* Step 1: Sum WORST_CASE(Omega(n^1),O(n^4))
+ Considered Problem:
- Strict TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus,le,log,minus,quot} 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_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus,le,log,minus,quot} 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^4))
+ Considered Problem:
- Strict TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus,le,log,minus,quot} and constructors {0,false,s
,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_minus#(true(),s(x),y) -> c_2()
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
le#(s(x),s(y)) -> c_5(le#(x,y))
log#(s(0())) -> c_6()
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(0(),y) -> c_8()
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
quot#(0(),s(y)) -> c_10()
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_minus#(true(),s(x),y) -> c_2()
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
le#(s(x),s(y)) -> c_5(le#(x,y))
log#(s(0())) -> c_6()
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(0(),y) -> c_8()
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
quot#(0(),s(y)) -> c_10()
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_minus#(true(),s(x),y) -> c_2()
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
le#(s(x),s(y)) -> c_5(le#(x,y))
log#(s(0())) -> c_6()
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(0(),y) -> c_8()
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
quot#(0(),s(y)) -> c_10()
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
if_minus#(true(),s(x),y) -> c_2()
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
le#(s(x),s(y)) -> c_5(le#(x,y))
log#(s(0())) -> c_6()
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(0(),y) -> c_8()
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
quot#(0(),s(y)) -> c_10()
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,3,4,6,8,10}
by application of
Pre({2,3,4,6,8,10}) = {1,5,7,9,11}.
Here rules are labelled as follows:
1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
2: if_minus#(true(),s(x),y) -> c_2()
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: log#(s(0())) -> c_6()
7: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
8: minus#(0(),y) -> c_8()
9: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
10: quot#(0(),s(y)) -> c_10()
11: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
if_minus#(true(),s(x),y) -> c_2()
le#(0(),y) -> c_3()
le#(s(x),0()) -> c_4()
log#(s(0())) -> c_6()
minus#(0(),y) -> c_8()
quot#(0(),s(y)) -> c_10()
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4
-->_1 minus#(0(),y) -> c_8():10
2:S:le#(s(x),s(y)) -> c_5(le#(x,y))
-->_1 le#(s(x),0()) -> c_4():8
-->_1 le#(0(),y) -> c_3():7
-->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2
3:S:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5
-->_2 quot#(0(),s(y)) -> c_10():11
-->_1 log#(s(0())) -> c_6():9
-->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):3
4:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_2 le#(s(x),0()) -> c_4():8
-->_1 if_minus#(true(),s(x),y) -> c_2():6
-->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
5:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(0(),s(y)) -> c_10():11
-->_2 minus#(0(),y) -> c_8():10
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5
-->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4
6:W:if_minus#(true(),s(x),y) -> c_2()
7:W:le#(0(),y) -> c_3()
8:W:le#(s(x),0()) -> c_4()
9:W:log#(s(0())) -> c_6()
10:W:minus#(0(),y) -> c_8()
11:W:quot#(0(),s(y)) -> c_10()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: log#(s(0())) -> c_6()
11: quot#(0(),s(y)) -> c_10()
10: minus#(0(),y) -> c_8()
7: le#(0(),y) -> c_3()
6: if_minus#(true(),s(x),y) -> c_2()
8: le#(s(x),0()) -> c_4()
** Step 1.b:5: Decompose WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ 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:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
Problem (S)
- Strict DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
*** Step 1.b:5.a:1: DecomposeDG WORST_CASE(?,O(n^4))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
and a lower component
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
Further, following extension rules are added to the lower component.
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
**** Step 1.b:5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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:
1: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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_7) = {1}
Following symbols are considered usable:
{if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if_minus) = [1] x2 + [0]
p(le) = [8] x2 + [0]
p(log) = [2] x1 + [1]
p(minus) = [1] x1 + [0]
p(quot) = [1] x1 + [0]
p(s) = [1] x1 + [2]
p(true) = [0]
p(if_minus#) = [1] x1 + [1] x2 + [1] x3 + [8]
p(le#) = [1] x2 + [1]
p(log#) = [4] x1 + [0]
p(minus#) = [1] x1 + [2] x2 + [1]
p(quot#) = [1] x1 + [1] x2 + [2]
p(c_1) = [1] x1 + [8]
p(c_2) = [8]
p(c_3) = [0]
p(c_4) = [1]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [1] x1 + [6]
p(c_8) = [1]
p(c_9) = [1]
p(c_10) = [1]
p(c_11) = [1] x1 + [1]
Following rules are strictly oriented:
log#(s(s(x))) = [4] x + [16]
> [4] x + [14]
= c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
Following rules are (at-least) weakly oriented:
if_minus(false(),s(x),y) = [1] x + [2]
>= [1] x + [2]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [2]
>= [0]
= 0()
minus(0(),y) = [0]
>= [0]
= 0()
minus(s(x),y) = [1] x + [2]
>= [1] x + [2]
= if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) = [0]
>= [0]
= 0()
quot(s(x),s(y)) = [1] x + [2]
>= [1] x + [2]
= s(quot(minus(x,y),s(y)))
***** Step 1.b:5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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:5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
-->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
***** Step 1.b:5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 1.b:5.a:1.b:1: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
and a lower component
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
Further, following extension rules are added to the lower component.
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
***** Step 1.b:5.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
The strictly oriented rules are moved into the weak component.
****** Step 1.b:5.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
NaturalMI {miDimension = 3, 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 (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_11) = {1}
Following symbols are considered usable:
{if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(false) = [0]
[0]
[0]
p(if_minus) = [1 0 0] [0]
[0 0 0] x2 + [0]
[0 0 0] [0]
p(le) = [0]
[0]
[0]
p(log) = [0]
[0]
[0]
p(minus) = [1 0 0] [0]
[0 0 1] x1 + [0]
[1 0 0] [0]
p(quot) = [1 0 0] [0 0 0] [0]
[0 1 0] x1 + [0 0 0] x2 + [1]
[0 0 0] [1 0 0] [1]
p(s) = [1 0 0] [1]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(true) = [0]
[0]
[0]
p(if_minus#) = [0]
[0]
[0]
p(le#) = [0]
[0]
[0]
p(log#) = [1 0 0] [0]
[0 0 0] x1 + [0]
[1 0 0] [1]
p(minus#) = [0]
[0]
[0]
p(quot#) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_1) = [0]
[0]
[0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [0]
[0]
[0]
p(c_4) = [0]
[0]
[0]
p(c_5) = [0]
[0]
[0]
p(c_6) = [0]
[0]
[0]
p(c_7) = [0]
[0]
[0]
p(c_8) = [0]
[0]
[0]
p(c_9) = [0]
[0]
[0]
p(c_10) = [0]
[0]
[0]
p(c_11) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
Following rules are strictly oriented:
quot#(s(x),s(y)) = [1 0 0] [1]
[0 0 0] x + [0]
[0 0 0] [0]
> [1 0 0] [0]
[0 0 0] x + [0]
[0 0 0] [0]
= c_11(quot#(minus(x,y),s(y)),minus#(x,y))
Following rules are (at-least) weakly oriented:
log#(s(s(x))) = [1 0 0] [2]
[0 0 0] x + [0]
[1 0 0] [3]
>= [1 0 0] [1]
[0 0 0] x + [0]
[1 0 0] [2]
= log#(s(quot(x,s(s(0())))))
log#(s(s(x))) = [1 0 0] [2]
[0 0 0] x + [0]
[1 0 0] [3]
>= [1 0 0] [0]
[0 0 0] x + [0]
[0 0 0] [0]
= quot#(x,s(s(0())))
if_minus(false(),s(x),y) = [1 0 0] [1]
[0 0 0] x + [0]
[0 0 0] [0]
>= [1 0 0] [1]
[0 0 0] x + [0]
[0 0 0] [0]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1 0 0] [1]
[0 0 0] x + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= 0()
minus(0(),y) = [0]
[0]
[0]
>= [0]
[0]
[0]
= 0()
minus(s(x),y) = [1 0 0] [1]
[0 0 0] x + [0]
[1 0 0] [1]
>= [1 0 0] [1]
[0 0 0] x + [0]
[0 0 0] [0]
= if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) = [0 0 0] [0]
[0 0 0] y + [1]
[1 0 0] [2]
>= [0]
[0]
[0]
= 0()
quot(s(x),s(y)) = [1 0 0] [0 0 0] [1]
[0 0 0] x + [0 0 0] y + [1]
[0 0 0] [1 0 0] [2]
>= [1 0 0] [1]
[0 0 0] x + [0]
[0 0 0] [0]
= s(quot(minus(x,y),s(y)))
****** Step 1.b:5.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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:5.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
-->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):2
-->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):1
2:W:log#(s(s(x))) -> quot#(x,s(s(0())))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):3
3:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
2: log#(s(s(x))) -> quot#(x,s(s(0())))
3: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
****** Step 1.b:5.a:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
***** Step 1.b:5.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ 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}}
+ Details:
We first use the processor NaturalMI {miDimension = 2, 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:5.a:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
NaturalMI {miDimension = 2, 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:
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_5) = {1},
uargs(c_9) = {1,2}
Following symbols are considered usable:
{if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(false) = [0]
[0]
p(if_minus) = [1 0] x2 + [0]
[0 1] [0]
p(le) = [0]
[0]
p(log) = [0 2] x1 + [0]
[2 2] [0]
p(minus) = [1 0] x1 + [0]
[0 1] [0]
p(quot) = [1 0] x1 + [0]
[0 1] [0]
p(s) = [1 2] x1 + [0]
[0 1] [1]
p(true) = [0]
[0]
p(if_minus#) = [1 0] x2 + [0 0] x3 + [0]
[0 3] [0 3] [0]
p(le#) = [0 1] x1 + [0]
[0 1] [0]
p(log#) = [1 0] x1 + [3]
[1 0] [3]
p(minus#) = [1 2] x1 + [0]
[0 1] [2]
p(quot#) = [1 0] x1 + [0]
[1 2] [1]
p(c_1) = [1 0] x1 + [0]
[0 0] [3]
p(c_2) = [0]
[0]
p(c_3) = [0]
[0]
p(c_4) = [0]
[0]
p(c_5) = [1 0] x1 + [0]
[0 0] [0]
p(c_6) = [0]
[0]
p(c_7) = [2 0] x1 + [0 0] x2 + [0]
[0 2] [0 2] [1]
p(c_8) = [0]
[0]
p(c_9) = [1 0] x1 + [2 0] x2 + [0]
[0 0] [0 1] [2]
p(c_10) = [0]
[1]
p(c_11) = [0 0] x2 + [0]
[0 1] [0]
Following rules are strictly oriented:
le#(s(x),s(y)) = [0 1] x + [1]
[0 1] [1]
> [0 1] x + [0]
[0 0] [0]
= c_5(le#(x,y))
Following rules are (at-least) weakly oriented:
if_minus#(false(),s(x),y) = [1 2] x + [0 0] y + [0]
[0 3] [0 3] [3]
>= [1 2] x + [0]
[0 0] [3]
= c_1(minus#(x,y))
log#(s(s(x))) = [1 4] x + [5]
[1 4] [5]
>= [1 2] x + [3]
[1 2] [3]
= log#(s(quot(x,s(s(0())))))
log#(s(s(x))) = [1 4] x + [5]
[1 4] [5]
>= [1 0] x + [0]
[1 2] [1]
= quot#(x,s(s(0())))
minus#(s(x),y) = [1 4] x + [2]
[0 1] [3]
>= [1 4] x + [2]
[0 1] [3]
= c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
quot#(s(x),s(y)) = [1 2] x + [0]
[1 4] [3]
>= [1 2] x + [0]
[0 1] [2]
= minus#(x,y)
quot#(s(x),s(y)) = [1 2] x + [0]
[1 4] [3]
>= [1 0] x + [0]
[1 2] [1]
= quot#(minus(x,y),s(y))
if_minus(false(),s(x),y) = [1 2] x + [0]
[0 1] [1]
>= [1 2] x + [0]
[0 1] [1]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1 2] x + [0]
[0 1] [1]
>= [0]
[0]
= 0()
minus(0(),y) = [0]
[0]
>= [0]
[0]
= 0()
minus(s(x),y) = [1 2] x + [0]
[0 1] [1]
>= [1 2] x + [0]
[0 1] [1]
= if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) = [0]
[0]
>= [0]
[0]
= 0()
quot(s(x),s(y)) = [1 2] x + [0]
[0 1] [1]
>= [1 2] x + [0]
[0 1] [1]
= s(quot(minus(x,y),s(y)))
****** Step 1.b:5.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak DPs:
le#(s(x),s(y)) -> c_5(le#(x,y))
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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:5.a:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak DPs:
le#(s(x),s(y)) -> c_5(le#(x,y))
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):3
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
3:W:le#(s(x),s(y)) -> c_5(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):3
4:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
-->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):5
-->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):4
5:W:log#(s(s(x))) -> quot#(x,s(s(0())))
-->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):7
-->_1 quot#(s(x),s(y)) -> minus#(x,y):6
6:W:quot#(s(x),s(y)) -> minus#(x,y)
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
7:W:quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
-->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):7
-->_1 quot#(s(x),s(y)) -> minus#(x,y):6
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: le#(s(x),s(y)) -> c_5(le#(x,y))
****** Step 1.b:5.a:1.b:1.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
4:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
-->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):5
-->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):4
5:W:log#(s(s(x))) -> quot#(x,s(s(0())))
-->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):7
-->_1 quot#(s(x),s(y)) -> minus#(x,y):6
6:W:quot#(s(x),s(y)) -> minus#(x,y)
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
7:W:quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
-->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):7
-->_1 quot#(s(x),s(y)) -> minus#(x,y):6
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
****** Step 1.b:5.a:1.b:1.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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:
1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
2: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
The strictly oriented rules are moved into the weak component.
******* Step 1.b:5.a:1.b:1.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
- Weak DPs:
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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_1) = {1},
uargs(c_9) = {1}
Following symbols are considered usable:
{if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if_minus) = [1] x2 + [0]
p(le) = [0]
p(log) = [1] x1 + [0]
p(minus) = [1] x1 + [0]
p(quot) = [1] x1 + [0]
p(s) = [1] x1 + [4]
p(true) = [0]
p(if_minus#) = [2] x2 + [2]
p(le#) = [1] x1 + [1]
p(log#) = [2] x1 + [9]
p(minus#) = [2] x1 + [6]
p(quot#) = [2] x1 + [8]
p(c_1) = [1] x1 + [3]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [8]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [1] x1 + [1] x2 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [2]
p(c_10) = [2]
p(c_11) = [1] x1 + [1] x2 + [1]
Following rules are strictly oriented:
if_minus#(false(),s(x),y) = [2] x + [10]
> [2] x + [9]
= c_1(minus#(x,y))
minus#(s(x),y) = [2] x + [14]
> [2] x + [12]
= c_9(if_minus#(le(s(x),y),s(x),y))
Following rules are (at-least) weakly oriented:
log#(s(s(x))) = [2] x + [25]
>= [2] x + [17]
= log#(s(quot(x,s(s(0())))))
log#(s(s(x))) = [2] x + [25]
>= [2] x + [8]
= quot#(x,s(s(0())))
quot#(s(x),s(y)) = [2] x + [16]
>= [2] x + [6]
= minus#(x,y)
quot#(s(x),s(y)) = [2] x + [16]
>= [2] x + [8]
= quot#(minus(x,y),s(y))
if_minus(false(),s(x),y) = [1] x + [4]
>= [1] x + [4]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [4]
>= [0]
= 0()
minus(0(),y) = [0]
>= [0]
= 0()
minus(s(x),y) = [1] x + [4]
>= [1] x + [4]
= if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) = [0]
>= [0]
= 0()
quot(s(x),s(y)) = [1] x + [4]
>= [1] x + [4]
= s(quot(minus(x,y),s(y)))
******* Step 1.b:5.a:1.b:1.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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:5.a:1.b:1.b:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
log#(s(s(x))) -> quot#(x,s(s(0())))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)):4
2:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
-->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):3
-->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):2
3:W:log#(s(s(x))) -> quot#(x,s(s(0())))
-->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):6
-->_1 quot#(s(x),s(y)) -> minus#(x,y):5
4:W:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
5:W:quot#(s(x),s(y)) -> minus#(x,y)
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)):4
6:W:quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
-->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):6
-->_1 quot#(s(x),s(y)) -> minus#(x,y):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
3: log#(s(s(x))) -> quot#(x,s(s(0())))
6: quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
5: quot#(s(x),s(y)) -> minus#(x,y)
1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
4: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
******* Step 1.b:5.a:1.b:1.b:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
le#(s(x),s(y)) -> c_5(le#(x,y))
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
2:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):5
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
3:W:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
-->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):5
4:W:le#(s(x),s(y)) -> c_5(le#(x,y))
-->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):4
5:W:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
-->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):4
-->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
3: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
4: le#(s(x),s(y)) -> c_5(le#(x,y))
*** Step 1.b:5.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
2:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
*** Step 1.b:5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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:
1: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
2: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:5.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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_7) = {1,2},
uargs(c_11) = {1}
Following symbols are considered usable:
{if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#}
TcT has computed the following interpretation:
p(0) = 0
p(false) = 0
p(if_minus) = x2
p(le) = 2
p(log) = 1 + x1^2
p(minus) = x1
p(quot) = x1
p(s) = 1 + x1
p(true) = 0
p(if_minus#) = 2*x2 + x2^2 + 2*x3^2
p(le#) = 0
p(log#) = 3 + x1^2
p(minus#) = 2*x1*x2 + 2*x2^2
p(quot#) = x1
p(c_1) = 1
p(c_2) = 0
p(c_3) = 0
p(c_4) = 0
p(c_5) = 0
p(c_6) = 0
p(c_7) = x1 + x2
p(c_8) = 1
p(c_9) = 0
p(c_10) = 0
p(c_11) = x1
Following rules are strictly oriented:
log#(s(s(x))) = 7 + 4*x + x^2
> 4 + 3*x + x^2
= c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) = 1 + x
> x
= c_11(quot#(minus(x,y),s(y)))
Following rules are (at-least) weakly oriented:
if_minus(false(),s(x),y) = 1 + x
>= 1 + x
= s(minus(x,y))
if_minus(true(),s(x),y) = 1 + x
>= 0
= 0()
minus(0(),y) = 0
>= 0
= 0()
minus(s(x),y) = 1 + x
>= 1 + x
= if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) = 0
>= 0
= 0()
quot(s(x),s(y)) = 1 + x
>= 1 + x
= s(quot(minus(x,y),s(y)))
**** Step 1.b:5.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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:5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):2
-->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
2:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
2: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
**** Step 1.b:5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
if_minus(false(),s(x),y) -> s(minus(x,y))
if_minus(true(),s(x),y) -> 0()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
minus(0(),y) -> 0()
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0
,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} 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^4))