*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Weak DP Rules:
Weak TRS Rules:
Signature:
{active/1,f/2,g/1,proper/1,top/1} / {mark/1,ok/1}
Obligation:
Innermost
basic terms: {active,f,g,proper,top}/{mark,ok}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
Strict TRS Rules:
Weak DP Rules:
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Problem (S)
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Strict TRS Rules:
Weak DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
*** 1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
Strict TRS Rules:
Weak DP Rules:
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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:
7: g#(ok(X)) -> c_7(g#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
Strict TRS Rules:
Weak DP Rules:
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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_1) = {1,2},
uargs(c_2) = {1,3},
uargs(c_3) = {1,2},
uargs(c_4) = {1},
uargs(c_5) = {1},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1,2,3},
uargs(c_9) = {1,2},
uargs(c_10) = {1,2},
uargs(c_11) = {1,2}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [0 0] x1 + [0]
[0 1] [0]
p(f) = [2 0] x1 + [0]
[0 1] [0]
p(g) = [2 0] x1 + [0]
[0 2] [0]
p(mark) = [0 0] x1 + [0]
[0 1] [0]
p(ok) = [0 2] x1 + [0]
[0 1] [2]
p(proper) = [0]
[0]
p(top) = [0 1] x1 + [0]
[0 0] [1]
p(active#) = [0 2] x1 + [0]
[0 1] [0]
p(f#) = [0 0] x1 + [0]
[0 1] [0]
p(g#) = [0 1] x1 + [0]
[1 0] [0]
p(proper#) = [0]
[2]
p(top#) = [1 0] x1 + [2]
[1 1] [3]
p(c_1) = [2 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
p(c_2) = [1 2] x1 + [0 0] x2 + [2
0] x3 + [0]
[0 1] [1 0] [1
0] [0]
p(c_3) = [1 1] x1 + [1 1] x2 + [0]
[0 0] [1 0] [0]
p(c_4) = [2 0] x1 + [0]
[1 0] [0]
p(c_5) = [1 0] x1 + [0]
[0 1] [0]
p(c_6) = [1 0] x1 + [0]
[0 0] [0]
p(c_7) = [1 0] x1 + [0]
[1 0] [0]
p(c_8) = [2 0] x1 + [2 0] x2 + [2
0] x3 + [0]
[2 1] [0 0] [0
0] [1]
p(c_9) = [1 1] x1 + [2 0] x2 + [0]
[2 1] [0 0] [2]
p(c_10) = [1 0] x1 + [1 0] x2 + [0]
[1 0] [0 0] [1]
p(c_11) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [1 0] [0]
Following rules are strictly oriented:
g#(ok(X)) = [0 1] X + [2]
[0 2] [0]
> [0 1] X + [0]
[0 1] [0]
= c_7(g#(X))
Following rules are (at-least) weakly oriented:
active#(f(X1,X2)) = [0 2] X1 + [0]
[0 1] [0]
>= [0 2] X1 + [0]
[0 1] [0]
= c_1(f#(active(X1),X2)
,active#(X1))
active#(f(g(X),Y)) = [0 4] X + [0]
[0 2] [0]
>= [0 4] X + [0]
[0 2] [0]
= c_2(f#(X,f(g(X),Y))
,f#(g(X),Y)
,g#(X))
active#(g(X)) = [0 4] X + [0]
[0 2] [0]
>= [0 4] X + [0]
[0 2] [0]
= c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) = [0 0] X1 + [0]
[0 1] [0]
>= [0]
[0]
= c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) = [0 0] X1 + [0]
[0 1] [2]
>= [0 0] X1 + [0]
[0 1] [0]
= c_5(f#(X1,X2))
g#(mark(X)) = [0 1] X + [0]
[0 0] [0]
>= [0 1] X + [0]
[0 0] [0]
= c_6(g#(X))
proper#(f(X1,X2)) = [0]
[2]
>= [0]
[1]
= c_8(f#(proper(X1),proper(X2))
,proper#(X1)
,proper#(X2))
proper#(g(X)) = [0]
[2]
>= [0]
[2]
= c_9(g#(proper(X)),proper#(X))
top#(mark(X)) = [0 0] X + [2]
[0 1] [3]
>= [2]
[3]
= c_10(top#(proper(X)),proper#(X))
top#(ok(X)) = [0 2] X + [2]
[0 3] [5]
>= [0 2] X + [2]
[0 3] [3]
= c_11(top#(active(X)),active#(X))
active(f(X1,X2)) = [0 0] X1 + [0]
[0 1] [0]
>= [0 0] X1 + [0]
[0 1] [0]
= f(active(X1),X2)
active(f(g(X),Y)) = [0 0] X + [0]
[0 2] [0]
>= [0 0] X + [0]
[0 1] [0]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [0 0] X + [0]
[0 2] [0]
>= [0 0] X + [0]
[0 2] [0]
= g(active(X))
f(mark(X1),X2) = [0 0] X1 + [0]
[0 1] [0]
>= [0 0] X1 + [0]
[0 1] [0]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [0 4] X1 + [0]
[0 1] [2]
>= [0 2] X1 + [0]
[0 1] [2]
= ok(f(X1,X2))
g(mark(X)) = [0 0] X + [0]
[0 2] [0]
>= [0 0] X + [0]
[0 2] [0]
= mark(g(X))
g(ok(X)) = [0 4] X + [0]
[0 2] [4]
>= [0 4] X + [0]
[0 2] [2]
= ok(g(X))
proper(f(X1,X2)) = [0]
[0]
>= [0]
[0]
= f(proper(X1),proper(X2))
proper(g(X)) = [0]
[0]
>= [0]
[0]
= g(proper(X))
*** 1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
Strict TRS Rules:
Weak DP Rules:
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
Strict TRS Rules:
Weak DP Rules:
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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:
5: f#(ok(X1),ok(X2)) -> c_5(f#(X1
,X2))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
Strict TRS Rules:
Weak DP Rules:
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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_1) = {1,2},
uargs(c_2) = {1,3},
uargs(c_3) = {1,2},
uargs(c_4) = {1},
uargs(c_5) = {1},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1,2,3},
uargs(c_9) = {1,2},
uargs(c_10) = {1,2},
uargs(c_11) = {1,2}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [0 0] x1 + [0]
[0 2] [0]
p(f) = [2 0] x1 + [0]
[0 2] [0]
p(g) = [2 0] x1 + [0]
[0 1] [0]
p(mark) = [0 0] x1 + [0]
[0 1] [0]
p(ok) = [0 1] x1 + [2]
[0 1] [2]
p(proper) = [0]
[0]
p(top) = [2 2] x1 + [2]
[2 0] [1]
p(active#) = [0 2] x1 + [0]
[0 0] [2]
p(f#) = [0 1] x1 + [0]
[0 3] [0]
p(g#) = [0 0] x1 + [0]
[0 1] [0]
p(proper#) = [0]
[0]
p(top#) = [2 0] x1 + [0]
[0 0] [1]
p(c_1) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
p(c_2) = [1 0] x1 + [0 1] x2 + [2
0] x3 + [0]
[0 0] [0 0] [0
0] [0]
p(c_3) = [2 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
p(c_4) = [1 0] x1 + [0]
[0 0] [0]
p(c_5) = [1 0] x1 + [0]
[2 0] [0]
p(c_6) = [2 0] x1 + [0]
[0 0] [0]
p(c_7) = [2 0] x1 + [0]
[0 0] [0]
p(c_8) = [2 0] x1 + [2 0] x2 + [2
0] x3 + [0]
[0 0] [0 1] [0
0] [0]
p(c_9) = [2 0] x1 + [1 0] x2 + [0]
[0 0] [1 0] [0]
p(c_10) = [2 0] x1 + [1 0] x2 + [0]
[0 1] [1 0] [0]
p(c_11) = [1 0] x1 + [1 2] x2 + [0]
[0 1] [0 0] [0]
Following rules are strictly oriented:
f#(ok(X1),ok(X2)) = [0 1] X1 + [2]
[0 3] [6]
> [0 1] X1 + [0]
[0 2] [0]
= c_5(f#(X1,X2))
Following rules are (at-least) weakly oriented:
active#(f(X1,X2)) = [0 4] X1 + [0]
[0 0] [2]
>= [0 4] X1 + [0]
[0 0] [0]
= c_1(f#(active(X1),X2)
,active#(X1))
active#(f(g(X),Y)) = [0 4] X + [0]
[0 0] [2]
>= [0 4] X + [0]
[0 0] [0]
= c_2(f#(X,f(g(X),Y))
,f#(g(X),Y)
,g#(X))
active#(g(X)) = [0 2] X + [0]
[0 0] [2]
>= [0 2] X + [0]
[0 0] [2]
= c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) = [0 1] X1 + [0]
[0 3] [0]
>= [0 1] X1 + [0]
[0 0] [0]
= c_4(f#(X1,X2))
g#(mark(X)) = [0 0] X + [0]
[0 1] [0]
>= [0]
[0]
= c_6(g#(X))
g#(ok(X)) = [0 0] X + [0]
[0 1] [2]
>= [0]
[0]
= c_7(g#(X))
proper#(f(X1,X2)) = [0]
[0]
>= [0]
[0]
= c_8(f#(proper(X1),proper(X2))
,proper#(X1)
,proper#(X2))
proper#(g(X)) = [0]
[0]
>= [0]
[0]
= c_9(g#(proper(X)),proper#(X))
top#(mark(X)) = [0]
[1]
>= [0]
[1]
= c_10(top#(proper(X)),proper#(X))
top#(ok(X)) = [0 2] X + [4]
[0 0] [1]
>= [0 2] X + [4]
[0 0] [1]
= c_11(top#(active(X)),active#(X))
active(f(X1,X2)) = [0 0] X1 + [0]
[0 4] [0]
>= [0 0] X1 + [0]
[0 4] [0]
= f(active(X1),X2)
active(f(g(X),Y)) = [0 0] X + [0]
[0 4] [0]
>= [0 0] X + [0]
[0 2] [0]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [0 0] X + [0]
[0 2] [0]
>= [0 0] X + [0]
[0 2] [0]
= g(active(X))
f(mark(X1),X2) = [0 0] X1 + [0]
[0 2] [0]
>= [0 0] X1 + [0]
[0 2] [0]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [0 2] X1 + [4]
[0 2] [4]
>= [0 2] X1 + [2]
[0 2] [2]
= ok(f(X1,X2))
g(mark(X)) = [0 0] X + [0]
[0 1] [0]
>= [0 0] X + [0]
[0 1] [0]
= mark(g(X))
g(ok(X)) = [0 2] X + [4]
[0 1] [2]
>= [0 1] X + [2]
[0 1] [2]
= ok(g(X))
proper(f(X1,X2)) = [0]
[0]
>= [0]
[0]
= f(proper(X1),proper(X2))
proper(g(X)) = [0]
[0]
>= [0]
[0]
= g(proper(X))
*** 1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
Strict TRS Rules:
Weak DP Rules:
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
Strict TRS Rules:
Weak DP Rules:
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
and a lower component
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
Further, following extension rules are added to the lower component.
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
*** 1.1.1.1.2.2.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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: top#(ok(X)) ->
c_11(top#(active(X)),active#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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_10) = {1},
uargs(c_11) = {1}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [1]
p(f) = [1] x1 + [0]
p(g) = [1] x1 + [0]
p(mark) = [1]
p(ok) = [2]
p(proper) = [0]
p(top) = [4] x1 + [1]
p(active#) = [2]
p(f#) = [0]
p(g#) = [2] x1 + [1]
p(proper#) = [1]
p(top#) = [8] x1 + [1]
p(c_1) = [1] x1 + [4] x2 + [4]
p(c_2) = [2] x3 + [2]
p(c_3) = [1] x1 + [1] x2 + [0]
p(c_4) = [1]
p(c_5) = [1] x1 + [1]
p(c_6) = [4] x1 + [1]
p(c_7) = [4] x1 + [1]
p(c_8) = [8] x1 + [2] x2 + [1]
p(c_9) = [1] x1 + [1] x2 + [8]
p(c_10) = [8] x1 + [1] x2 + [0]
p(c_11) = [1] x1 + [0]
Following rules are strictly oriented:
top#(ok(X)) = [17]
> [9]
= c_11(top#(active(X)),active#(X))
Following rules are (at-least) weakly oriented:
top#(mark(X)) = [9]
>= [9]
= c_10(top#(proper(X)),proper#(X))
active(f(X1,X2)) = [1]
>= [1]
= f(active(X1),X2)
active(f(g(X),Y)) = [1]
>= [1]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [1]
>= [1]
= g(active(X))
f(mark(X1),X2) = [1]
>= [1]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [2]
>= [2]
= ok(f(X1,X2))
g(mark(X)) = [1]
>= [1]
= mark(g(X))
g(ok(X)) = [2]
>= [2]
= ok(g(X))
proper(f(X1,X2)) = [0]
>= [0]
= f(proper(X1),proper(X2))
proper(g(X)) = [0]
>= [0]
= g(proper(X))
*** 1.1.1.1.2.2.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
Strict TRS Rules:
Weak DP Rules:
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2.2.1.2 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
Strict TRS Rules:
Weak DP Rules:
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: top#(mark(X)) ->
c_10(top#(proper(X)),proper#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.2.2.1.2.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
Strict TRS Rules:
Weak DP Rules:
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, 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 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_10) = {1},
uargs(c_11) = {1}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [1]
p(f) = [1] x1 + [0]
p(g) = [1] x1 + [0]
p(mark) = [1]
p(ok) = [2]
p(proper) = [0]
p(top) = [0]
p(active#) = [8]
p(f#) = [0]
p(g#) = [0]
p(proper#) = [0]
p(top#) = [8] x1 + [1]
p(c_1) = [1] x1 + [4] x2 + [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [2] x2 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [2] x1 + [1]
p(c_7) = [8] x1 + [1]
p(c_8) = [2] x2 + [0]
p(c_9) = [2] x1 + [2]
p(c_10) = [4] x1 + [2]
p(c_11) = [1] x1 + [1] x2 + [0]
Following rules are strictly oriented:
top#(mark(X)) = [9]
> [6]
= c_10(top#(proper(X)),proper#(X))
Following rules are (at-least) weakly oriented:
top#(ok(X)) = [17]
>= [17]
= c_11(top#(active(X)),active#(X))
active(f(X1,X2)) = [1]
>= [1]
= f(active(X1),X2)
active(f(g(X),Y)) = [1]
>= [1]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [1]
>= [1]
= g(active(X))
f(mark(X1),X2) = [1]
>= [1]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [2]
>= [2]
= ok(f(X1,X2))
g(mark(X)) = [1]
>= [1]
= mark(g(X))
g(ok(X)) = [2]
>= [2]
= ok(g(X))
proper(f(X1,X2)) = [0]
>= [0]
= f(proper(X1),proper(X2))
proper(g(X)) = [0]
>= [0]
= g(proper(X))
*** 1.1.1.1.2.2.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2.2.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
-->_1 top#(ok(X)) -> c_11(top#(active(X)),active#(X)):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):1
2:W:top#(ok(X)) -> c_11(top#(active(X)),active#(X))
-->_1 top#(ok(X)) -> c_11(top#(active(X)),active#(X)):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: top#(mark(X)) ->
c_10(top#(proper(X)),proper#(X))
2: top#(ok(X)) ->
c_11(top#(active(X)),active#(X))
*** 1.1.1.1.2.2.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.2.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
Strict TRS Rules:
Weak DP Rules:
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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:
2: active#(f(g(X),Y)) -> c_2(f#(X
,f(g(X),Y))
,f#(g(X),Y)
,g#(X))
3: active#(g(X)) ->
c_3(g#(active(X)),active#(X))
5: g#(mark(X)) -> c_6(g#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.2.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
Strict TRS Rules:
Weak DP Rules:
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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_1) = {1,2},
uargs(c_2) = {1,3},
uargs(c_3) = {1,2},
uargs(c_4) = {1},
uargs(c_5) = {1},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1,2,3},
uargs(c_9) = {1,2}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [1 2] x1 + [0]
[0 1] [0]
p(f) = [1 0] x1 + [0]
[0 1] [0]
p(g) = [3 0] x1 + [0]
[0 3] [2]
p(mark) = [1 0] x1 + [2]
[0 0] [0]
p(ok) = [1 2] x1 + [2]
[0 0] [0]
p(proper) = [0 0] x1 + [0]
[0 2] [0]
p(top) = [0]
[1]
p(active#) = [1 2] x1 + [1]
[0 0] [0]
p(f#) = [0 0] x1 + [0 0] x2 + [0]
[1 0] [1 0] [0]
p(g#) = [2 0] x1 + [0]
[0 2] [0]
p(proper#) = [0]
[0]
p(top#) = [1 0] x1 + [2]
[0 0] [0]
p(c_1) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
p(c_2) = [2 0] x1 + [0 0] x2 + [1
2] x3 + [0]
[1 0] [2 0] [0
0] [0]
p(c_3) = [1 0] x1 + [1 1] x2 + [1]
[0 0] [0 2] [0]
p(c_4) = [2 0] x1 + [0]
[0 1] [1]
p(c_5) = [2 0] x1 + [0]
[1 0] [2]
p(c_6) = [1 0] x1 + [3]
[0 0] [0]
p(c_7) = [1 2] x1 + [1]
[0 0] [0]
p(c_8) = [2 0] x1 + [1 2] x2 + [1
1] x3 + [0]
[0 0] [2 0] [0
2] [0]
p(c_9) = [1 0] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
p(c_10) = [0]
[1]
p(c_11) = [0 1] x1 + [0 1] x2 + [0]
[1 1] [1 0] [0]
Following rules are strictly oriented:
active#(f(g(X),Y)) = [3 6] X + [5]
[0 0] [0]
> [2 4] X + [0]
[0 0] [0]
= c_2(f#(X,f(g(X),Y))
,f#(g(X),Y)
,g#(X))
active#(g(X)) = [3 6] X + [5]
[0 0] [0]
> [3 6] X + [2]
[0 0] [0]
= c_3(g#(active(X)),active#(X))
g#(mark(X)) = [2 0] X + [4]
[0 0] [0]
> [2 0] X + [3]
[0 0] [0]
= c_6(g#(X))
Following rules are (at-least) weakly oriented:
active#(f(X1,X2)) = [1 2] X1 + [1]
[0 0] [0]
>= [1 2] X1 + [1]
[0 0] [0]
= c_1(f#(active(X1),X2)
,active#(X1))
f#(mark(X1),X2) = [0 0] X1 + [0 0] X2 + [0]
[1 0] [1 0] [2]
>= [0 0] X1 + [0 0] X2 + [0]
[1 0] [1 0] [1]
= c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) = [0 0] X1 + [0 0] X2 + [0]
[1 2] [1 2] [4]
>= [0]
[2]
= c_5(f#(X1,X2))
g#(ok(X)) = [2 4] X + [4]
[0 0] [0]
>= [2 4] X + [1]
[0 0] [0]
= c_7(g#(X))
proper#(f(X1,X2)) = [0]
[0]
>= [0]
[0]
= c_8(f#(proper(X1),proper(X2))
,proper#(X1)
,proper#(X2))
proper#(g(X)) = [0]
[0]
>= [0]
[0]
= c_9(g#(proper(X)),proper#(X))
top#(mark(X)) = [1 0] X + [4]
[0 0] [0]
>= [0]
[0]
= proper#(X)
top#(mark(X)) = [1 0] X + [4]
[0 0] [0]
>= [2]
[0]
= top#(proper(X))
top#(ok(X)) = [1 2] X + [4]
[0 0] [0]
>= [1 2] X + [1]
[0 0] [0]
= active#(X)
top#(ok(X)) = [1 2] X + [4]
[0 0] [0]
>= [1 2] X + [2]
[0 0] [0]
= top#(active(X))
active(f(X1,X2)) = [1 2] X1 + [0]
[0 1] [0]
>= [1 2] X1 + [0]
[0 1] [0]
= f(active(X1),X2)
active(f(g(X),Y)) = [3 6] X + [4]
[0 3] [2]
>= [1 0] X + [2]
[0 0] [0]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [3 6] X + [4]
[0 3] [2]
>= [3 6] X + [0]
[0 3] [2]
= g(active(X))
f(mark(X1),X2) = [1 0] X1 + [2]
[0 0] [0]
>= [1 0] X1 + [2]
[0 0] [0]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [1 2] X1 + [2]
[0 0] [0]
>= [1 2] X1 + [2]
[0 0] [0]
= ok(f(X1,X2))
g(mark(X)) = [3 0] X + [6]
[0 0] [2]
>= [3 0] X + [2]
[0 0] [0]
= mark(g(X))
g(ok(X)) = [3 6] X + [6]
[0 0] [2]
>= [3 6] X + [6]
[0 0] [0]
= ok(g(X))
proper(f(X1,X2)) = [0 0] X1 + [0]
[0 2] [0]
>= [0 0] X1 + [0]
[0 2] [0]
= f(proper(X1),proper(X2))
proper(g(X)) = [0 0] X + [0]
[0 6] [4]
>= [0 0] X + [0]
[0 6] [2]
= g(proper(X))
*** 1.1.1.1.2.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
Strict TRS Rules:
Weak DP Rules:
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2.2.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
Strict TRS Rules:
Weak DP Rules:
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_2 active#(g(X)) -> c_3(g#(active(X)),active#(X)):4
-->_2 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X)):3
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
-->_2 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):1
2:S:f#(mark(X1),X2) -> c_4(f#(X1,X2))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
3:W:active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
-->_3 g#(ok(X)) -> c_7(g#(X)):7
-->_3 g#(mark(X)) -> c_6(g#(X)):6
-->_2 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_2 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
4:W:active#(g(X)) -> c_3(g#(active(X)),active#(X))
-->_1 g#(ok(X)) -> c_7(g#(X)):7
-->_1 g#(mark(X)) -> c_6(g#(X)):6
-->_2 active#(g(X)) -> c_3(g#(active(X)),active#(X)):4
-->_2 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X)):3
-->_2 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):1
5:W:f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
6:W:g#(mark(X)) -> c_6(g#(X))
-->_1 g#(ok(X)) -> c_7(g#(X)):7
-->_1 g#(mark(X)) -> c_6(g#(X)):6
7:W:g#(ok(X)) -> c_7(g#(X))
-->_1 g#(ok(X)) -> c_7(g#(X)):7
-->_1 g#(mark(X)) -> c_6(g#(X)):6
8:W:proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
-->_3 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):9
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):9
-->_3 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):8
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):8
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
9:W:proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):9
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):8
-->_1 g#(ok(X)) -> c_7(g#(X)):7
-->_1 g#(mark(X)) -> c_6(g#(X)):6
10:W:top#(mark(X)) -> proper#(X)
-->_1 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):9
-->_1 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):8
11:W:top#(mark(X)) -> top#(proper(X))
-->_1 top#(ok(X)) -> top#(active(X)):13
-->_1 top#(ok(X)) -> active#(X):12
-->_1 top#(mark(X)) -> top#(proper(X)):11
-->_1 top#(mark(X)) -> proper#(X):10
12:W:top#(ok(X)) -> active#(X)
-->_1 active#(g(X)) -> c_3(g#(active(X)),active#(X)):4
-->_1 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X)):3
-->_1 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):1
13:W:top#(ok(X)) -> top#(active(X))
-->_1 top#(ok(X)) -> top#(active(X)):13
-->_1 top#(ok(X)) -> active#(X):12
-->_1 top#(mark(X)) -> top#(proper(X)):11
-->_1 top#(mark(X)) -> proper#(X):10
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: g#(ok(X)) -> c_7(g#(X))
6: g#(mark(X)) -> c_6(g#(X))
*** 1.1.1.1.2.2.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
Strict TRS Rules:
Weak DP Rules:
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_2 active#(g(X)) -> c_3(g#(active(X)),active#(X)):4
-->_2 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X)):3
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
-->_2 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):1
2:S:f#(mark(X1),X2) -> c_4(f#(X1,X2))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
3:W:active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
-->_2 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_2 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
4:W:active#(g(X)) -> c_3(g#(active(X)),active#(X))
-->_2 active#(g(X)) -> c_3(g#(active(X)),active#(X)):4
-->_2 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X)):3
-->_2 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):1
5:W:f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
8:W:proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
-->_3 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):9
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):9
-->_3 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):8
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):8
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):2
9:W:proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):9
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):8
10:W:top#(mark(X)) -> proper#(X)
-->_1 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):9
-->_1 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):8
11:W:top#(mark(X)) -> top#(proper(X))
-->_1 top#(ok(X)) -> top#(active(X)):13
-->_1 top#(ok(X)) -> active#(X):12
-->_1 top#(mark(X)) -> top#(proper(X)):11
-->_1 top#(mark(X)) -> proper#(X):10
12:W:top#(ok(X)) -> active#(X)
-->_1 active#(g(X)) -> c_3(g#(active(X)),active#(X)):4
-->_1 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X)):3
-->_1 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):1
13:W:top#(ok(X)) -> top#(active(X))
-->_1 top#(ok(X)) -> top#(active(X)):13
-->_1 top#(ok(X)) -> active#(X):12
-->_1 top#(mark(X)) -> top#(proper(X)):11
-->_1 top#(mark(X)) -> proper#(X):10
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y))
active#(g(X)) -> c_3(active#(X))
proper#(g(X)) -> c_9(proper#(X))
*** 1.1.1.1.2.2.2.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
Strict TRS Rules:
Weak DP Rules:
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y))
active#(g(X)) -> c_3(active#(X))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/2,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/1,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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: active#(f(X1,X2)) ->
c_1(f#(active(X1),X2)
,active#(X1))
2: f#(mark(X1),X2) -> c_4(f#(X1
,X2))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.2.2.2.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
Strict TRS Rules:
Weak DP Rules:
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y))
active#(g(X)) -> c_3(active#(X))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/2,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/1,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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_1) = {1,2},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1,2,3},
uargs(c_9) = {1}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [1 1] x1 + [0]
[0 1] [0]
p(f) = [3 0] x1 + [0]
[0 3] [1]
p(g) = [2 0] x1 + [0]
[0 2] [1]
p(mark) = [1 0] x1 + [2]
[0 0] [0]
p(ok) = [1 1] x1 + [2]
[0 0] [0]
p(proper) = [0 0] x1 + [0]
[0 1] [0]
p(top) = [2 0] x1 + [2]
[2 1] [1]
p(active#) = [1 1] x1 + [0]
[0 0] [2]
p(f#) = [2 0] x1 + [0]
[0 1] [0]
p(g#) = [0 0] x1 + [0]
[0 1] [2]
p(proper#) = [0]
[2]
p(top#) = [1 0] x1 + [2]
[0 0] [2]
p(c_1) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
p(c_2) = [2 0] x1 + [2]
[0 0] [1]
p(c_3) = [1 0] x1 + [1]
[0 0] [0]
p(c_4) = [1 0] x1 + [2]
[0 0] [0]
p(c_5) = [1 2] x1 + [2]
[0 0] [0]
p(c_6) = [2 0] x1 + [0]
[0 1] [0]
p(c_7) = [0 2] x1 + [2]
[2 1] [2]
p(c_8) = [2 0] x1 + [1 0] x2 + [1
0] x3 + [0]
[0 0] [0 0] [0
1] [0]
p(c_9) = [1 0] x1 + [0]
[1 1] [0]
p(c_10) = [0 0] x1 + [0]
[0 2] [2]
p(c_11) = [0]
[0]
Following rules are strictly oriented:
active#(f(X1,X2)) = [3 3] X1 + [1]
[0 0] [2]
> [3 3] X1 + [0]
[0 0] [2]
= c_1(f#(active(X1),X2)
,active#(X1))
f#(mark(X1),X2) = [2 0] X1 + [4]
[0 0] [0]
> [2 0] X1 + [2]
[0 0] [0]
= c_4(f#(X1,X2))
Following rules are (at-least) weakly oriented:
active#(f(g(X),Y)) = [6 6] X + [4]
[0 0] [2]
>= [4 0] X + [2]
[0 0] [1]
= c_2(f#(X,f(g(X),Y)),f#(g(X),Y))
active#(g(X)) = [2 2] X + [1]
[0 0] [2]
>= [1 1] X + [1]
[0 0] [0]
= c_3(active#(X))
f#(ok(X1),ok(X2)) = [2 2] X1 + [4]
[0 0] [0]
>= [2 2] X1 + [2]
[0 0] [0]
= c_5(f#(X1,X2))
proper#(f(X1,X2)) = [0]
[2]
>= [0]
[2]
= c_8(f#(proper(X1),proper(X2))
,proper#(X1)
,proper#(X2))
proper#(g(X)) = [0]
[2]
>= [0]
[2]
= c_9(proper#(X))
top#(mark(X)) = [1 0] X + [4]
[0 0] [2]
>= [0]
[2]
= proper#(X)
top#(mark(X)) = [1 0] X + [4]
[0 0] [2]
>= [2]
[2]
= top#(proper(X))
top#(ok(X)) = [1 1] X + [4]
[0 0] [2]
>= [1 1] X + [0]
[0 0] [2]
= active#(X)
top#(ok(X)) = [1 1] X + [4]
[0 0] [2]
>= [1 1] X + [2]
[0 0] [2]
= top#(active(X))
active(f(X1,X2)) = [3 3] X1 + [1]
[0 3] [1]
>= [3 3] X1 + [0]
[0 3] [1]
= f(active(X1),X2)
active(f(g(X),Y)) = [6 6] X + [4]
[0 6] [4]
>= [3 0] X + [2]
[0 0] [0]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [2 2] X + [1]
[0 2] [1]
>= [2 2] X + [0]
[0 2] [1]
= g(active(X))
f(mark(X1),X2) = [3 0] X1 + [6]
[0 0] [1]
>= [3 0] X1 + [2]
[0 0] [0]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [3 3] X1 + [6]
[0 0] [1]
>= [3 3] X1 + [3]
[0 0] [0]
= ok(f(X1,X2))
g(mark(X)) = [2 0] X + [4]
[0 0] [1]
>= [2 0] X + [2]
[0 0] [0]
= mark(g(X))
g(ok(X)) = [2 2] X + [4]
[0 0] [1]
>= [2 2] X + [3]
[0 0] [0]
= ok(g(X))
proper(f(X1,X2)) = [0 0] X1 + [0]
[0 3] [1]
>= [0 0] X1 + [0]
[0 3] [1]
= f(proper(X1),proper(X2))
proper(g(X)) = [0 0] X + [0]
[0 2] [1]
>= [0 0] X + [0]
[0 2] [1]
= g(proper(X))
*** 1.1.1.1.2.2.2.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y))
active#(g(X)) -> c_3(active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/2,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/1,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2.2.2.2.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y))
active#(g(X)) -> c_3(active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> active#(X)
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/2,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/1,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):4
-->_2 active#(g(X)) -> c_3(active#(X)):3
-->_2 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y)):2
-->_2 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):1
2:W:active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y))
-->_2 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_2 f#(mark(X1),X2) -> c_4(f#(X1,X2)):4
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):4
3:W:active#(g(X)) -> c_3(active#(X))
-->_1 active#(g(X)) -> c_3(active#(X)):3
-->_1 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y)):2
-->_1 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):1
4:W:f#(mark(X1),X2) -> c_4(f#(X1,X2))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):4
5:W:f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):4
6:W:proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
-->_3 proper#(g(X)) -> c_9(proper#(X)):7
-->_2 proper#(g(X)) -> c_9(proper#(X)):7
-->_3 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):6
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):6
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):5
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):4
7:W:proper#(g(X)) -> c_9(proper#(X))
-->_1 proper#(g(X)) -> c_9(proper#(X)):7
-->_1 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):6
8:W:top#(mark(X)) -> proper#(X)
-->_1 proper#(g(X)) -> c_9(proper#(X)):7
-->_1 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):6
9:W:top#(mark(X)) -> top#(proper(X))
-->_1 top#(ok(X)) -> top#(active(X)):11
-->_1 top#(ok(X)) -> active#(X):10
-->_1 top#(mark(X)) -> top#(proper(X)):9
-->_1 top#(mark(X)) -> proper#(X):8
10:W:top#(ok(X)) -> active#(X)
-->_1 active#(g(X)) -> c_3(active#(X)):3
-->_1 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y)):2
-->_1 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):1
11:W:top#(ok(X)) -> top#(active(X))
-->_1 top#(ok(X)) -> top#(active(X)):11
-->_1 top#(ok(X)) -> active#(X):10
-->_1 top#(mark(X)) -> top#(proper(X)):9
-->_1 top#(mark(X)) -> proper#(X):8
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: top#(mark(X)) -> top#(proper(X))
11: top#(ok(X)) -> top#(active(X))
10: top#(ok(X)) -> active#(X)
8: top#(mark(X)) -> proper#(X)
6: proper#(f(X1,X2)) ->
c_8(f#(proper(X1),proper(X2))
,proper#(X1)
,proper#(X2))
7: proper#(g(X)) -> c_9(proper#(X))
1: active#(f(X1,X2)) ->
c_1(f#(active(X1),X2)
,active#(X1))
3: active#(g(X)) -> c_3(active#(X))
2: active#(f(g(X),Y)) -> c_2(f#(X
,f(g(X),Y))
,f#(g(X),Y))
5: f#(ok(X1),ok(X2)) -> c_5(f#(X1
,X2))
4: f#(mark(X1),X2) -> c_4(f#(X1
,X2))
*** 1.1.1.1.2.2.2.2.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/2,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/1,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Strict TRS Rules:
Weak DP Rules:
active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
active#(g(X)) -> c_3(g#(active(X)),active#(X))
f#(mark(X1),X2) -> c_4(f#(X1,X2))
f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
g#(mark(X)) -> c_6(g#(X))
g#(ok(X)) -> c_7(g#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):9
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):8
-->_3 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):2
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):2
-->_3 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):1
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):1
2:S:proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
-->_1 g#(ok(X)) -> c_7(g#(X)):11
-->_1 g#(mark(X)) -> c_6(g#(X)):10
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):2
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):1
3:S:top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
-->_1 top#(ok(X)) -> c_11(top#(active(X)),active#(X)):4
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):3
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):2
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):1
4:S:top#(ok(X)) -> c_11(top#(active(X)),active#(X))
-->_2 active#(g(X)) -> c_3(g#(active(X)),active#(X)):7
-->_2 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X)):6
-->_2 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):5
-->_1 top#(ok(X)) -> c_11(top#(active(X)),active#(X)):4
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):3
5:W:active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):9
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):8
-->_2 active#(g(X)) -> c_3(g#(active(X)),active#(X)):7
-->_2 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X)):6
-->_2 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):5
6:W:active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X))
-->_3 g#(ok(X)) -> c_7(g#(X)):11
-->_3 g#(mark(X)) -> c_6(g#(X)):10
-->_2 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):9
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):9
-->_2 f#(mark(X1),X2) -> c_4(f#(X1,X2)):8
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):8
7:W:active#(g(X)) -> c_3(g#(active(X)),active#(X))
-->_1 g#(ok(X)) -> c_7(g#(X)):11
-->_1 g#(mark(X)) -> c_6(g#(X)):10
-->_2 active#(g(X)) -> c_3(g#(active(X)),active#(X)):7
-->_2 active#(f(g(X),Y)) -> c_2(f#(X,f(g(X),Y)),f#(g(X),Y),g#(X)):6
-->_2 active#(f(X1,X2)) -> c_1(f#(active(X1),X2),active#(X1)):5
8:W:f#(mark(X1),X2) -> c_4(f#(X1,X2))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):9
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):8
9:W:f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2))
-->_1 f#(ok(X1),ok(X2)) -> c_5(f#(X1,X2)):9
-->_1 f#(mark(X1),X2) -> c_4(f#(X1,X2)):8
10:W:g#(mark(X)) -> c_6(g#(X))
-->_1 g#(ok(X)) -> c_7(g#(X)):11
-->_1 g#(mark(X)) -> c_6(g#(X)):10
11:W:g#(ok(X)) -> c_7(g#(X))
-->_1 g#(ok(X)) -> c_7(g#(X)):11
-->_1 g#(mark(X)) -> c_6(g#(X)):10
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: active#(g(X)) ->
c_3(g#(active(X)),active#(X))
5: active#(f(X1,X2)) ->
c_1(f#(active(X1),X2)
,active#(X1))
6: active#(f(g(X),Y)) -> c_2(f#(X
,f(g(X),Y))
,f#(g(X),Y)
,g#(X))
11: g#(ok(X)) -> c_7(g#(X))
10: g#(mark(X)) -> c_6(g#(X))
9: f#(ok(X1),ok(X2)) -> c_5(f#(X1
,X2))
8: f#(mark(X1),X2) -> c_4(f#(X1
,X2))
*** 1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)),active#(X))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/3,c_9/2,c_10/2,c_11/2}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2))
-->_3 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):2
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):2
-->_3 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):1
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):1
2:S:proper#(g(X)) -> c_9(g#(proper(X)),proper#(X))
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):2
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):1
3:S:top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
-->_1 top#(ok(X)) -> c_11(top#(active(X)),active#(X)):4
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):3
-->_2 proper#(g(X)) -> c_9(g#(proper(X)),proper#(X)):2
-->_2 proper#(f(X1,X2)) -> c_8(f#(proper(X1),proper(X2)),proper#(X1),proper#(X2)):1
4:S:top#(ok(X)) -> c_11(top#(active(X)),active#(X))
-->_1 top#(ok(X)) -> c_11(top#(active(X)),active#(X)):4
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):3
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
*** 1.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Problem (S)
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
Strict TRS Rules:
Weak DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
*** 1.1.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
and a lower component
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
Further, following extension rules are added to the lower component.
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> top#(active(X))
*** 1.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
Strict TRS Rules:
Weak DP Rules:
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: top#(mark(X)) ->
c_10(top#(proper(X)),proper#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
Strict TRS Rules:
Weak DP Rules:
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, 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 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_10) = {1},
uargs(c_11) = {1}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [4]
p(f) = [1] x1 + [0]
p(g) = [1] x1 + [0]
p(mark) = [4]
p(ok) = [7]
p(proper) = [0]
p(top) = [2] x1 + [2]
p(active#) = [1] x1 + [2]
p(f#) = [2] x2 + [1]
p(g#) = [4] x1 + [1]
p(proper#) = [2]
p(top#) = [4] x1 + [1]
p(c_1) = [0]
p(c_2) = [2] x2 + [2] x3 + [1]
p(c_3) = [1] x1 + [1]
p(c_4) = [8] x1 + [0]
p(c_5) = [1] x1 + [1]
p(c_6) = [1] x1 + [1]
p(c_7) = [0]
p(c_8) = [1]
p(c_9) = [1] x1 + [1]
p(c_10) = [8] x1 + [2] x2 + [0]
p(c_11) = [1] x1 + [12]
Following rules are strictly oriented:
top#(mark(X)) = [17]
> [12]
= c_10(top#(proper(X)),proper#(X))
Following rules are (at-least) weakly oriented:
top#(ok(X)) = [29]
>= [29]
= c_11(top#(active(X)))
active(f(X1,X2)) = [4]
>= [4]
= f(active(X1),X2)
active(f(g(X),Y)) = [4]
>= [4]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [4]
>= [4]
= g(active(X))
f(mark(X1),X2) = [4]
>= [4]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [7]
>= [7]
= ok(f(X1,X2))
g(mark(X)) = [4]
>= [4]
= mark(g(X))
g(ok(X)) = [7]
>= [7]
= ok(g(X))
proper(f(X1,X2)) = [0]
>= [0]
= f(proper(X1),proper(X2))
proper(g(X)) = [0]
>= [0]
= g(proper(X))
*** 1.1.1.2.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.2.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
-->_1 top#(ok(X)) -> c_11(top#(active(X))):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):1
2:W:top#(ok(X)) -> c_11(top#(active(X)))
-->_1 top#(ok(X)) -> c_11(top#(active(X))):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: top#(mark(X)) ->
c_10(top#(proper(X)),proper#(X))
2: top#(ok(X)) ->
c_11(top#(active(X)))
*** 1.1.1.2.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.2.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: proper#(g(X)) -> c_9(proper#(X))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.2.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
NaturalMI {miDimension = 2, 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:
The following argument positions are considered usable:
uargs(c_8) = {1,2},
uargs(c_9) = {1}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [1 0] x1 + [0]
[0 3] [2]
p(f) = [1 0] x1 + [2 0] x2 + [0]
[0 1] [0 2] [0]
p(g) = [4 0] x1 + [0]
[0 2] [1]
p(mark) = [0 0] x1 + [0]
[0 1] [1]
p(ok) = [1 2] x1 + [1]
[0 1] [1]
p(proper) = [0 0] x1 + [0]
[0 1] [0]
p(top) = [0]
[0]
p(active#) = [0 0] x1 + [1]
[2 0] [0]
p(f#) = [1 4] x1 + [0 0] x2 + [1]
[1 1] [0 2] [1]
p(g#) = [0 1] x1 + [0]
[2 0] [0]
p(proper#) = [0 1] x1 + [0]
[0 0] [0]
p(top#) = [1 1] x1 + [4]
[4 1] [4]
p(c_1) = [1 1] x2 + [4]
[1 0] [0]
p(c_2) = [0 2] x1 + [1 1] x2 + [1]
[0 1] [1 0] [0]
p(c_3) = [1 1] x1 + [1 2] x2 + [1]
[0 0] [2 0] [1]
p(c_4) = [0 1] x1 + [0]
[0 1] [1]
p(c_5) = [0 0] x1 + [1]
[4 2] [4]
p(c_6) = [0 1] x1 + [0]
[0 4] [0]
p(c_7) = [1 0] x1 + [4]
[0 0] [0]
p(c_8) = [1 0] x1 + [2 0] x2 + [0]
[0 2] [0 2] [0]
p(c_9) = [1 0] x1 + [0]
[0 0] [0]
p(c_10) = [1 1] x2 + [4]
[0 1] [0]
p(c_11) = [0 4] x1 + [0]
[1 0] [0]
Following rules are strictly oriented:
proper#(g(X)) = [0 2] X + [1]
[0 0] [0]
> [0 1] X + [0]
[0 0] [0]
= c_9(proper#(X))
Following rules are (at-least) weakly oriented:
proper#(f(X1,X2)) = [0 1] X1 + [0 2] X2 + [0]
[0 0] [0 0] [0]
>= [0 1] X1 + [0 2] X2 + [0]
[0 0] [0 0] [0]
= c_8(proper#(X1),proper#(X2))
top#(mark(X)) = [0 1] X + [5]
[0 1] [5]
>= [0 1] X + [0]
[0 0] [0]
= proper#(X)
top#(mark(X)) = [0 1] X + [5]
[0 1] [5]
>= [0 1] X + [4]
[0 1] [4]
= top#(proper(X))
top#(ok(X)) = [1 3] X + [6]
[4 9] [9]
>= [1 3] X + [6]
[4 3] [6]
= top#(active(X))
active(f(X1,X2)) = [1 0] X1 + [2 0] X2 + [0]
[0 3] [0 6] [2]
>= [1 0] X1 + [2 0] X2 + [0]
[0 3] [0 2] [2]
= f(active(X1),X2)
active(f(g(X),Y)) = [4 0] X + [2 0] Y + [0]
[0 6] [0 6] [5]
>= [0 0] X + [0 0] Y + [0]
[0 5] [0 4] [3]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [4 0] X + [0]
[0 6] [5]
>= [4 0] X + [0]
[0 6] [5]
= g(active(X))
f(mark(X1),X2) = [0 0] X1 + [2 0] X2 + [0]
[0 1] [0 2] [1]
>= [0 0] X1 + [0 0] X2 + [0]
[0 1] [0 2] [1]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [1 2] X1 + [2 4] X2 + [3]
[0 1] [0 2] [3]
>= [1 2] X1 + [2 4] X2 + [1]
[0 1] [0 2] [1]
= ok(f(X1,X2))
g(mark(X)) = [0 0] X + [0]
[0 2] [3]
>= [0 0] X + [0]
[0 2] [2]
= mark(g(X))
g(ok(X)) = [4 8] X + [4]
[0 2] [3]
>= [4 4] X + [3]
[0 2] [2]
= ok(g(X))
proper(f(X1,X2)) = [0 0] X1 + [0 0] X2 + [0]
[0 1] [0 2] [0]
>= [0 0] X1 + [0 0] X2 + [0]
[0 1] [0 2] [0]
= f(proper(X1),proper(X2))
proper(g(X)) = [0 0] X + [0]
[0 2] [1]
>= [0 0] X + [0]
[0 2] [1]
= g(proper(X))
*** 1.1.1.2.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
Strict TRS Rules:
Weak DP Rules:
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.2.1.1.1.2.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
Strict TRS Rules:
Weak DP Rules:
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: proper#(f(X1,X2)) ->
c_8(proper#(X1),proper#(X2))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.2.1.1.1.2.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
Strict TRS Rules:
Weak DP Rules:
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
NaturalMI {miDimension = 2, 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:
The following argument positions are considered usable:
uargs(c_8) = {1,2},
uargs(c_9) = {1}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [1 0] x1 + [0]
[0 4] [1]
p(f) = [1 0] x1 + [2 0] x2 + [0]
[0 1] [0 2] [1]
p(g) = [2 0] x1 + [0]
[0 1] [1]
p(mark) = [0 0] x1 + [0]
[0 1] [4]
p(ok) = [1 1] x1 + [2]
[0 0] [1]
p(proper) = [0 0] x1 + [0]
[0 1] [0]
p(top) = [0 0] x1 + [1]
[0 1] [0]
p(active#) = [1 1] x1 + [1]
[0 0] [2]
p(f#) = [0 0] x1 + [0]
[1 0] [2]
p(g#) = [0 1] x1 + [1]
[4 2] [1]
p(proper#) = [0 1] x1 + [0]
[0 1] [0]
p(top#) = [4 1] x1 + [6]
[6 1] [0]
p(c_1) = [1 1] x2 + [1]
[4 0] [0]
p(c_2) = [0 0] x1 + [1 4] x3 + [0]
[1 2] [0 1] [4]
p(c_3) = [1 0] x1 + [0 0] x2 + [0]
[0 4] [0 1] [2]
p(c_4) = [4 2] x1 + [0]
[0 1] [1]
p(c_5) = [4 1] x1 + [1]
[0 1] [4]
p(c_6) = [2 0] x1 + [0]
[2 0] [0]
p(c_7) = [1]
[0]
p(c_8) = [1 0] x1 + [1 1] x2 + [0]
[1 0] [1 1] [1]
p(c_9) = [1 0] x1 + [1]
[0 1] [1]
p(c_10) = [0 0] x2 + [0]
[1 0] [1]
p(c_11) = [1 0] x1 + [0]
[1 1] [1]
Following rules are strictly oriented:
proper#(f(X1,X2)) = [0 1] X1 + [0 2] X2 + [1]
[0 1] [0 2] [1]
> [0 1] X1 + [0 2] X2 + [0]
[0 1] [0 2] [1]
= c_8(proper#(X1),proper#(X2))
Following rules are (at-least) weakly oriented:
proper#(g(X)) = [0 1] X + [1]
[0 1] [1]
>= [0 1] X + [1]
[0 1] [1]
= c_9(proper#(X))
top#(mark(X)) = [0 1] X + [10]
[0 1] [4]
>= [0 1] X + [0]
[0 1] [0]
= proper#(X)
top#(mark(X)) = [0 1] X + [10]
[0 1] [4]
>= [0 1] X + [6]
[0 1] [0]
= top#(proper(X))
top#(ok(X)) = [4 4] X + [15]
[6 6] [13]
>= [4 4] X + [7]
[6 4] [1]
= top#(active(X))
active(f(X1,X2)) = [1 0] X1 + [2 0] X2 + [0]
[0 4] [0 8] [5]
>= [1 0] X1 + [2 0] X2 + [0]
[0 4] [0 2] [2]
= f(active(X1),X2)
active(f(g(X),Y)) = [2 0] X + [2 0] Y + [0]
[0 4] [0 8] [9]
>= [0 0] X + [0 0] Y + [0]
[0 3] [0 4] [9]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [2 0] X + [0]
[0 4] [5]
>= [2 0] X + [0]
[0 4] [2]
= g(active(X))
f(mark(X1),X2) = [0 0] X1 + [2 0] X2 + [0]
[0 1] [0 2] [5]
>= [0 0] X1 + [0 0] X2 + [0]
[0 1] [0 2] [5]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [1 1] X1 + [2 2] X2 + [6]
[0 0] [0 0] [4]
>= [1 1] X1 + [2 2] X2 + [3]
[0 0] [0 0] [1]
= ok(f(X1,X2))
g(mark(X)) = [0 0] X + [0]
[0 1] [5]
>= [0 0] X + [0]
[0 1] [5]
= mark(g(X))
g(ok(X)) = [2 2] X + [4]
[0 0] [2]
>= [2 1] X + [3]
[0 0] [1]
= ok(g(X))
proper(f(X1,X2)) = [0 0] X1 + [0 0] X2 + [0]
[0 1] [0 2] [1]
>= [0 0] X1 + [0 0] X2 + [0]
[0 1] [0 2] [1]
= f(proper(X1),proper(X2))
proper(g(X)) = [0 0] X + [0]
[0 1] [1]
>= [0 0] X + [0]
[0 1] [1]
= g(proper(X))
*** 1.1.1.2.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.2.1.1.1.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
top#(mark(X)) -> proper#(X)
top#(mark(X)) -> top#(proper(X))
top#(ok(X)) -> top#(active(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
-->_2 proper#(g(X)) -> c_9(proper#(X)):2
-->_1 proper#(g(X)) -> c_9(proper#(X)):2
-->_2 proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2)):1
-->_1 proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2)):1
2:W:proper#(g(X)) -> c_9(proper#(X))
-->_1 proper#(g(X)) -> c_9(proper#(X)):2
-->_1 proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2)):1
3:W:top#(mark(X)) -> proper#(X)
-->_1 proper#(g(X)) -> c_9(proper#(X)):2
-->_1 proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2)):1
4:W:top#(mark(X)) -> top#(proper(X))
-->_1 top#(ok(X)) -> top#(active(X)):5
-->_1 top#(mark(X)) -> top#(proper(X)):4
-->_1 top#(mark(X)) -> proper#(X):3
5:W:top#(ok(X)) -> top#(active(X))
-->_1 top#(ok(X)) -> top#(active(X)):5
-->_1 top#(mark(X)) -> top#(proper(X)):4
-->_1 top#(mark(X)) -> proper#(X):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: top#(mark(X)) -> top#(proper(X))
5: top#(ok(X)) -> top#(active(X))
3: top#(mark(X)) -> proper#(X)
1: proper#(f(X1,X2)) ->
c_8(proper#(X1),proper#(X2))
2: proper#(g(X)) -> c_9(proper#(X))
*** 1.1.1.2.1.1.1.2.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
Strict TRS Rules:
Weak DP Rules:
proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
proper#(g(X)) -> c_9(proper#(X))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
-->_2 proper#(g(X)) -> c_9(proper#(X)):4
-->_2 proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2)):3
-->_1 top#(ok(X)) -> c_11(top#(active(X))):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):1
2:S:top#(ok(X)) -> c_11(top#(active(X)))
-->_1 top#(ok(X)) -> c_11(top#(active(X))):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):1
3:W:proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2))
-->_2 proper#(g(X)) -> c_9(proper#(X)):4
-->_1 proper#(g(X)) -> c_9(proper#(X)):4
-->_2 proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2)):3
-->_1 proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2)):3
4:W:proper#(g(X)) -> c_9(proper#(X))
-->_1 proper#(g(X)) -> c_9(proper#(X)):4
-->_1 proper#(f(X1,X2)) -> c_8(proper#(X1),proper#(X2)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: proper#(g(X)) -> c_9(proper#(X))
3: proper#(f(X1,X2)) ->
c_8(proper#(X1),proper#(X2))
*** 1.1.1.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
top#(ok(X)) -> c_11(top#(active(X)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/2,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:top#(mark(X)) -> c_10(top#(proper(X)),proper#(X))
-->_1 top#(ok(X)) -> c_11(top#(active(X))):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):1
2:S:top#(ok(X)) -> c_11(top#(active(X)))
-->_1 top#(ok(X)) -> c_11(top#(active(X))):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X)),proper#(X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
top#(mark(X)) -> c_10(top#(proper(X)))
*** 1.1.1.2.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)))
top#(ok(X)) -> c_11(top#(active(X)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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: top#(ok(X)) ->
c_11(top#(active(X)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)))
top#(ok(X)) -> c_11(top#(active(X)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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_10) = {1},
uargs(c_11) = {1}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [1] x1 + [0]
p(f) = [4] x2 + [0]
p(g) = [1] x1 + [0]
p(mark) = [0]
p(ok) = [1] x1 + [4]
p(proper) = [0]
p(top) = [2]
p(active#) = [1]
p(f#) = [8] x2 + [1]
p(g#) = [0]
p(proper#) = [1]
p(top#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [0]
p(c_3) = [1]
p(c_4) = [2]
p(c_5) = [2]
p(c_6) = [1]
p(c_7) = [1]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [4] x1 + [0]
p(c_11) = [1] x1 + [0]
Following rules are strictly oriented:
top#(ok(X)) = [1] X + [4]
> [1] X + [0]
= c_11(top#(active(X)))
Following rules are (at-least) weakly oriented:
top#(mark(X)) = [0]
>= [0]
= c_10(top#(proper(X)))
active(f(X1,X2)) = [4] X2 + [0]
>= [4] X2 + [0]
= f(active(X1),X2)
active(f(g(X),Y)) = [4] Y + [0]
>= [0]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [1] X + [0]
>= [1] X + [0]
= g(active(X))
f(mark(X1),X2) = [4] X2 + [0]
>= [0]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [4] X2 + [16]
>= [4] X2 + [4]
= ok(f(X1,X2))
g(mark(X)) = [0]
>= [0]
= mark(g(X))
g(ok(X)) = [1] X + [4]
>= [1] X + [4]
= ok(g(X))
proper(f(X1,X2)) = [0]
>= [0]
= f(proper(X1),proper(X2))
proper(g(X)) = [0]
>= [0]
= g(proper(X))
*** 1.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)))
Strict TRS Rules:
Weak DP Rules:
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.2.1.1.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)))
Strict TRS Rules:
Weak DP Rules:
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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: top#(mark(X)) ->
c_10(top#(proper(X)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.2.1.1.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)))
Strict TRS Rules:
Weak DP Rules:
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
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_10) = {1},
uargs(c_11) = {1}
Following symbols are considered usable:
{active,f,g,proper,active#,f#,g#,proper#,top#}
TcT has computed the following interpretation:
p(active) = [1] x1 + [1]
p(f) = [1] x1 + [2]
p(g) = [1] x1 + [6]
p(mark) = [1] x1 + [6]
p(ok) = [1] x1 + [1]
p(proper) = [1] x1 + [4]
p(top) = [2]
p(active#) = [4]
p(f#) = [1] x1 + [0]
p(g#) = [1]
p(proper#) = [1] x1 + [0]
p(top#) = [3] x1 + [12]
p(c_1) = [1] x2 + [8]
p(c_2) = [1] x3 + [0]
p(c_3) = [1] x1 + [4] x2 + [2]
p(c_4) = [2] x1 + [8]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [0]
p(c_8) = [1] x1 + [1] x2 + [4]
p(c_9) = [1]
p(c_10) = [1] x1 + [4]
p(c_11) = [1] x1 + [0]
Following rules are strictly oriented:
top#(mark(X)) = [3] X + [30]
> [3] X + [28]
= c_10(top#(proper(X)))
Following rules are (at-least) weakly oriented:
top#(ok(X)) = [3] X + [15]
>= [3] X + [15]
= c_11(top#(active(X)))
active(f(X1,X2)) = [1] X1 + [3]
>= [1] X1 + [3]
= f(active(X1),X2)
active(f(g(X),Y)) = [1] X + [9]
>= [1] X + [8]
= mark(f(X,f(g(X),Y)))
active(g(X)) = [1] X + [7]
>= [1] X + [7]
= g(active(X))
f(mark(X1),X2) = [1] X1 + [8]
>= [1] X1 + [8]
= mark(f(X1,X2))
f(ok(X1),ok(X2)) = [1] X1 + [3]
>= [1] X1 + [3]
= ok(f(X1,X2))
g(mark(X)) = [1] X + [12]
>= [1] X + [12]
= mark(g(X))
g(ok(X)) = [1] X + [7]
>= [1] X + [7]
= ok(g(X))
proper(f(X1,X2)) = [1] X1 + [6]
>= [1] X1 + [6]
= f(proper(X1),proper(X2))
proper(g(X)) = [1] X + [10]
>= [1] X + [10]
= g(proper(X))
*** 1.1.1.2.1.1.2.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)))
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.2.1.1.2.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
top#(mark(X)) -> c_10(top#(proper(X)))
top#(ok(X)) -> c_11(top#(active(X)))
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:top#(mark(X)) -> c_10(top#(proper(X)))
-->_1 top#(ok(X)) -> c_11(top#(active(X))):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X))):1
2:W:top#(ok(X)) -> c_11(top#(active(X)))
-->_1 top#(ok(X)) -> c_11(top#(active(X))):2
-->_1 top#(mark(X)) -> c_10(top#(proper(X))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: top#(mark(X)) ->
c_10(top#(proper(X)))
2: top#(ok(X)) ->
c_11(top#(active(X)))
*** 1.1.1.2.1.1.2.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
active(f(X1,X2)) -> f(active(X1),X2)
active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
active(g(X)) -> g(active(X))
f(mark(X1),X2) -> mark(f(X1,X2))
f(ok(X1),ok(X2)) -> ok(f(X1,X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1,X2)) -> f(proper(X1),proper(X2))
proper(g(X)) -> g(proper(X))
Signature:
{active/1,f/2,g/1,proper/1,top/1,active#/1,f#/2,g#/1,proper#/1,top#/1} / {mark/1,ok/1,c_1/2,c_2/3,c_3/2,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1,c_10/1,c_11/1}
Obligation:
Innermost
basic terms: {active#,f#,g#,proper#,top#}/{mark,ok}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).