We consider the following Problem:
Strict Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, bits(0()) -> 0()
, bits(s(x)) -> s(bits(half(s(x))))}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, bits(0()) -> 0()
, bits(s(x)) -> s(bits(half(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:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, bits(0()) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(half) = {}, Uargs(s) = {1}, Uargs(bits) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
half(x1) = [0 0] x1 + [1]
[0 0] [1]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [1]
bits(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:
{ half(s(s(x))) -> s(half(x))
, bits(s(x)) -> s(bits(half(s(x))))}
Weak Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, bits(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {bits(s(x)) -> s(bits(half(s(x))))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(half) = {}, Uargs(s) = {1}, Uargs(bits) = {1}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
half(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [0]
[0 0] [3]
bits(x1) = [1 2] x1 + [2]
[0 0] [3]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs: {half(s(s(x))) -> s(half(x))}
Weak Trs:
{ bits(s(x)) -> s(bits(half(s(x))))
, half(0()) -> 0()
, half(s(0())) -> 0()
, bits(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict Trs: {half(s(s(x))) -> s(half(x))}
Weak Trs:
{ bits(s(x)) -> s(bits(half(s(x))))
, half(0()) -> 0()
, half(s(0())) -> 0()
, bits(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We have computed the following dependency pairs
Strict DPs: {half^#(s(s(x))) -> half^#(x)}
Weak DPs:
{ bits^#(s(x)) -> bits^#(half(s(x)))
, half^#(0()) -> c_3()
, half^#(s(0())) -> c_4()
, bits^#(0()) -> c_5()}
We consider the following Problem:
Strict DPs: {half^#(s(s(x))) -> half^#(x)}
Strict Trs: {half(s(s(x))) -> s(half(x))}
Weak DPs:
{ bits^#(s(x)) -> bits^#(half(s(x)))
, half^#(0()) -> c_3()
, half^#(s(0())) -> c_4()
, bits^#(0()) -> c_5()}
Weak Trs:
{ bits(s(x)) -> s(bits(half(s(x))))
, half(0()) -> 0()
, half(s(0())) -> 0()
, bits(0()) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We replace strict/weak-rules by the corresponding usable rules:
Strict Usable Rules: {half(s(s(x))) -> s(half(x))}
Weak Usable Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()}
We consider the following Problem:
Strict DPs: {half^#(s(s(x))) -> half^#(x)}
Strict Trs: {half(s(s(x))) -> s(half(x))}
Weak DPs:
{ bits^#(s(x)) -> bits^#(half(s(x)))
, half^#(0()) -> c_3()
, half^#(s(0())) -> c_4()
, bits^#(0()) -> c_5()}
Weak Trs:
{ half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {half(s(s(x))) -> s(half(x))}
Interpretation of constant growth:
----------------------------------
The following argument positions are usable:
Uargs(half) = {}, Uargs(s) = {1}, Uargs(bits) = {},
Uargs(half^#) = {}, Uargs(bits^#) = {1}
We have the following constructor-based EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
half(x1) = [0 1] x1 + [0]
[1 0] [0]
0() = [0]
[0]
s(x1) = [1 0] x1 + [1]
[0 1] [1]
bits(x1) = [0 0] x1 + [0]
[0 0] [0]
half^#(x1) = [0 0] x1 + [1]
[0 1] [0]
bits^#(x1) = [1 1] x1 + [1]
[0 0] [1]
c_3() = [0]
[0]
c_4() = [0]
[0]
c_5() = [0]
[0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict DPs: {half^#(s(s(x))) -> half^#(x)}
Weak DPs:
{ bits^#(s(x)) -> bits^#(half(s(x)))
, half^#(0()) -> c_3()
, half^#(s(0())) -> c_4()
, bits^#(0()) -> c_5()}
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We use following congruence DG for path analysis
->3:{1} [ YES(?,O(n^1)) ]
|
|->4:{3} [ YES(O(1),O(1)) ]
|
`->5:{4} [ YES(O(1),O(1)) ]
->1:{2} [ subsumed ]
|
`->2:{5} [ YES(O(1),O(1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{1: half^#(s(s(x))) -> half^#(x)}
WeakDPs DPs:
{ 2: bits^#(s(x)) -> bits^#(half(s(x)))
, 3: half^#(0()) -> c_3()
, 4: half^#(s(0())) -> c_4()
, 5: bits^#(0()) -> c_5()}
* Path 3:{1}: YES(?,O(n^1))
-------------------------
We consider the following Problem:
Strict DPs: {half^#(s(s(x))) -> half^#(x)}
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {half^#(s(s(x))) -> half^#(x)}
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
We consider the following Problem:
Strict DPs: {half^#(s(s(x))) -> half^#(x)}
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
No rule is usable.
We consider the following Problem:
Strict DPs: {half^#(s(s(x))) -> half^#(x)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^1))
Proof:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ s_0(2) -> 2
, half^#_0(2) -> 1
, half^#_1(2) -> 1}
* Path 3:{1}->4:{3}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Weak DPs: {half^#(s(s(x))) -> half^#(x)}
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: half^#(s(s(x))) -> half^#(x)
-->_1 half^#(s(s(x))) -> half^#(x) :1
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: half^#(s(s(x))) -> half^#(x)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: half^#(s(s(x))) -> half^#(x)}
We consider the following Problem:
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
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}->5:{4}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Weak DPs: {half^#(s(s(x))) -> half^#(x)}
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: half^#(s(s(x))) -> half^#(x)
-->_1 half^#(s(s(x))) -> half^#(x) :1
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: half^#(s(s(x))) -> half^#(x)}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: half^#(s(s(x))) -> half^#(x)}
We consider the following Problem:
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
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}: subsumed
--------------------
This path is subsumed by the proof of paths 1:{2}->2:{5}.
* Path 1:{2}->2:{5}: YES(O(1),O(1))
---------------------------------
We consider the following Problem:
Weak DPs: {bits^#(s(x)) -> bits^#(half(s(x)))}
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the the dependency-graph
1: bits^#(s(x)) -> bits^#(half(s(x)))
-->_1 bits^#(s(x)) -> bits^#(half(s(x))) :1
together with the congruence-graph
->1:{1} Weak SCC
Here dependency-pairs are as follows:
WeakDPs DPs:
{1: bits^#(s(x)) -> bits^#(half(s(x)))}
The following rules are either leafs or part of trailing weak paths, and thus they can be removed:
{1: bits^#(s(x)) -> bits^#(half(s(x)))}
We consider the following Problem:
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(O(1),O(1))
Proof:
We consider the following Problem:
Weak Trs:
{ half(s(s(x))) -> s(half(x))
, half(0()) -> 0()
, half(s(0())) -> 0()}
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))