), M, N)
, plus^#(N, 0()) -> c_5()}
Weak Trs: {activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->2:{1,4,3} [ YES(O(1),O(1)) ]
|
`->3:{5} [ YES(O(1),O(1)) ]
->1:{2} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{1: U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
WeakDPs DPs:
{ 2: activate^#(X) -> c_2()
, 3: U12^#(tt(), M, N) -> plus^#(activate(N), activate(M))
, 4: plus^#(N, s(M)) -> U11^#(tt(), M, N)
, 5: plus^#(N, 0()) -> c_5()}
* Path 2:{1,4,3}: YES(O(1),O(1))
------------------------------
We consider the following Problem:
Strict DPs:
{U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
Weak Trs: {activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))
together with the congruence-graph
->1:{1} Noncyclic, trivial, SCC
Here dependency-pairs are as follows:
Strict DPs:
{1: U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
We consider the following Problem:
Weak Trs: {activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs: {activate(X) -> 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:{1,4,3}->3:{5}: YES(O(1),O(1))
-------------------------------------
We consider the following Problem:
Weak DPs:
{ plus^#(N, s(M)) -> U11^#(tt(), M, N)
, U12^#(tt(), M, N) -> plus^#(activate(N), activate(M))
, U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
Weak Trs: {activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: plus^#(N, s(M)) -> U11^#(tt(), M, N)
-->_1 U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N)) :3
2: U12^#(tt(), M, N) -> plus^#(activate(N), activate(M))
-->_1 plus^#(N, s(M)) -> U11^#(tt(), M, N) :1
3: U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))
-->_1 U12^#(tt(), M, N) -> plus^#(activate(N), activate(M)) :2
together with the congruence-graph
->1:{1,2,3} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{ 1: plus^#(N, s(M)) -> U11^#(tt(), M, N)
, 2: U12^#(tt(), M, N) -> plus^#(activate(N), activate(M))
, 3: U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{ 1: plus^#(N, s(M)) -> U11^#(tt(), M, N)
, 2: U12^#(tt(), M, N) -> plus^#(activate(N), activate(M))
, 3: U11^#(tt(), M, N) -> U12^#(tt(), activate(M), activate(N))}
We consider the following Problem:
Weak Trs: {activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs: {activate(X) -> 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:{2}: YES(O(1),O(1))
--------------------------
We consider the following Problem:
Weak Trs: {activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs: {activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs: {activate(X) -> 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))