WORST_CASE(Omega(n^1),O(n^4))
* 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: 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)
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:
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:3: 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)
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:
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:4: UsableRules 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)
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))
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))
** Step 1.b:5: 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))
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:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, 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: SimplifyRHS 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:
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()))))
-->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))))
*** Step 1.b:5.a:2: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_7(log#(s(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_7/1,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 any 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) = [1]
p(false) = [1]
p(if_minus) = [1] x2 + [0]
p(le) = [14]
p(log) = [1] x1 + [1]
p(minus) = [1] x1 + [0]
p(quot) = [1] x1 + [0]
p(s) = [1] x1 + [3]
p(true) = [0]
p(if_minus#) = [1] x1 + [1]
p(le#) = [1] x1 + [1] x2 + [1]
p(log#) = [3] x1 + [0]
p(minus#) = [4] x1 + [1] x2 + [0]
p(quot#) = [1] x1 + [1]
p(c_1) = [2] x1 + [1]
p(c_2) = [0]
p(c_3) = [2]
p(c_4) = [2]
p(c_5) = [1]
p(c_6) = [2]
p(c_7) = [1] x1 + [8]
p(c_8) = [2]
p(c_9) = [2] x1 + [1]
p(c_10) = [1]
p(c_11) = [1]
Following rules are strictly oriented:
log#(s(s(x))) = [3] x + [18]
> [3] x + [17]
= c_7(log#(s(quot(x,s(s(0()))))))
Following rules are (at-least) weakly oriented:
if_minus(false(),s(x),y) = [1] x + [3]
>= [1] x + [3]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [3]
>= [1]
= 0()
minus(0(),y) = [1]
>= [1]
= 0()
minus(s(x),y) = [1] x + [3]
>= [1] x + [3]
= if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) = [1]
>= [1]
= 0()
quot(s(x),s(y)) = [1] x + [3]
>= [1] x + [3]
= s(quot(minus(x,y),s(y)))
*** Step 1.b:5.a:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
log#(s(s(x))) -> c_7(log#(s(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_7/1,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.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))
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:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, 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.b:1.a:1: SimplifyRHS 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:
SimplifyRHS
+ Details:
Consider the dependency graph
1: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)):1
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)) -> c_11(quot#(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:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
**** Step 1.b:5.b:1.a:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(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_11/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0
,false,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(if_minus) = {1},
uargs(quot) = {1},
uargs(s) = {1},
uargs(log#) = {1},
uargs(quot#) = {1},
uargs(c_11) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if_minus) = [1] x1 + [1] x2 + [0]
p(le) = [0]
p(log) = [2] x1 + [1]
p(minus) = [1] x1 + [0]
p(quot) = [1] x1 + [1]
p(s) = [1] x1 + [2]
p(true) = [0]
p(if_minus#) = [2] x1 + [4] x3 + [1]
p(le#) = [2]
p(log#) = [1] x1 + [4]
p(minus#) = [1] x1 + [0]
p(quot#) = [1] x1 + [7]
p(c_1) = [2]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [2]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [1]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
Following rules are strictly oriented:
quot#(s(x),s(y)) = [1] x + [9]
> [1] x + [7]
= c_11(quot#(minus(x,y),s(y)))
Following rules are (at-least) weakly oriented:
log#(s(s(x))) = [1] x + [8]
>= [1] x + [7]
= log#(s(quot(x,s(s(0())))))
log#(s(s(x))) = [1] x + [8]
>= [1] x + [7]
= quot#(x,s(s(0())))
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()
le(0(),y) = [0]
>= [0]
= true()
le(s(x),0()) = [0]
>= [0]
= false()
le(s(x),s(y)) = [0]
>= [0]
= le(x,y)
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)) = [1]
>= [0]
= 0()
quot(s(x),s(y)) = [1] x + [3]
>= [1] x + [3]
= s(quot(minus(x,y),s(y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
**** Step 1.b:5.b:1.a:3: EmptyProcessor 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)))
- 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).
**** Step 1.b:5.b:1.b:1: DecomposeDG 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:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
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),le#(s(x),y))
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
and a lower component
le#(s(x),s(y)) -> c_5(le#(x,y))
Further, following extension rules are added to the lower component.
if_minus#(false(),s(x),y) -> 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) -> if_minus#(le(s(x),y),s(x),y)
minus#(s(x),y) -> le#(s(x),y)
quot#(s(x),s(y)) -> minus#(x,y)
quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
***** Step 1.b:5.b:1.b:1.a:1: 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
3:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
-->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):4
-->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):3
4: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
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),le#(s(x),y)):2
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
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.b:1.b:1.a:2: WeightGap 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:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(if_minus) = {1},
uargs(quot) = {1},
uargs(s) = {1},
uargs(if_minus#) = {1},
uargs(log#) = {1},
uargs(quot#) = {1},
uargs(c_1) = {1},
uargs(c_9) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if_minus) = [1] x1 + [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 + [0]
p(true) = [0]
p(if_minus#) = [1] x1 + [1] x3 + [4]
p(le#) = [1] x1 + [1] x2 + [4]
p(log#) = [1] x1 + [1]
p(minus#) = [1] x2 + [1]
p(quot#) = [1] x1 + [1] x2 + [1]
p(c_1) = [1] x1 + [1]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [2] x1 + [1] x2 + [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [1] x2 + [0]
Following rules are strictly oriented:
if_minus#(false(),s(x),y) = [1] y + [4]
> [1] y + [2]
= c_1(minus#(x,y))
Following rules are (at-least) weakly oriented:
log#(s(s(x))) = [1] x + [1]
>= [1] x + [1]
= log#(s(quot(x,s(s(0())))))
log#(s(s(x))) = [1] x + [1]
>= [1] x + [1]
= quot#(x,s(s(0())))
minus#(s(x),y) = [1] y + [1]
>= [1] y + [4]
= c_9(if_minus#(le(s(x),y),s(x),y))
quot#(s(x),s(y)) = [1] x + [1] y + [1]
>= [1] y + [1]
= minus#(x,y)
quot#(s(x),s(y)) = [1] x + [1] y + [1]
>= [1] x + [1] y + [1]
= quot#(minus(x,y),s(y))
if_minus(false(),s(x),y) = [1] x + [0]
>= [1] x + [0]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [0]
>= [0]
= 0()
le(0(),y) = [0]
>= [0]
= true()
le(s(x),0()) = [0]
>= [0]
= false()
le(s(x),s(y)) = [0]
>= [0]
= le(x,y)
minus(0(),y) = [0]
>= [0]
= 0()
minus(s(x),y) = [1] x + [0]
>= [1] x + [0]
= if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) = [0]
>= [0]
= 0()
quot(s(x),s(y)) = [1] x + [0]
>= [1] x + [0]
= s(quot(minus(x,y),s(y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
***** Step 1.b:5.b:1.b:1.a:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
- 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())))
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:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(if_minus) = {1},
uargs(quot) = {1},
uargs(s) = {1},
uargs(if_minus#) = {1},
uargs(log#) = {1},
uargs(quot#) = {1},
uargs(c_1) = {1},
uargs(c_9) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(false) = [0]
p(if_minus) = [1] x1 + [1] x2 + [0]
p(le) = [0]
p(log) = [1]
p(minus) = [1] x1 + [0]
p(quot) = [1] x1 + [1]
p(s) = [1] x1 + [1]
p(true) = [0]
p(if_minus#) = [1] x1 + [1] x2 + [1]
p(le#) = [1] x1 + [1] x2 + [0]
p(log#) = [1] x1 + [2]
p(minus#) = [1] x1 + [2]
p(quot#) = [1] x1 + [3]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [1] x1 + [1]
p(c_6) = [2]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [2]
Following rules are strictly oriented:
minus#(s(x),y) = [1] x + [3]
> [1] x + [2]
= c_9(if_minus#(le(s(x),y),s(x),y))
Following rules are (at-least) weakly oriented:
if_minus#(false(),s(x),y) = [1] x + [2]
>= [1] x + [2]
= c_1(minus#(x,y))
log#(s(s(x))) = [1] x + [4]
>= [1] x + [4]
= log#(s(quot(x,s(s(0())))))
log#(s(s(x))) = [1] x + [4]
>= [1] x + [3]
= quot#(x,s(s(0())))
quot#(s(x),s(y)) = [1] x + [4]
>= [1] x + [2]
= minus#(x,y)
quot#(s(x),s(y)) = [1] x + [4]
>= [1] x + [3]
= quot#(minus(x,y),s(y))
if_minus(false(),s(x),y) = [1] x + [1]
>= [1] x + [1]
= s(minus(x,y))
if_minus(true(),s(x),y) = [1] x + [1]
>= [0]
= 0()
le(0(),y) = [0]
>= [0]
= true()
le(s(x),0()) = [0]
>= [0]
= false()
le(s(x),s(y)) = [0]
>= [0]
= le(x,y)
minus(0(),y) = [0]
>= [0]
= 0()
minus(s(x),y) = [1] x + [1]
>= [1] x + [1]
= if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) = [1]
>= [0]
= 0()
quot(s(x),s(y)) = [1] x + [2]
>= [1] x + [2]
= s(quot(minus(x,y),s(y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
***** Step 1.b:5.b:1.b:1.a:4: EmptyProcessor 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:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
***** Step 1.b:5.b:1.b:1.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
le#(s(x),s(y)) -> c_5(le#(x,y))
- Weak DPs:
if_minus#(false(),s(x),y) -> 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) -> if_minus#(le(s(x),y),s(x),y)
minus#(s(x),y) -> le#(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_9/2,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:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(if_minus) = {1},
uargs(quot) = {1},
uargs(s) = {1},
uargs(if_minus#) = {1},
uargs(log#) = {1},
uargs(quot#) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(false) = [0]
p(if_minus) = [1] x1 + [1] x2 + [0]
p(le) = [0]
p(log) = [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 + [0]
p(le#) = [1] x1 + [1]
p(log#) = [1] x1 + [1]
p(minus#) = [1] x1 + [2]
p(quot#) = [1] x1 + [4]
p(c_1) = [1]
p(c_2) = [2]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [1]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1]
p(c_10) = [1]
p(c_11) = [1] x2 + [0]
Following rules are strictly oriented:
le#(s(x),s(y)) = [1] x + [3]
> [1] x + [2]
= c_5(le#(x,y))
Following rules are (at-least) weakly oriented:
if_minus#(false(),s(x),y) = [1] x + [2]
>= [1] x + [2]
= minus#(x,y)
log#(s(s(x))) = [1] x + [5]
>= [1] x + [3]
= log#(s(quot(x,s(s(0())))))
log#(s(s(x))) = [1] x + [5]
>= [1] x + [4]
= quot#(x,s(s(0())))
minus#(s(x),y) = [1] x + [4]
>= [1] x + [2]
= if_minus#(le(s(x),y),s(x),y)
minus#(s(x),y) = [1] x + [4]
>= [1] x + [3]
= le#(s(x),y)
quot#(s(x),s(y)) = [1] x + [6]
>= [1] x + [2]
= minus#(x,y)
quot#(s(x),s(y)) = [1] x + [6]
>= [1] x + [4]
= quot#(minus(x,y),s(y))
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]
>= [2]
= 0()
le(0(),y) = [0]
>= [0]
= true()
le(s(x),0()) = [0]
>= [0]
= false()
le(s(x),s(y)) = [0]
>= [0]
= le(x,y)
minus(0(),y) = [2]
>= [2]
= 0()
minus(s(x),y) = [1] x + [2]
>= [1] x + [2]
= if_minus(le(s(x),y),s(x),y)
quot(0(),s(y)) = [2]
>= [2]
= 0()
quot(s(x),s(y)) = [1] x + [2]
>= [1] x + [2]
= s(quot(minus(x,y),s(y)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
***** Step 1.b:5.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
if_minus#(false(),s(x),y) -> minus#(x,y)
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())))
minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
minus#(s(x),y) -> le#(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_9/2,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).
WORST_CASE(Omega(n^1),O(n^4))