*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
max(L(x)) -> x
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Weak DP Rules:
Weak TRS Rules:
Signature:
{max/1} / {0/0,L/1,N/2,s/1}
Obligation:
Full
basic terms: {max}/{0,L,N,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^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
max(L(x)) -> x
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Weak DP Rules:
Weak TRS Rules:
Signature:
{max/1} / {0/0,L/1,N/2,s/1}
Obligation:
Innermost
basic terms: {max}/{0,L,N,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
max#(L(x)) -> c_1()
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
max#(N(L(0()),L(y))) -> c_3()
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Weak DPs
and mark the set of starting terms.
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
max#(L(x)) -> c_1()
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
max#(N(L(0()),L(y))) -> c_3()
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
max(L(x)) -> x
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
max#(L(x)) -> c_1()
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
max#(N(L(0()),L(y))) -> c_3()
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
max#(L(x)) -> c_1()
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
max#(N(L(0()),L(y))) -> c_3()
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,3}
by application of
Pre({1,3}) = {2,4}.
Here rules are labelled as follows:
1: max#(L(x)) -> c_1()
2: max#(N(L(x),N(y,z))) ->
c_2(max#(N(L(x),L(max(N(y,z)))))
,max#(N(y,z)))
3: max#(N(L(0()),L(y))) -> c_3()
4: max#(N(L(s(x)),L(s(y)))) ->
c_4(max#(N(L(x),L(y))))
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Strict TRS Rules:
Weak DP Rules:
max#(L(x)) -> c_1()
max#(N(L(0()),L(y))) -> c_3()
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
-->_2 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):2
-->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):2
-->_2 max#(N(L(0()),L(y))) -> c_3():4
-->_1 max#(N(L(0()),L(y))) -> c_3():4
-->_2 max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z))):1
2:S:max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
-->_1 max#(N(L(0()),L(y))) -> c_3():4
-->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):2
3:W:max#(L(x)) -> c_1()
4:W:max#(N(L(0()),L(y))) -> c_3()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: max#(L(x)) -> c_1()
4: max#(N(L(0()),L(y))) -> c_3()
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
Proof:
We decompose the input problem according to the dependency graph into the upper component
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
and a lower component
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Further, following extension rules are added to the lower component.
max#(N(L(x),N(y,z))) -> max#(N(y,z))
max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: max#(N(L(x),N(y,z))) ->
c_2(max#(N(L(x),L(max(N(y,z)))))
,max#(N(y,z)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
NaturalMI {miDimension = 2, 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) = {1,2}
Following symbols are considered usable:
{max#}
TcT has computed the following interpretation:
p(0) = [1]
[1]
p(L) = [0]
[0]
p(N) = [0 1] x2 + [0]
[0 1] [2]
p(max) = [0 0] x1 + [0]
[1 0] [0]
p(s) = [0]
[0]
p(max#) = [1 0] x1 + [0]
[0 0] [1]
p(c_1) = [1]
[2]
p(c_2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 1] [0]
p(c_3) = [2]
[0]
p(c_4) = [2]
[2]
Following rules are strictly oriented:
max#(N(L(x),N(y,z))) = [0 1] z + [2]
[0 0] [1]
> [0 1] z + [1]
[0 0] [1]
= c_2(max#(N(L(x),L(max(N(y,z)))))
,max#(N(y,z)))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z)))
-->_2 max#(N(L(x),N(y,z))) -> c_2(max#(N(L(x),L(max(N(y,z))))),max#(N(y,z))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: max#(N(L(x),N(y,z))) ->
c_2(max#(N(L(x),L(max(N(y,z)))))
,max#(N(y,z)))
*** 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:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Strict TRS Rules:
Weak DP Rules:
max#(N(L(x),N(y,z))) -> max#(N(y,z))
max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,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: max#(N(L(s(x)),L(s(y)))) ->
c_4(max#(N(L(x),L(y))))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Strict TRS Rules:
Weak DP Rules:
max#(N(L(x),N(y,z))) -> max#(N(y,z))
max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,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_4) = {1}
Following symbols are considered usable:
{max,max#}
TcT has computed the following interpretation:
p(0) = [2]
p(L) = [1] x1 + [0]
p(N) = [1] x1 + [1] x2 + [0]
p(max) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(max#) = [8] x1 + [4]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [1]
p(c_3) = [1]
p(c_4) = [1] x1 + [15]
Following rules are strictly oriented:
max#(N(L(s(x)),L(s(y)))) = [8] x + [8] y + [20]
> [8] x + [8] y + [19]
= c_4(max#(N(L(x),L(y))))
Following rules are (at-least) weakly oriented:
max#(N(L(x),N(y,z))) = [8] x + [8] y + [8] z + [4]
>= [8] y + [8] z + [4]
= max#(N(y,z))
max#(N(L(x),N(y,z))) = [8] x + [8] y + [8] z + [4]
>= [8] x + [8] y + [8] z + [4]
= max#(N(L(x),L(max(N(y,z)))))
max(N(L(x),N(y,z))) = [1] x + [1] y + [1] z + [0]
>= [1] x + [1] y + [1] z + [0]
= max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) = [1] y + [2]
>= [1] y + [0]
= y
max(N(L(s(x)),L(s(y)))) = [1] x + [1] y + [2]
>= [1] x + [1] y + [1]
= s(max(N(L(x),L(y))))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
max#(N(L(x),N(y,z))) -> max#(N(y,z))
max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
max#(N(L(x),N(y,z))) -> max#(N(y,z))
max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:max#(N(L(x),N(y,z))) -> max#(N(y,z))
-->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):3
-->_1 max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z))))):2
-->_1 max#(N(L(x),N(y,z))) -> max#(N(y,z)):1
2:W:max#(N(L(x),N(y,z))) -> max#(N(L(x),L(max(N(y,z)))))
-->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):3
3:W:max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y))))
-->_1 max#(N(L(s(x)),L(s(y)))) -> c_4(max#(N(L(x),L(y)))):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: max#(N(L(x),N(y,z))) -> max#(N(y
,z))
2: max#(N(L(x),N(y,z))) ->
max#(N(L(x),L(max(N(y,z)))))
3: max#(N(L(s(x)),L(s(y)))) ->
c_4(max#(N(L(x),L(y))))
*** 1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
max(N(L(0()),L(y))) -> y
max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
Signature:
{max/1,max#/1} / {0/0,L/1,N/2,s/1,c_1/0,c_2/2,c_3/0,c_4/1}
Obligation:
Innermost
basic terms: {max#}/{0,L,N,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).