*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
-(x,0()) -> x
-(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
-(0(),y) -> 0()
p(0()) -> 0()
p(s(x)) -> x
Weak DP Rules:
Weak TRS Rules:
Signature:
{-/2,p/1} / {0/0,greater/2,if/3,s/1}
Obligation:
Full
basic terms: {-,p}/{0,greater,if,s}
Applied Processor:
DependencyPairs {dpKind_ = WIDP}
Proof:
We add the following weak dependency pairs:
Strict DPs
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-#(0(),y) -> c_3()
p#(0()) -> c_4()
p#(s(x)) -> c_5(x)
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-#(0(),y) -> c_3()
p#(0()) -> c_4()
p#(s(x)) -> c_5(x)
Strict TRS Rules:
-(x,0()) -> x
-(x,s(y)) -> if(greater(x,s(y)),s(-(x,p(s(y)))),0())
-(0(),y) -> 0()
p(0()) -> 0()
p(s(x)) -> x
Weak DP Rules:
Weak TRS Rules:
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
p(s(x)) -> x
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-#(0(),y) -> c_3()
p#(0()) -> c_4()
p#(s(x)) -> c_5(x)
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-#(0(),y) -> c_3()
p#(0()) -> c_4()
p#(s(x)) -> c_5(x)
Strict TRS Rules:
p(s(x)) -> x
Weak DP Rules:
Weak TRS Rules:
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,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(-#) = {2},
uargs(c_2) = {3}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(-) = [0]
p(0) = [0]
p(greater) = [1] x1 + [1] x2 + [0]
p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
p(p) = [1] x1 + [0]
p(s) = [1] x1 + [3]
p(-#) = [1] x2 + [0]
p(p#) = [0]
p(c_1) = [0]
p(c_2) = [1] x3 + [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
Following rules are strictly oriented:
p(s(x)) = [1] x + [3]
> [1] x + [0]
= x
Following rules are (at-least) weakly oriented:
-#(x,0()) = [0]
>= [0]
= c_1(x)
-#(x,s(y)) = [1] y + [3]
>= [1] y + [3]
= c_2(x,y,-#(x,p(s(y))))
-#(0(),y) = [1] y + [0]
>= [0]
= c_3()
p#(0()) = [0]
>= [0]
= c_4()
p#(s(x)) = [0]
>= [0]
= c_5(x)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-#(0(),y) -> c_3()
p#(0()) -> c_4()
p#(s(x)) -> c_5(x)
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{3,4}
by application of
Pre({3,4}) = {1,2,5}.
Here rules are labelled as follows:
1: -#(x,0()) -> c_1(x)
2: -#(x,s(y)) -> c_2(x
,y
,-#(x,p(s(y))))
3: -#(0(),y) -> c_3()
4: p#(0()) -> c_4()
5: p#(s(x)) -> c_5(x)
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
Strict TRS Rules:
Weak DP Rules:
-#(0(),y) -> c_3()
p#(0()) -> c_4()
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:-#(x,0()) -> c_1(x)
-->_1 p#(s(x)) -> c_5(x):3
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 p#(0()) -> c_4():5
-->_1 -#(0(),y) -> c_3():4
-->_1 -#(x,0()) -> c_1(x):1
2:S:-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-->_2 p#(s(x)) -> c_5(x):3
-->_1 p#(s(x)) -> c_5(x):3
-->_2 p#(0()) -> c_4():5
-->_1 p#(0()) -> c_4():5
-->_3 -#(0(),y) -> c_3():4
-->_2 -#(0(),y) -> c_3():4
-->_1 -#(0(),y) -> c_3():4
-->_3 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_2 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_3 -#(x,0()) -> c_1(x):1
-->_2 -#(x,0()) -> c_1(x):1
-->_1 -#(x,0()) -> c_1(x):1
3:S:p#(s(x)) -> c_5(x)
-->_1 p#(0()) -> c_4():5
-->_1 -#(0(),y) -> c_3():4
-->_1 p#(s(x)) -> c_5(x):3
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,0()) -> c_1(x):1
4:W:-#(0(),y) -> c_3()
5:W:p#(0()) -> c_4()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: -#(0(),y) -> c_3()
5: p#(0()) -> c_4()
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,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:
3: p#(s(x)) -> c_5(x)
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,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_2) = {3}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(-) = [2] x1 + [8] x2 + [8]
p(0) = [1]
p(greater) = [4]
p(if) = [1] x1 + [0]
p(p) = [1] x1 + [7]
p(s) = [1] x1 + [1]
p(-#) = [0]
p(p#) = [9]
p(c_1) = [0]
p(c_2) = [8] x3 + [0]
p(c_3) = [1]
p(c_4) = [0]
p(c_5) = [1]
Following rules are strictly oriented:
p#(s(x)) = [9]
> [1]
= c_5(x)
Following rules are (at-least) weakly oriented:
-#(x,0()) = [0]
>= [0]
= c_1(x)
-#(x,s(y)) = [0]
>= [0]
= c_2(x,y,-#(x,p(s(y))))
p(s(x)) = [1] x + [8]
>= [1] x + [0]
= x
*** 1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
Strict TRS Rules:
Weak DP Rules:
p#(s(x)) -> c_5(x)
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
Strict TRS Rules:
Weak DP Rules:
p#(s(x)) -> c_5(x)
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,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: -#(x,0()) -> c_1(x)
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:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
Strict TRS Rules:
Weak DP Rules:
p#(s(x)) -> c_5(x)
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,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_2) = {3}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(-) = [2] x1 + [2] x2 + [0]
p(0) = [2]
p(greater) = [1] x1 + [1] x2 + [1]
p(if) = [1] x3 + [1]
p(p) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(-#) = [2] x1 + [8] x2 + [0]
p(p#) = [9] x1 + [0]
p(c_1) = [2] x1 + [1]
p(c_2) = [1] x3 + [0]
p(c_3) = [0]
p(c_4) = [8]
p(c_5) = [0]
Following rules are strictly oriented:
-#(x,0()) = [2] x + [16]
> [2] x + [1]
= c_1(x)
Following rules are (at-least) weakly oriented:
-#(x,s(y)) = [2] x + [8] y + [0]
>= [2] x + [8] y + [0]
= c_2(x,y,-#(x,p(s(y))))
p#(s(x)) = [9] x + [0]
>= [0]
= c_5(x)
p(s(x)) = [1] x + [0]
>= [1] x + [0]
= x
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
Strict TRS Rules:
Weak DP Rules:
-#(x,0()) -> c_1(x)
p#(s(x)) -> c_5(x)
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
Strict TRS Rules:
Weak DP Rules:
-#(x,0()) -> c_1(x)
p#(s(x)) -> c_5(x)
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: -#(x,s(y)) -> c_2(x
,y
,-#(x,p(s(y))))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
Strict TRS Rules:
Weak DP Rules:
-#(x,0()) -> c_1(x)
p#(s(x)) -> c_5(x)
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, 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 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_2) = {3}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(-) = [0 0 2] [1 0 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[1 2 1] [1 0 1] [2]
p(0) = [1]
[3]
[1]
p(greater) = [1 0 0] [0 2 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [1]
[0 0 0] [0 0 0] [1]
p(if) = [0 1 0] [0 0 1] [1]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
p(p) = [1 0 0] [0]
[1 0 0] x1 + [1]
[0 1 0] [0]
p(s) = [1 2 0] [0]
[0 0 1] x1 + [0]
[0 0 1] [2]
p(-#) = [0 0 0] [0 0 2] [0]
[0 0 2] x1 + [0 1 0] x2 + [1]
[0 3 0] [2 1 1] [0]
p(p#) = [2 0 0] [0]
[2 3 1] x1 + [3]
[1 0 2] [2]
p(c_1) = [2]
[0]
[1]
p(c_2) = [0 0 0] [0 0 0] [1 0
0] [3]
[0 0 2] x1 + [0 0 1] x2 + [0 0
0] x3 + [1]
[0 1 0] [2 3 2] [0 0
0] [0]
p(c_3) = [1]
[2]
[0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [1 0 0] [0]
[0 0 0] x1 + [0]
[1 0 1] [1]
Following rules are strictly oriented:
-#(x,s(y)) = [0 0 0] [0 0 2] [4]
[0 0 2] x + [0 0 1] y + [1]
[0 3 0] [2 4 2] [2]
> [0 0 0] [0 0 2] [3]
[0 0 2] x + [0 0 1] y + [1]
[0 1 0] [2 3 2] [0]
= c_2(x,y,-#(x,p(s(y))))
Following rules are (at-least) weakly oriented:
-#(x,0()) = [0 0 0] [2]
[0 0 2] x + [4]
[0 3 0] [6]
>= [2]
[0]
[1]
= c_1(x)
p#(s(x)) = [2 4 0] [0]
[2 4 4] x + [5]
[1 2 2] [6]
>= [1 0 0] [0]
[0 0 0] x + [0]
[1 0 1] [1]
= c_5(x)
p(s(x)) = [1 2 0] [0]
[1 2 0] x + [1]
[0 0 1] [0]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
*** 1.1.1.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
-#(x,0()) -> c_1(x)
-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
p#(s(x)) -> c_5(x)
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:-#(x,0()) -> c_1(x)
-->_1 p#(s(x)) -> c_5(x):3
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,0()) -> c_1(x):1
2:W:-#(x,s(y)) -> c_2(x,y,-#(x,p(s(y))))
-->_2 p#(s(x)) -> c_5(x):3
-->_1 p#(s(x)) -> c_5(x):3
-->_3 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_2 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_3 -#(x,0()) -> c_1(x):1
-->_2 -#(x,0()) -> c_1(x):1
-->_1 -#(x,0()) -> c_1(x):1
3:W:p#(s(x)) -> c_5(x)
-->_1 p#(s(x)) -> c_5(x):3
-->_1 -#(x,s(y)) -> c_2(x,y,-#(x,p(s(y)))):2
-->_1 -#(x,0()) -> c_1(x):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: -#(x,0()) -> c_1(x)
3: p#(s(x)) -> c_5(x)
2: -#(x,s(y)) -> c_2(x
,y
,-#(x,p(s(y))))
*** 1.1.1.1.1.1.2.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
p(s(x)) -> x
Signature:
{-/2,p/1,-#/2,p#/1} / {0/0,greater/2,if/3,s/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/1}
Obligation:
Full
basic terms: {-#,p#}/{0,greater,if,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).