*** 1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
bits(0()) -> 0()
bits(s(x)) -> s(bits(half(s(x))))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Weak DP Rules:
Weak TRS Rules:
Signature:
{bits/1,half/1} / {0/0,s/1}
Obligation:
Full
basic terms: {bits,half}/{0,s}
Applied Processor:
ToInnermost
Proof:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
*** 1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
bits(0()) -> 0()
bits(s(x)) -> s(bits(half(s(x))))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Weak DP Rules:
Weak TRS Rules:
Signature:
{bits/1,half/1} / {0/0,s/1}
Obligation:
Innermost
basic terms: {bits,half}/{0,s}
Applied Processor:
DependencyPairs {dpKind_ = WIDP}
Proof:
We add the following weak innermost dependency pairs:
Strict DPs
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
Weak DPs
and mark the set of starting terms.
*** 1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
bits(0()) -> 0()
bits(s(x)) -> s(bits(half(s(x))))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Weak DP Rules:
Weak TRS Rules:
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
*** 1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Weak DP Rules:
Weak TRS Rules:
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
Proof:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(s) = {1},
uargs(bits#) = {1},
uargs(c_2) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(bits) = [0]
p(half) = [1] x1 + [1]
p(s) = [1] x1 + [10]
p(bits#) = [1] x1 + [0]
p(half#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
Following rules are strictly oriented:
half(0()) = [1]
> [0]
= 0()
half(s(0())) = [11]
> [0]
= 0()
half(s(s(x))) = [1] x + [21]
> [1] x + [11]
= s(half(x))
Following rules are (at-least) weakly oriented:
bits#(0()) = [0]
>= [0]
= c_1()
bits#(s(x)) = [1] x + [10]
>= [1] x + [11]
= c_2(bits#(half(s(x))))
half#(0()) = [0]
>= [0]
= c_3()
half#(s(0())) = [0]
>= [0]
= c_4()
half#(s(s(x))) = [0]
>= [0]
= c_5(half#(x))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,3,4}
by application of
Pre({1,3,4}) = {2,5}.
Here rules are labelled as follows:
1: bits#(0()) -> c_1()
2: bits#(s(x)) ->
c_2(bits#(half(s(x))))
3: half#(0()) -> c_3()
4: half#(s(0())) -> c_4()
5: half#(s(s(x))) -> c_5(half#(x))
*** 1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
Weak DP Rules:
bits#(0()) -> c_1()
half#(0()) -> c_3()
half#(s(0())) -> c_4()
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:bits#(s(x)) -> c_2(bits#(half(s(x))))
-->_1 bits#(0()) -> c_1():3
-->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):1
2:S:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(0())) -> c_4():5
-->_1 half#(0()) -> c_3():4
-->_1 half#(s(s(x))) -> c_5(half#(x)):2
3:W:bits#(0()) -> c_1()
4:W:half#(0()) -> c_3()
5:W:half#(s(0())) -> c_4()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: half#(0()) -> c_3()
5: half#(s(0())) -> c_4()
3: bits#(0()) -> c_1()
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
Strict TRS Rules:
Weak DP Rules:
half#(s(s(x))) -> c_5(half#(x))
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Problem (S)
Strict DP Rules:
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
Weak DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
Strict TRS Rules:
Weak DP Rules:
half#(s(s(x))) -> c_5(half#(x))
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:bits#(s(x)) -> c_2(bits#(half(s(x))))
-->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):1
2:W:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(s(x))) -> c_5(half#(x)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: half#(s(s(x))) -> c_5(half#(x))
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: bits#(s(x)) ->
c_2(bits#(half(s(x))))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_2) = {1}
Following symbols are considered usable:
{half,bits#,half#}
TcT has computed the following interpretation:
p(0) = [1]
[2]
[2]
p(bits) = [0]
[1]
[0]
p(half) = [0 1 0] [0]
[0 1 0] x1 + [1]
[1 0 0] [2]
p(s) = [0 1 0] [3]
[0 1 0] x1 + [2]
[0 0 0] [0]
p(bits#) = [1 0 0] [2]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(half#) = [2 0 0] [0]
[0 2 2] x1 + [0]
[0 0 0] [0]
p(c_1) = [2]
[1]
[1]
p(c_2) = [1 0 0] [0]
[0 2 2] x1 + [0]
[0 2 1] [0]
p(c_3) = [0]
[0]
[2]
p(c_4) = [0]
[1]
[0]
p(c_5) = [0 0 0] [1]
[0 1 1] x1 + [0]
[0 0 2] [0]
Following rules are strictly oriented:
bits#(s(x)) = [0 1 0] [5]
[0 0 0] x + [0]
[0 0 0] [0]
> [0 1 0] [4]
[0 0 0] x + [0]
[0 0 0] [0]
= c_2(bits#(half(s(x))))
Following rules are (at-least) weakly oriented:
half(0()) = [2]
[3]
[3]
>= [1]
[2]
[2]
= 0()
half(s(0())) = [4]
[5]
[7]
>= [1]
[2]
[2]
= 0()
half(s(s(x))) = [0 1 0] [4]
[0 1 0] x + [5]
[0 1 0] [7]
>= [0 1 0] [4]
[0 1 0] x + [3]
[0 0 0] [0]
= s(half(x))
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:bits#(s(x)) -> c_2(bits#(half(s(x))))
-->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: bits#(s(x)) ->
c_2(bits#(half(s(x))))
*** 1.1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
Weak DP Rules:
bits#(s(x)) -> c_2(bits#(half(s(x))))
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(s(x))) -> c_5(half#(x)):1
2:W:bits#(s(x)) -> c_2(bits#(half(s(x))))
-->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: bits#(s(x)) ->
c_2(bits#(half(s(x))))
*** 1.1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
half#(s(s(x))) -> c_5(half#(x))
*** 1.1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: half#(s(s(x))) -> c_5(half#(x))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
half#(s(s(x))) -> c_5(half#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_5) = {1}
Following symbols are considered usable:
{bits#,half#}
TcT has computed the following interpretation:
p(0) = [1]
p(bits) = [1] x1 + [1]
p(half) = [1]
p(s) = [1] x1 + [1]
p(bits#) = [2] x1 + [0]
p(half#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [1] x1 + [0]
p(c_3) = [0]
p(c_4) = [4]
p(c_5) = [1] x1 + [1]
Following rules are strictly oriented:
half#(s(s(x))) = [1] x + [2]
> [1] x + [1]
= c_5(half#(x))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
half#(s(s(x))) -> c_5(half#(x))
Weak TRS Rules:
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
half#(s(s(x))) -> c_5(half#(x))
Weak TRS Rules:
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(s(x))) -> c_5(half#(x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: half#(s(s(x))) -> c_5(half#(x))
*** 1.1.1.1.1.1.1.2.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
Obligation:
Innermost
basic terms: {bits#,half#}/{0,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).