* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod,pred} and constructors {0,false,s
,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod,pred} and constructors {0,false,s
,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
le(x,y){x -> s(x),y -> s(y)} =
le(s(x),s(y)) ->^+ le(x,y)
= C[le(x,y) = le(x,y){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod,pred} and constructors {0,false,s
,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
if_mod#(false(),s(x),s(y)) -> c_1()
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(x,0()) -> c_6()
minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
mod#(0(),y) -> c_8()
mod#(s(x),0()) -> c_9()
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
pred#(s(x)) -> c_11()
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(false(),s(x),s(y)) -> c_1()
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(x,0()) -> c_6()
minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
mod#(0(),y) -> c_8()
mod#(s(x),0()) -> c_9()
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
pred#(s(x)) -> c_11()
- Weak TRS:
if_mod(false(),s(x),s(y)) -> s(x)
if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
mod(0(),y) -> 0()
mod(s(x),0()) -> 0()
mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
if_mod#(false(),s(x),s(y)) -> c_1()
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(x,0()) -> c_6()
minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
mod#(0(),y) -> c_8()
mod#(s(x),0()) -> c_9()
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
pred#(s(x)) -> c_11()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(false(),s(x),s(y)) -> c_1()
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(x,0()) -> c_6()
minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
mod#(0(),y) -> c_8()
mod#(s(x),0()) -> c_9()
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
pred#(s(x)) -> c_11()
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,4,6,8,9,11}
by application of
Pre({1,3,4,6,8,9,11}) = {2,5,7,10}.
Here rules are labelled as follows:
1: if_mod#(false(),s(x),s(y)) -> c_1()
2: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
3: le#(0(),y) -> c_3()
4: le#(s(x),0()) -> c_4()
5: le#(s(x),s(y)) -> c_5(le#(x,y))
6: minus#(x,0()) -> c_6()
7: minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
8: mod#(0(),y) -> c_8()
9: mod#(s(x),0()) -> c_9()
10: mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
11: pred#(s(x)) -> c_11()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
- Weak DPs:
if_mod#(false(),s(x),s(y)) -> c_1()
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
minus#(x,0()) -> c_6()
mod#(0(),y) -> c_8()
mod#(s(x),0()) -> c_9()
pred#(s(x)) -> c_11()
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):4
-->_2 minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y)):3
-->_1 mod#(0(),y) -> c_8():9
-->_2 minus#(x,0()) -> c_6():8
2:S:le#(s(x),s(y)) -> c_5(le#(x,y))
-->_1 le#(s(x),0()) -> c_4():7
-->_1 le#(0(),y) -> c_3():6
-->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2
3:S:minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
-->_1 pred#(s(x)) -> c_11():11
-->_2 minus#(x,0()) -> c_6():8
-->_2 minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y)):3
4:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
-->_2 le#(s(x),0()) -> c_4():7
-->_2 le#(0(),y) -> c_3():6
-->_1 if_mod#(false(),s(x),s(y)) -> c_1():5
-->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2
-->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
5:W:if_mod#(false(),s(x),s(y)) -> c_1()
6:W:le#(0(),y) -> c_3()
7:W:le#(s(x),0()) -> c_4()
8:W:minus#(x,0()) -> c_6()
9:W:mod#(0(),y) -> c_8()
10:W:mod#(s(x),0()) -> c_9()
11:W:pred#(s(x)) -> c_11()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
10: mod#(s(x),0()) -> c_9()
9: mod#(0(),y) -> c_8()
8: minus#(x,0()) -> c_6()
11: pred#(s(x)) -> c_11()
5: if_mod#(false(),s(x),s(y)) -> c_1()
6: le#(0(),y) -> c_3()
7: le#(s(x),0()) -> c_4()
** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):4
-->_2 minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y)):3
2:S:le#(s(x),s(y)) -> c_5(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2
3:S:minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y)):3
4:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
-->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2
-->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
minus#(x,s(y)) -> c_7(minus#(x,y))
** Step 1.b:6: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(x,s(y)) -> c_7(minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: le#(s(x),s(y)) -> c_5(le#(x,y))
The strictly oriented rules are moved into the weak component.
*** Step 1.b:6.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(x,s(y)) -> c_7(minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_2) = {1,2},
uargs(c_5) = {1},
uargs(c_7) = {1},
uargs(c_10) = {1,2}
Following symbols are considered usable:
{le,minus,pred,if_mod#,le#,minus#,mod#,pred#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[1]
p(false) = [1]
[0]
[0]
p(if_mod) = [0]
[0]
[0]
p(le) = [0 0 1] [0 0 0] [0]
[0 0 1] x1 + [0 0 1] x2 + [0]
[0 0 0] [1 0 0] [0]
p(minus) = [1 0 0] [0 0 0] [0]
[1 1 1] x1 + [1 0 1] x2 + [0]
[0 0 1] [0 0 0] [0]
p(mod) = [0]
[0]
[0]
p(pred) = [1 0 0] [0]
[1 0 0] x1 + [0]
[0 0 1] [0]
p(s) = [1 1 1] [0]
[0 0 1] x1 + [1]
[0 0 1] [1]
p(true) = [1]
[1]
[0]
p(if_mod#) = [0 0 0] [1 0 0] [0 1 1] [1]
[1 0 0] x1 + [0 1 0] x2 + [0 0 0] x3 + [1]
[1 1 0] [1 0 0] [0 0 0] [0]
p(le#) = [0 0 1] [0]
[1 0 0] x2 + [0]
[0 0 1] [0]
p(minus#) = [0 1 0] [0]
[0 0 0] x1 + [1]
[0 0 1] [1]
p(mod#) = [1 0 1] [0 1 1] [0]
[1 1 0] x1 + [0 0 1] x2 + [1]
[0 0 0] [1 0 1] [0]
p(pred#) = [0]
[0]
[0]
p(c_1) = [0]
[0]
[0]
p(c_2) = [1 0 0] [1 1 0] [0]
[0 0 0] x1 + [0 1 1] x2 + [1]
[0 0 0] [0 0 1] [1]
p(c_3) = [0]
[0]
[0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [1 0 0] [0]
[1 1 0] x1 + [0]
[0 0 0] [1]
p(c_6) = [0]
[0]
[0]
p(c_7) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_8) = [0]
[0]
[0]
p(c_9) = [0]
[0]
[0]
p(c_10) = [1 0 0] [1 0 0] [0]
[0 1 0] x1 + [1 1 0] x2 + [1]
[0 0 0] [0 0 0] [0]
p(c_11) = [0]
[0]
[0]
Following rules are strictly oriented:
le#(s(x),s(y)) = [0 0 1] [1]
[1 1 1] y + [0]
[0 0 1] [1]
> [0 0 1] [0]
[1 0 1] y + [0]
[0 0 0] [1]
= c_5(le#(x,y))
Following rules are (at-least) weakly oriented:
if_mod#(true(),s(x),s(y)) = [1 1 1] [0 0 2] [3]
[0 0 1] x + [0 0 0] y + [3]
[1 1 1] [0 0 0] [2]
>= [1 1 1] [0 0 2] [3]
[0 0 1] x + [0 0 0] y + [3]
[0 0 1] [0 0 0] [2]
= c_2(mod#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) = [0 1 0] [0]
[0 0 0] x + [1]
[0 0 1] [1]
>= [0 1 0] [0]
[0 0 0] x + [0]
[0 0 0] [0]
= c_7(minus#(x,y))
mod#(s(x),s(y)) = [1 1 2] [0 0 2] [3]
[1 1 2] x + [0 0 1] y + [3]
[0 0 0] [1 1 2] [1]
>= [1 1 2] [0 0 2] [3]
[1 0 2] x + [0 0 1] y + [3]
[0 0 0] [0 0 0] [0]
= c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
le(0(),y) = [0 0 0] [1]
[0 0 1] y + [1]
[1 0 0] [0]
>= [1]
[1]
[0]
= true()
le(s(x),0()) = [0 0 1] [1]
[0 0 1] x + [2]
[0 0 0] [0]
>= [1]
[0]
[0]
= false()
le(s(x),s(y)) = [0 0 1] [0 0 0] [1]
[0 0 1] x + [0 0 1] y + [2]
[0 0 0] [1 1 1] [0]
>= [0 0 1] [0 0 0] [0]
[0 0 1] x + [0 0 1] y + [0]
[0 0 0] [1 0 0] [0]
= le(x,y)
minus(x,0()) = [1 0 0] [0]
[1 1 1] x + [1]
[0 0 1] [0]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
minus(x,s(y)) = [1 0 0] [0 0 0] [0]
[1 1 1] x + [1 1 2] y + [1]
[0 0 1] [0 0 0] [0]
>= [1 0 0] [0]
[1 0 0] x + [0]
[0 0 1] [0]
= pred(minus(x,y))
pred(s(x)) = [1 1 1] [0]
[1 1 1] x + [0]
[0 0 1] [1]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= x
*** Step 1.b:6.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) -> c_7(minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
- Weak DPs:
le#(s(x),s(y)) -> c_5(le#(x,y))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) -> c_7(minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
- Weak DPs:
le#(s(x),s(y)) -> c_5(le#(x,y))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):3
-->_2 minus#(x,s(y)) -> c_7(minus#(x,y)):2
2:S:minus#(x,s(y)) -> c_7(minus#(x,y))
-->_1 minus#(x,s(y)) -> c_7(minus#(x,y)):2
3:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
-->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):4
-->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
4:W:le#(s(x),s(y)) -> c_5(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: le#(s(x),s(y)) -> c_5(le#(x,y))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) -> c_7(minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):3
-->_2 minus#(x,s(y)) -> c_7(minus#(x,y)):2
2:S:minus#(x,s(y)) -> c_7(minus#(x,y))
-->_1 minus#(x,s(y)) -> c_7(minus#(x,y)):2
3:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
-->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
*** Step 1.b:6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) -> c_7(minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: minus#(x,s(y)) -> c_7(minus#(x,y))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
minus#(x,s(y)) -> c_7(minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_2) = {1,2},
uargs(c_7) = {1},
uargs(c_10) = {1}
Following symbols are considered usable:
{minus,pred,if_mod#,le#,minus#,mod#,pred#}
TcT has computed the following interpretation:
p(0) = 0
p(false) = 1
p(if_mod) = 2 + 4*x1*x3 + x1^2 + 4*x3
p(le) = 0
p(minus) = x1
p(mod) = 1 + 2*x1 + x1^2 + 4*x2^2
p(pred) = x1
p(s) = 1 + x1
p(true) = 0
p(if_mod#) = 6 + 2*x2*x3
p(le#) = 2 + x1 + 4*x1*x2
p(minus#) = 1 + x2
p(mod#) = 6 + 2*x1*x2
p(pred#) = 1 + 2*x1^2
p(c_1) = 0
p(c_2) = 1 + x1 + x2
p(c_3) = 0
p(c_4) = 1
p(c_5) = 1
p(c_6) = 1
p(c_7) = x1
p(c_8) = 0
p(c_9) = 0
p(c_10) = x1
p(c_11) = 1
Following rules are strictly oriented:
minus#(x,s(y)) = 2 + y
> 1 + y
= c_7(minus#(x,y))
Following rules are (at-least) weakly oriented:
if_mod#(true(),s(x),s(y)) = 8 + 2*x + 2*x*y + 2*y
>= 8 + 2*x + 2*x*y + y
= c_2(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) = 8 + 2*x + 2*x*y + 2*y
>= 8 + 2*x + 2*x*y + 2*y
= c_10(if_mod#(le(y,x),s(x),s(y)))
minus(x,0()) = x
>= x
= x
minus(x,s(y)) = x
>= x
= pred(minus(x,y))
pred(s(x)) = 1 + x
>= x
= x
**** Step 1.b:6.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
- Weak DPs:
minus#(x,s(y)) -> c_7(minus#(x,y))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
- Weak DPs:
minus#(x,s(y)) -> c_7(minus#(x,y))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_7(minus#(x,y)):3
-->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y))):2
2:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
-->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
3:W:minus#(x,s(y)) -> c_7(minus#(x,y))
-->_1 minus#(x,s(y)) -> c_7(minus#(x,y)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: minus#(x,s(y)) -> c_7(minus#(x,y))
**** Step 1.b:6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
-->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y))):2
2:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
-->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
**** Step 1.b:6.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ 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}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
Consider the set of all dependency pairs
1: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
2: mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{2}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
***** Step 1.b:6.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_10) = {1}
Following symbols are considered usable:
{le,minus,pred,if_mod#,le#,minus#,mod#,pred#}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [1]
p(if_mod) = [8] x2 + [1] x3 + [4]
p(le) = [1]
p(minus) = [1] x1 + [0]
p(mod) = [1] x1 + [1] x2 + [2]
p(pred) = [1] x1 + [0]
p(s) = [1] x1 + [2]
p(true) = [1]
p(if_mod#) = [4] x1 + [1] x2 + [9] x3 + [1]
p(le#) = [0]
p(minus#) = [2] x2 + [1]
p(mod#) = [1] x1 + [9] x2 + [7]
p(pred#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [1] x1 + [0]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [8]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
Following rules are strictly oriented:
mod#(s(x),s(y)) = [1] x + [9] y + [27]
> [1] x + [9] y + [25]
= c_10(if_mod#(le(y,x),s(x),s(y)))
Following rules are (at-least) weakly oriented:
if_mod#(true(),s(x),s(y)) = [1] x + [9] y + [25]
>= [1] x + [9] y + [25]
= c_2(mod#(minus(x,y),s(y)))
le(0(),y) = [1]
>= [1]
= true()
le(s(x),0()) = [1]
>= [1]
= false()
le(s(x),s(y)) = [1]
>= [1]
= le(x,y)
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(x,s(y)) = [1] x + [0]
>= [1] x + [0]
= pred(minus(x,y))
pred(s(x)) = [1] x + [2]
>= [1] x + [0]
= x
***** Step 1.b:6.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
- Weak DPs:
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
-->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y))):2
2:W:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
-->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
2: mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
***** Step 1.b:6.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
- Signature:
{if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))