* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {log,minus,pred,quot} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
log#(s(0())) -> c_1()
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(x,0()) -> c_3()
minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_5()
quot#(0(),s(y)) -> c_6()
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(0())) -> c_1()
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(x,0()) -> c_3()
minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_5()
quot#(0(),s(y)) -> c_6()
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
log#(s(0())) -> c_1()
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(x,0()) -> c_3()
minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_5()
quot#(0(),s(y)) -> c_6()
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(0())) -> c_1()
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(x,0()) -> c_3()
minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
pred#(s(x)) -> c_5()
quot#(0(),s(y)) -> c_6()
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,5,6}
by application of
Pre({1,3,5,6}) = {2,4,7}.
Here rules are labelled as follows:
1: log#(s(0())) -> c_1()
2: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
3: minus#(x,0()) -> c_3()
4: minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
5: pred#(s(x)) -> c_5()
6: quot#(0(),s(y)) -> c_6()
7: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak DPs:
log#(s(0())) -> c_1()
minus#(x,0()) -> c_3()
pred#(s(x)) -> c_5()
quot#(0(),s(y)) -> c_6()
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
-->_2 quot#(0(),s(y)) -> c_6():7
-->_1 log#(s(0())) -> c_1():4
-->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
2:S:minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
-->_1 pred#(s(x)) -> c_5():6
-->_2 minus#(x,0()) -> c_3():5
-->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
3:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(0(),s(y)) -> c_6():7
-->_2 minus#(x,0()) -> c_3():5
-->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
-->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
4:W:log#(s(0())) -> c_1()
5:W:minus#(x,0()) -> c_3()
6:W:pred#(s(x)) -> c_5()
7:W:quot#(0(),s(y)) -> c_6()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: log#(s(0())) -> c_1()
6: pred#(s(x)) -> c_5()
5: minus#(x,0()) -> c_3()
7: quot#(0(),s(y)) -> c_6()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
2:S:minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y))
-->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
3:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
-->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
minus#(x,s(y)) -> c_4(minus#(x,y))
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(x,s(y)) -> c_4(minus#(x,y))
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: minus#(x,s(y)) -> c_4(minus#(x,y))
3: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus#(x,s(y)) -> c_4(minus#(x,y))
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_2) = {1,2},
uargs(c_4) = {1},
uargs(c_7) = {1,2}
Following symbols are considered usable:
{minus,pred,quot,log#,minus#,pred#,quot#}
TcT has computed the following interpretation:
p(0) = 0
p(log) = 2 + 2*x1^2
p(minus) = x1
p(pred) = x1
p(quot) = x1
p(s) = 1 + x1
p(log#) = x1 + x1^2
p(minus#) = x2
p(pred#) = 1
p(quot#) = x1*x2 + 2*x2
p(c_1) = 0
p(c_2) = x1 + x2
p(c_3) = 0
p(c_4) = x1
p(c_5) = 0
p(c_6) = 0
p(c_7) = x1 + x2
Following rules are strictly oriented:
minus#(x,s(y)) = 1 + y
> y
= c_4(minus#(x,y))
quot#(s(x),s(y)) = 3 + x + x*y + 3*y
> 2 + x + x*y + 3*y
= c_7(quot#(minus(x,y),s(y)),minus#(x,y))
Following rules are (at-least) weakly oriented:
log#(s(s(x))) = 6 + 5*x + x^2
>= 6 + 5*x + x^2
= c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
minus(x,0()) = x
>= x
= x
minus(x,s(y)) = x
>= x
= pred(minus(x,y))
pred(s(x)) = 1 + x
>= x
= x
quot(0(),s(y)) = 0
>= 0
= 0()
quot(s(x),s(y)) = 1 + x
>= 1 + x
= s(quot(minus(x,y),s(y)))
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
- Weak DPs:
minus#(x,s(y)) -> c_4(minus#(x,y))
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
- Weak DPs:
minus#(x,s(y)) -> c_4(minus#(x,y))
quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
-->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
-->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1
2:W:minus#(x,s(y)) -> c_4(minus#(x,y))
-->_1 minus#(x,s(y)) -> c_4(minus#(x,y)):2
3:W:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3
-->_2 minus#(x,s(y)) -> c_4(minus#(x,y)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y))
2: minus#(x,s(y)) -> c_4(minus#(x,y))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0()))))
-->_1 log#(s(s(x))) -> c_2(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_2(log#(s(quot(x,s(s(0()))))))
** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ 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_2(log#(s(quot(x,s(s(0()))))))
The strictly oriented rules are moved into the weak component.
*** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1}
Following symbols are considered usable:
{minus,pred,quot,log#,minus#,pred#,quot#}
TcT has computed the following interpretation:
p(0) = [0]
p(log) = [0]
p(minus) = [1] x1 + [0]
p(pred) = [1] x1 + [0]
p(quot) = [1] x1 + [0]
p(s) = [1] x1 + [8]
p(log#) = [1] x1 + [8]
p(minus#) = [1] x1 + [0]
p(pred#) = [2] x1 + [1]
p(quot#) = [1] x1 + [1] x2 + [0]
p(c_1) = [1]
p(c_2) = [1] x1 + [7]
p(c_3) = [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1]
p(c_6) = [0]
p(c_7) = [1] x1 + [1] x2 + [1]
Following rules are strictly oriented:
log#(s(s(x))) = [1] x + [24]
> [1] x + [23]
= c_2(log#(s(quot(x,s(s(0()))))))
Following rules are (at-least) weakly oriented:
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(x,s(y)) = [1] x + [0]
>= [1] x + [0]
= pred(minus(x,y))
pred(s(x)) = [1] x + [8]
>= [1] x + [0]
= x
quot(0(),s(y)) = [0]
>= [0]
= 0()
quot(s(x),s(y)) = [1] x + [8]
>= [1] x + [8]
= s(quot(minus(x,y),s(y)))
*** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))))
-->_1 log#(s(s(x))) -> c_2(log#(s(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_2(log#(s(quot(x,s(s(0()))))))
*** Step 6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
minus(x,0()) -> x
minus(x,s(y)) -> pred(minus(x,y))
pred(s(x)) -> x
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
- Signature:
{log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0
,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))