WORST_CASE(?,O(n^1))
* Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
d(0()) -> 0()
d(s(X)) -> s(s(d(X)))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
p(X,0()) -> X
p(0(),X) -> X
p(s(X),s(Y)) -> s(s(p(X,Y)))
q(0()) -> 0()
q(s(X)) -> s(p(q(X),d(X)))
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
d(s(X)) -> s(s(d(X)))
f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
p(s(X),s(Y)) -> s(s(p(X,Y)))
q(s(X)) -> s(p(q(X),d(X)))
All above mentioned rules can be savely removed.
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
d(0()) -> 0()
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
p(X,0()) -> X
p(0(),X) -> X
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
a#(X) -> c_1()
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
d#(0()) -> c_5()
f#(X1,X2) -> c_6()
f#(0(),X) -> c_7()
p#(X,0()) -> c_8()
p#(0(),X) -> c_9()
q#(0()) -> c_10()
s#(X) -> c_11()
t#(N) -> c_12(q#(N))
t#(X) -> c_13()
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a#(X) -> c_1()
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
d#(0()) -> c_5()
f#(X1,X2) -> c_6()
f#(0(),X) -> c_7()
p#(X,0()) -> c_8()
p#(0(),X) -> c_9()
q#(0()) -> c_10()
s#(X) -> c_11()
t#(N) -> c_12(q#(N))
t#(X) -> c_13()
- Strict TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
d(0()) -> 0()
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
p(X,0()) -> X
p(0(),X) -> X
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
a#(X) -> c_1()
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
d#(0()) -> c_5()
f#(X1,X2) -> c_6()
f#(0(),X) -> c_7()
p#(X,0()) -> c_8()
p#(0(),X) -> c_9()
q#(0()) -> c_10()
s#(X) -> c_11()
t#(N) -> c_12(q#(N))
t#(X) -> c_13()
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a#(X) -> c_1()
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
d#(0()) -> c_5()
f#(X1,X2) -> c_6()
f#(0(),X) -> c_7()
p#(X,0()) -> c_8()
p#(0(),X) -> c_9()
q#(0()) -> c_10()
s#(X) -> c_11()
t#(N) -> c_12(q#(N))
t#(X) -> c_13()
- Strict TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cs) = {1},
uargs(f) = {1,2},
uargs(r) = {1},
uargs(s) = {1},
uargs(t) = {1},
uargs(f#) = {1,2},
uargs(s#) = {1},
uargs(t#) = {1},
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(a) = [4] x1 + [1]
p(cs) = [1] x1 + [0]
p(d) = [0]
p(f) = [1] x1 + [1] x2 + [14]
p(nf) = [1] x1 + [1] x2 + [4]
p(nil) = [0]
p(ns) = [1] x1 + [4]
p(nt) = [1] x1 + [6]
p(p) = [1] x2 + [0]
p(q) = [1] x1 + [2]
p(r) = [1] x1 + [3]
p(s) = [1] x1 + [8]
p(t) = [1] x1 + [8]
p(a#) = [5] x1 + [0]
p(d#) = [0]
p(f#) = [1] x1 + [1] x2 + [0]
p(p#) = [5]
p(q#) = [1] x1 + [13]
p(s#) = [1] x1 + [0]
p(t#) = [1] x1 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [8]
p(c_8) = [2]
p(c_9) = [2]
p(c_10) = [2]
p(c_11) = [1]
p(c_12) = [1] x1 + [5]
p(c_13) = [0]
Following rules are strictly oriented:
a#(nf(X1,X2)) = [5] X1 + [5] X2 + [20]
> [4] X1 + [4] X2 + [2]
= c_2(f#(a(X1),a(X2)))
a#(ns(X)) = [5] X + [20]
> [4] X + [1]
= c_3(s#(a(X)))
a#(nt(X)) = [5] X + [30]
> [4] X + [1]
= c_4(t#(a(X)))
p#(X,0()) = [5]
> [2]
= c_8()
p#(0(),X) = [5]
> [2]
= c_9()
q#(0()) = [13]
> [2]
= c_10()
a(X) = [4] X + [1]
> [1] X + [0]
= X
a(nf(X1,X2)) = [4] X1 + [4] X2 + [17]
> [4] X1 + [4] X2 + [16]
= f(a(X1),a(X2))
a(ns(X)) = [4] X + [17]
> [4] X + [9]
= s(a(X))
a(nt(X)) = [4] X + [25]
> [4] X + [9]
= t(a(X))
f(X1,X2) = [1] X1 + [1] X2 + [14]
> [1] X1 + [1] X2 + [4]
= nf(X1,X2)
f(0(),X) = [1] X + [14]
> [0]
= nil()
q(0()) = [2]
> [0]
= 0()
s(X) = [1] X + [8]
> [1] X + [4]
= ns(X)
t(N) = [1] N + [8]
> [1] N + [5]
= cs(r(q(N)),nt(ns(N)))
t(X) = [1] X + [8]
> [1] X + [6]
= nt(X)
Following rules are (at-least) weakly oriented:
a#(X) = [5] X + [0]
>= [0]
= c_1()
d#(0()) = [0]
>= [1]
= c_5()
f#(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1]
= c_6()
f#(0(),X) = [1] X + [0]
>= [8]
= c_7()
s#(X) = [1] X + [0]
>= [1]
= c_11()
t#(N) = [1] N + [0]
>= [1] N + [18]
= c_12(q#(N))
t#(X) = [1] X + [0]
>= [0]
= c_13()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
a#(X) -> c_1()
d#(0()) -> c_5()
f#(X1,X2) -> c_6()
f#(0(),X) -> c_7()
s#(X) -> c_11()
t#(N) -> c_12(q#(N))
t#(X) -> c_13()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
p#(X,0()) -> c_8()
p#(0(),X) -> c_9()
q#(0()) -> c_10()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2}
by application of
Pre({1,2}) = {}.
Here rules are labelled as follows:
1: a#(X) -> c_1()
2: d#(0()) -> c_5()
3: f#(X1,X2) -> c_6()
4: f#(0(),X) -> c_7()
5: s#(X) -> c_11()
6: t#(N) -> c_12(q#(N))
7: t#(X) -> c_13()
8: a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
9: a#(ns(X)) -> c_3(s#(a(X)))
10: a#(nt(X)) -> c_4(t#(a(X)))
11: p#(X,0()) -> c_8()
12: p#(0(),X) -> c_9()
13: q#(0()) -> c_10()
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X1,X2) -> c_6()
f#(0(),X) -> c_7()
s#(X) -> c_11()
t#(N) -> c_12(q#(N))
t#(X) -> c_13()
- Weak DPs:
a#(X) -> c_1()
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
d#(0()) -> c_5()
p#(X,0()) -> c_8()
p#(0(),X) -> c_9()
q#(0()) -> c_10()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(X1,X2) -> c_6()
2:S:f#(0(),X) -> c_7()
3:S:s#(X) -> c_11()
4:S:t#(N) -> c_12(q#(N))
-->_1 q#(0()) -> c_10():13
5:S:t#(X) -> c_13()
6:W:a#(X) -> c_1()
7:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
-->_1 f#(0(),X) -> c_7():2
-->_1 f#(X1,X2) -> c_6():1
8:W:a#(ns(X)) -> c_3(s#(a(X)))
-->_1 s#(X) -> c_11():3
9:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(X) -> c_13():5
-->_1 t#(N) -> c_12(q#(N)):4
10:W:d#(0()) -> c_5()
11:W:p#(X,0()) -> c_8()
12:W:p#(0(),X) -> c_9()
13:W:q#(0()) -> c_10()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
12: p#(0(),X) -> c_9()
11: p#(X,0()) -> c_8()
10: d#(0()) -> c_5()
6: a#(X) -> c_1()
13: q#(0()) -> c_10()
* Step 7: SimplifyRHS WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X1,X2) -> c_6()
f#(0(),X) -> c_7()
s#(X) -> c_11()
t#(N) -> c_12(q#(N))
t#(X) -> c_13()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f#(X1,X2) -> c_6()
2:S:f#(0(),X) -> c_7()
3:S:s#(X) -> c_11()
4:S:t#(N) -> c_12(q#(N))
5:S:t#(X) -> c_13()
7:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
-->_1 f#(0(),X) -> c_7():2
-->_1 f#(X1,X2) -> c_6():1
8:W:a#(ns(X)) -> c_3(s#(a(X)))
-->_1 s#(X) -> c_11():3
9:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(X) -> c_13():5
-->_1 t#(N) -> c_12(q#(N)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
t#(N) -> c_12()
* Step 8: Decompose WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X1,X2) -> c_6()
f#(0(),X) -> c_7()
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
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 DPs:
f#(X1,X2) -> c_6()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
f#(0(),X) -> c_7()
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
,nt,r}
Problem (S)
- Strict DPs:
f#(0(),X) -> c_7()
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
f#(X1,X2) -> c_6()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
,nt,r}
** Step 8.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X1,X2) -> c_6()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
f#(0(),X) -> c_7()
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(X1,X2) -> c_6()
2:W:f#(0(),X) -> c_7()
3:W:s#(X) -> c_11()
4:W:t#(N) -> c_12()
5:W:t#(X) -> c_13()
6:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
-->_1 f#(X1,X2) -> c_6():1
-->_1 f#(0(),X) -> c_7():2
7:W:a#(ns(X)) -> c_3(s#(a(X)))
-->_1 s#(X) -> c_11():3
8:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(N) -> c_12():4
-->_1 t#(X) -> c_13():5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
8: a#(nt(X)) -> c_4(t#(a(X)))
7: a#(ns(X)) -> c_3(s#(a(X)))
5: t#(X) -> c_13()
4: t#(N) -> c_12()
3: s#(X) -> c_11()
2: f#(0(),X) -> c_7()
** Step 8.a:2: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(X1,X2) -> c_6()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:f#(X1,X2) -> c_6()
6:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
-->_1 f#(X1,X2) -> c_6():1
The dependency graph contains no loops, we remove all dependency pairs.
** Step 8.a:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(0(),X) -> c_7()
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
f#(X1,X2) -> c_6()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(0(),X) -> c_7()
2:S:s#(X) -> c_11()
3:S:t#(N) -> c_12()
4:S:t#(X) -> c_13()
5:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
-->_1 f#(X1,X2) -> c_6():8
-->_1 f#(0(),X) -> c_7():1
6:W:a#(ns(X)) -> c_3(s#(a(X)))
-->_1 s#(X) -> c_11():2
7:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(X) -> c_13():4
-->_1 t#(N) -> c_12():3
8:W:f#(X1,X2) -> c_6()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
8: f#(X1,X2) -> c_6()
** Step 8.b:2: Decompose WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(0(),X) -> c_7()
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
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 DPs:
f#(0(),X) -> c_7()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
,nt,r}
Problem (S)
- Strict DPs:
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
f#(0(),X) -> c_7()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
,nt,r}
*** Step 8.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(0(),X) -> c_7()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(0(),X) -> c_7()
2:W:s#(X) -> c_11()
3:W:t#(N) -> c_12()
4:W:t#(X) -> c_13()
5:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
-->_1 f#(0(),X) -> c_7():1
6:W:a#(ns(X)) -> c_3(s#(a(X)))
-->_1 s#(X) -> c_11():2
7:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(N) -> c_12():3
-->_1 t#(X) -> c_13():4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: a#(nt(X)) -> c_4(t#(a(X)))
6: a#(ns(X)) -> c_3(s#(a(X)))
4: t#(X) -> c_13()
3: t#(N) -> c_12()
2: s#(X) -> c_11()
*** Step 8.b:2.a:2: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(0(),X) -> c_7()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:f#(0(),X) -> c_7()
5:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
-->_1 f#(0(),X) -> c_7():1
The dependency graph contains no loops, we remove all dependency pairs.
*** Step 8.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 8.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
f#(0(),X) -> c_7()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:s#(X) -> c_11()
2:S:t#(N) -> c_12()
3:S:t#(X) -> c_13()
4:W:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
-->_1 f#(0(),X) -> c_7():7
5:W:a#(ns(X)) -> c_3(s#(a(X)))
-->_1 s#(X) -> c_11():1
6:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(X) -> c_13():3
-->_1 t#(N) -> c_12():2
7:W:f#(0(),X) -> c_7()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)))
7: f#(0(),X) -> c_7()
*** Step 8.b:2.b:2: Decompose WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
s#(X) -> c_11()
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
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 DPs:
s#(X) -> c_11()
- Weak DPs:
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
t#(N) -> c_12()
t#(X) -> c_13()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
,nt,r}
Problem (S)
- Strict DPs:
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
s#(X) -> c_11()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
,nt,r}
**** Step 8.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
s#(X) -> c_11()
- Weak DPs:
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
t#(N) -> c_12()
t#(X) -> c_13()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:s#(X) -> c_11()
2:W:t#(N) -> c_12()
3:W:t#(X) -> c_13()
5:W:a#(ns(X)) -> c_3(s#(a(X)))
-->_1 s#(X) -> c_11():1
6:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(N) -> c_12():2
-->_1 t#(X) -> c_13():3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: a#(nt(X)) -> c_4(t#(a(X)))
3: t#(X) -> c_13()
2: t#(N) -> c_12()
**** Step 8.b:2.b:2.a:2: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
s#(X) -> c_11()
- Weak DPs:
a#(ns(X)) -> c_3(s#(a(X)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:s#(X) -> c_11()
5:W:a#(ns(X)) -> c_3(s#(a(X)))
-->_1 s#(X) -> c_11():1
The dependency graph contains no loops, we remove all dependency pairs.
**** Step 8.b:2.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 8.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(ns(X)) -> c_3(s#(a(X)))
a#(nt(X)) -> c_4(t#(a(X)))
s#(X) -> c_11()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:t#(N) -> c_12()
2:S:t#(X) -> c_13()
3:W:a#(ns(X)) -> c_3(s#(a(X)))
-->_1 s#(X) -> c_11():5
4:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(X) -> c_13():2
-->_1 t#(N) -> c_12():1
5:W:s#(X) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: a#(ns(X)) -> c_3(s#(a(X)))
5: s#(X) -> c_11()
**** Step 8.b:2.b:2.b:2: Decompose WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
t#(N) -> c_12()
t#(X) -> c_13()
- Weak DPs:
a#(nt(X)) -> c_4(t#(a(X)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
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 DPs:
t#(N) -> c_12()
- Weak DPs:
a#(nt(X)) -> c_4(t#(a(X)))
t#(X) -> c_13()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
,nt,r}
Problem (S)
- Strict DPs:
t#(X) -> c_13()
- Weak DPs:
a#(nt(X)) -> c_4(t#(a(X)))
t#(N) -> c_12()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1
,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns
,nt,r}
***** Step 8.b:2.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
t#(N) -> c_12()
- Weak DPs:
a#(nt(X)) -> c_4(t#(a(X)))
t#(X) -> c_13()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:t#(N) -> c_12()
2:W:t#(X) -> c_13()
4:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(N) -> c_12():1
-->_1 t#(X) -> c_13():2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: t#(X) -> c_13()
***** Step 8.b:2.b:2.b:2.a:2: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
t#(N) -> c_12()
- Weak DPs:
a#(nt(X)) -> c_4(t#(a(X)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:t#(N) -> c_12()
4:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(N) -> c_12():1
The dependency graph contains no loops, we remove all dependency pairs.
***** Step 8.b:2.b:2.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
***** Step 8.b:2.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
t#(X) -> c_13()
- Weak DPs:
a#(nt(X)) -> c_4(t#(a(X)))
t#(N) -> c_12()
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:t#(X) -> c_13()
2:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(N) -> c_12():3
-->_1 t#(X) -> c_13():1
3:W:t#(N) -> c_12()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: t#(N) -> c_12()
***** Step 8.b:2.b:2.b:2.b:2: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
t#(X) -> c_13()
- Weak DPs:
a#(nt(X)) -> c_4(t#(a(X)))
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:t#(X) -> c_13()
2:W:a#(nt(X)) -> c_4(t#(a(X)))
-->_1 t#(X) -> c_13():1
The dependency graph contains no loops, we remove all dependency pairs.
***** Step 8.b:2.b:2.b:2.b:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
a(X) -> X
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(ns(X)) -> s(a(X))
a(nt(X)) -> t(a(X))
f(X1,X2) -> nf(X1,X2)
f(0(),X) -> nil()
q(0()) -> 0()
s(X) -> ns(X)
t(N) -> cs(r(q(N)),nt(ns(N)))
t(X) -> nt(X)
- Signature:
{a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
,r}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))