*** 1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
D(+(x,y)) -> +(D(x),D(y))
D(-(x,y)) -> -(D(x),D(y))
D(constant()) -> 0()
D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2())))
D(ln(x)) -> div(D(x),x)
D(minus(x)) -> minus(D(x))
D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))
D(t()) -> 1()
Weak DP Rules:
Weak TRS Rules:
Signature:
{D/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0}
Obligation:
Innermost
basic terms: {D}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
DependencyPairs {dpKind_ = WIDP}
Proof:
We add the following weak innermost dependency pairs:
Strict DPs
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(constant()) -> c_4()
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
D#(t()) -> c_9()
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(constant()) -> c_4()
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
D#(t()) -> c_9()
Strict TRS Rules:
D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
D(+(x,y)) -> +(D(x),D(y))
D(-(x,y)) -> -(D(x),D(y))
D(constant()) -> 0()
D(div(x,y)) -> -(div(D(x),y),div(*(x,D(y)),pow(y,2())))
D(ln(x)) -> div(D(x),x)
D(minus(x)) -> minus(D(x))
D(pow(x,y)) -> +(*(*(y,pow(x,-(y,1()))),D(x)),*(*(pow(x,y),ln(x)),D(y)))
D(t()) -> 1()
Weak DP Rules:
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(constant()) -> c_4()
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
D#(t()) -> c_9()
*** 1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(constant()) -> c_4()
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
D#(t()) -> c_9()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
Succeeding
Proof:
()
*** 1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(constant()) -> c_4()
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
D#(t()) -> c_9()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{4,9}
by application of
Pre({4,9}) = {1,2,3,5,6,7,8}.
Here rules are labelled as follows:
1: D#(*(x,y)) -> c_1(D#(x),D#(y))
2: D#(+(x,y)) -> c_2(D#(x),D#(y))
3: D#(-(x,y)) -> c_3(D#(x),D#(y))
4: D#(constant()) -> c_4()
5: D#(div(x,y)) -> c_5(D#(x),D#(y))
6: D#(ln(x)) -> c_6(D#(x))
7: D#(minus(x)) -> c_7(D#(x))
8: D#(pow(x,y)) -> c_8(D#(x),D#(y))
9: D#(t()) -> c_9()
*** 1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Strict TRS Rules:
Weak DP Rules:
D#(constant()) -> c_4()
D#(t()) -> c_9()
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:D#(*(x,y)) -> c_1(D#(x),D#(y))
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(t()) -> c_9():9
-->_1 D#(t()) -> c_9():9
-->_2 D#(constant()) -> c_4():8
-->_1 D#(constant()) -> c_4():8
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
2:S:D#(+(x,y)) -> c_2(D#(x),D#(y))
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(t()) -> c_9():9
-->_1 D#(t()) -> c_9():9
-->_2 D#(constant()) -> c_4():8
-->_1 D#(constant()) -> c_4():8
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
3:S:D#(-(x,y)) -> c_3(D#(x),D#(y))
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(t()) -> c_9():9
-->_1 D#(t()) -> c_9():9
-->_2 D#(constant()) -> c_4():8
-->_1 D#(constant()) -> c_4():8
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
4:S:D#(div(x,y)) -> c_5(D#(x),D#(y))
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(t()) -> c_9():9
-->_1 D#(t()) -> c_9():9
-->_2 D#(constant()) -> c_4():8
-->_1 D#(constant()) -> c_4():8
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
5:S:D#(ln(x)) -> c_6(D#(x))
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(t()) -> c_9():9
-->_1 D#(constant()) -> c_4():8
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
6:S:D#(minus(x)) -> c_7(D#(x))
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(t()) -> c_9():9
-->_1 D#(constant()) -> c_4():8
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
7:S:D#(pow(x,y)) -> c_8(D#(x),D#(y))
-->_2 D#(t()) -> c_9():9
-->_1 D#(t()) -> c_9():9
-->_2 D#(constant()) -> c_4():8
-->_1 D#(constant()) -> c_4():8
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
8:W:D#(constant()) -> c_4()
9:W:D#(t()) -> c_9()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
8: D#(constant()) -> c_4()
9: D#(t()) -> c_9()
*** 1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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: D#(*(x,y)) -> c_1(D#(x),D#(y))
2: D#(+(x,y)) -> c_2(D#(x),D#(y))
7: D#(pow(x,y)) -> c_8(D#(x),D#(y))
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:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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_1) = {1,2},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{D#}
TcT has computed the following interpretation:
p(*) = [1] x1 + [1] x2 + [3]
p(+) = [1] x1 + [1] x2 + [3]
p(-) = [1] x1 + [1] x2 + [1]
p(0) = [0]
p(1) = [1]
p(2) = [1]
p(D) = [0]
p(constant) = [8]
p(div) = [1] x1 + [1] x2 + [1]
p(ln) = [1] x1 + [0]
p(minus) = [1] x1 + [0]
p(pow) = [1] x1 + [1] x2 + [3]
p(t) = [0]
p(D#) = [8] x1 + [4]
p(c_1) = [1] x1 + [1] x2 + [13]
p(c_2) = [1] x1 + [1] x2 + [10]
p(c_3) = [1] x1 + [1] x2 + [4]
p(c_4) = [1]
p(c_5) = [1] x1 + [1] x2 + [4]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [1] x1 + [1] x2 + [12]
p(c_9) = [0]
Following rules are strictly oriented:
D#(*(x,y)) = [8] x + [8] y + [28]
> [8] x + [8] y + [21]
= c_1(D#(x),D#(y))
D#(+(x,y)) = [8] x + [8] y + [28]
> [8] x + [8] y + [18]
= c_2(D#(x),D#(y))
D#(pow(x,y)) = [8] x + [8] y + [28]
> [8] x + [8] y + [20]
= c_8(D#(x),D#(y))
Following rules are (at-least) weakly oriented:
D#(-(x,y)) = [8] x + [8] y + [12]
>= [8] x + [8] y + [12]
= c_3(D#(x),D#(y))
D#(div(x,y)) = [8] x + [8] y + [12]
>= [8] x + [8] y + [12]
= c_5(D#(x),D#(y))
D#(ln(x)) = [8] x + [4]
>= [8] x + [4]
= c_6(D#(x))
D#(minus(x)) = [8] x + [4]
>= [8] x + [4]
= c_7(D#(x))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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: D#(ln(x)) -> c_6(D#(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:
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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_1) = {1,2},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{D#}
TcT has computed the following interpretation:
p(*) = [1] x1 + [1] x2 + [3]
p(+) = [1] x1 + [1] x2 + [0]
p(-) = [1] x1 + [1] x2 + [0]
p(0) = [1]
p(1) = [1]
p(2) = [0]
p(D) = [2]
p(constant) = [1]
p(div) = [1] x1 + [1] x2 + [0]
p(ln) = [1] x1 + [1]
p(minus) = [1] x1 + [0]
p(pow) = [1] x1 + [1] x2 + [0]
p(t) = [0]
p(D#) = [8] x1 + [0]
p(c_1) = [1] x1 + [1] x2 + [13]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1] x1 + [1] x2 + [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [1] x2 + [0]
p(c_6) = [1] x1 + [1]
p(c_7) = [1] x1 + [0]
p(c_8) = [1] x1 + [1] x2 + [0]
p(c_9) = [0]
Following rules are strictly oriented:
D#(ln(x)) = [8] x + [8]
> [8] x + [1]
= c_6(D#(x))
Following rules are (at-least) weakly oriented:
D#(*(x,y)) = [8] x + [8] y + [24]
>= [8] x + [8] y + [13]
= c_1(D#(x),D#(y))
D#(+(x,y)) = [8] x + [8] y + [0]
>= [8] x + [8] y + [0]
= c_2(D#(x),D#(y))
D#(-(x,y)) = [8] x + [8] y + [0]
>= [8] x + [8] y + [0]
= c_3(D#(x),D#(y))
D#(div(x,y)) = [8] x + [8] y + [0]
>= [8] x + [8] y + [0]
= c_5(D#(x),D#(y))
D#(minus(x)) = [8] x + [0]
>= [8] x + [0]
= c_7(D#(x))
D#(pow(x,y)) = [8] x + [8] y + [0]
>= [8] x + [8] y + [0]
= c_8(D#(x),D#(y))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(minus(x)) -> c_7(D#(x))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(minus(x)) -> c_7(D#(x))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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:
2: D#(div(x,y)) -> c_5(D#(x),D#(y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(minus(x)) -> c_7(D#(x))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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_1) = {1,2},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{D#}
TcT has computed the following interpretation:
p(*) = [1] x1 + [1] x2 + [2]
p(+) = [1] x1 + [1] x2 + [0]
p(-) = [1] x1 + [1] x2 + [0]
p(0) = [0]
p(1) = [0]
p(2) = [2]
p(D) = [1]
p(constant) = [0]
p(div) = [1] x1 + [1] x2 + [2]
p(ln) = [1] x1 + [0]
p(minus) = [1] x1 + [0]
p(pow) = [1] x1 + [1] x2 + [0]
p(t) = [2]
p(D#) = [8] x1 + [0]
p(c_1) = [1] x1 + [1] x2 + [10]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1] x1 + [1] x2 + [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [1] x2 + [11]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [1] x1 + [1] x2 + [0]
p(c_9) = [1]
Following rules are strictly oriented:
D#(div(x,y)) = [8] x + [8] y + [16]
> [8] x + [8] y + [11]
= c_5(D#(x),D#(y))
Following rules are (at-least) weakly oriented:
D#(*(x,y)) = [8] x + [8] y + [16]
>= [8] x + [8] y + [10]
= c_1(D#(x),D#(y))
D#(+(x,y)) = [8] x + [8] y + [0]
>= [8] x + [8] y + [0]
= c_2(D#(x),D#(y))
D#(-(x,y)) = [8] x + [8] y + [0]
>= [8] x + [8] y + [0]
= c_3(D#(x),D#(y))
D#(ln(x)) = [8] x + [0]
>= [8] x + [0]
= c_6(D#(x))
D#(minus(x)) = [8] x + [0]
>= [8] x + [0]
= c_7(D#(x))
D#(pow(x,y)) = [8] x + [8] y + [0]
>= [8] x + [8] y + [0]
= c_8(D#(x),D#(y))
*** 1.1.1.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(minus(x)) -> c_7(D#(x))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(minus(x)) -> c_7(D#(x))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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:
2: D#(minus(x)) -> c_7(D#(x))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(minus(x)) -> c_7(D#(x))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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_1) = {1,2},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{D#}
TcT has computed the following interpretation:
p(*) = [1] x1 + [1] x2 + [1]
p(+) = [1] x1 + [1] x2 + [0]
p(-) = [1] x1 + [1] x2 + [0]
p(0) = [0]
p(1) = [0]
p(2) = [0]
p(D) = [4]
p(constant) = [0]
p(div) = [1] x1 + [1] x2 + [2]
p(ln) = [1] x1 + [2]
p(minus) = [1] x1 + [4]
p(pow) = [1] x1 + [1] x2 + [3]
p(t) = [0]
p(D#) = [4] x1 + [0]
p(c_1) = [1] x1 + [1] x2 + [1]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1] x1 + [1] x2 + [0]
p(c_4) = [2]
p(c_5) = [1] x1 + [1] x2 + [8]
p(c_6) = [1] x1 + [1]
p(c_7) = [1] x1 + [15]
p(c_8) = [1] x1 + [1] x2 + [1]
p(c_9) = [1]
Following rules are strictly oriented:
D#(minus(x)) = [4] x + [16]
> [4] x + [15]
= c_7(D#(x))
Following rules are (at-least) weakly oriented:
D#(*(x,y)) = [4] x + [4] y + [4]
>= [4] x + [4] y + [1]
= c_1(D#(x),D#(y))
D#(+(x,y)) = [4] x + [4] y + [0]
>= [4] x + [4] y + [0]
= c_2(D#(x),D#(y))
D#(-(x,y)) = [4] x + [4] y + [0]
>= [4] x + [4] y + [0]
= c_3(D#(x),D#(y))
D#(div(x,y)) = [4] x + [4] y + [8]
>= [4] x + [4] y + [8]
= c_5(D#(x),D#(y))
D#(ln(x)) = [4] x + [8]
>= [4] x + [1]
= c_6(D#(x))
D#(pow(x,y)) = [4] x + [4] y + [12]
>= [4] x + [4] y + [1]
= c_8(D#(x),D#(y))
*** 1.1.1.1.1.1.2.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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: D#(-(x,y)) -> c_3(D#(x),D#(y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.2.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
D#(-(x,y)) -> c_3(D#(x),D#(y))
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
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_1) = {1,2},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{D#}
TcT has computed the following interpretation:
p(*) = [1] x1 + [1] x2 + [2]
p(+) = [1] x1 + [1] x2 + [0]
p(-) = [1] x1 + [1] x2 + [2]
p(0) = [8]
p(1) = [1]
p(2) = [0]
p(D) = [1]
p(constant) = [0]
p(div) = [1] x1 + [1] x2 + [1]
p(ln) = [1] x1 + [2]
p(minus) = [1] x1 + [0]
p(pow) = [1] x1 + [1] x2 + [2]
p(t) = [1]
p(D#) = [8] x1 + [0]
p(c_1) = [1] x1 + [1] x2 + [2]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1] x1 + [1] x2 + [8]
p(c_4) = [1]
p(c_5) = [1] x1 + [1] x2 + [4]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [1] x1 + [1] x2 + [4]
p(c_9) = [1]
Following rules are strictly oriented:
D#(-(x,y)) = [8] x + [8] y + [16]
> [8] x + [8] y + [8]
= c_3(D#(x),D#(y))
Following rules are (at-least) weakly oriented:
D#(*(x,y)) = [8] x + [8] y + [16]
>= [8] x + [8] y + [2]
= c_1(D#(x),D#(y))
D#(+(x,y)) = [8] x + [8] y + [0]
>= [8] x + [8] y + [0]
= c_2(D#(x),D#(y))
D#(div(x,y)) = [8] x + [8] y + [8]
>= [8] x + [8] y + [4]
= c_5(D#(x),D#(y))
D#(ln(x)) = [8] x + [16]
>= [8] x + [0]
= c_6(D#(x))
D#(minus(x)) = [8] x + [0]
>= [8] x + [0]
= c_7(D#(x))
D#(pow(x,y)) = [8] x + [8] y + [16]
>= [8] x + [8] y + [4]
= c_8(D#(x),D#(y))
*** 1.1.1.1.1.1.2.2.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.2.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
D#(*(x,y)) -> c_1(D#(x),D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(div(x,y)) -> c_5(D#(x),D#(y))
D#(ln(x)) -> c_6(D#(x))
D#(minus(x)) -> c_7(D#(x))
D#(pow(x,y)) -> c_8(D#(x),D#(y))
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:D#(*(x,y)) -> c_1(D#(x),D#(y))
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
2:W:D#(+(x,y)) -> c_2(D#(x),D#(y))
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
3:W:D#(-(x,y)) -> c_3(D#(x),D#(y))
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
4:W:D#(div(x,y)) -> c_5(D#(x),D#(y))
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
5:W:D#(ln(x)) -> c_6(D#(x))
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
6:W:D#(minus(x)) -> c_7(D#(x))
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
7:W:D#(pow(x,y)) -> c_8(D#(x),D#(y))
-->_2 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_1 D#(pow(x,y)) -> c_8(D#(x),D#(y)):7
-->_2 D#(minus(x)) -> c_7(D#(x)):6
-->_1 D#(minus(x)) -> c_7(D#(x)):6
-->_2 D#(ln(x)) -> c_6(D#(x)):5
-->_1 D#(ln(x)) -> c_6(D#(x)):5
-->_2 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_1 D#(div(x,y)) -> c_5(D#(x),D#(y)):4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
-->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: D#(*(x,y)) -> c_1(D#(x),D#(y))
7: D#(pow(x,y)) -> c_8(D#(x),D#(y))
6: D#(minus(x)) -> c_7(D#(x))
5: D#(ln(x)) -> c_6(D#(x))
4: D#(div(x,y)) -> c_5(D#(x),D#(y))
3: D#(-(x,y)) -> c_3(D#(x),D#(y))
2: D#(+(x,y)) -> c_2(D#(x),D#(y))
*** 1.1.1.1.1.1.2.2.2.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,2/0,constant/0,div/2,ln/1,minus/1,pow/2,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1,c_8/2,c_9/0}
Obligation:
Innermost
basic terms: {D#}/{*,+,-,0,1,2,constant,div,ln,minus,pow,t}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).