* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(0(),y,0(),u) -> true()
f(0(),y,s(z),u) -> false()
f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
perfectp(0()) -> false()
perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
- Signature:
{f/4,perfectp/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,perfectp} and constructors {0,false,if,le,minus,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
f#(0(),y,0(),u) -> c_1()
f#(0(),y,s(z),u) -> c_2()
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
perfectp#(0()) -> c_5()
perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(0(),y,0(),u) -> c_1()
f#(0(),y,s(z),u) -> c_2()
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
perfectp#(0()) -> c_5()
perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
- Strict TRS:
f(0(),y,0(),u) -> true()
f(0(),y,s(z),u) -> false()
f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
perfectp(0()) -> false()
perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(0(),y,0(),u) -> c_1()
f#(0(),y,s(z),u) -> c_2()
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
perfectp#(0()) -> c_5()
perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(0(),y,0(),u) -> c_1()
f#(0(),y,s(z),u) -> c_2()
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
perfectp#(0()) -> c_5()
perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,5}
by application of
Pre({1,2,5}) = {4,6}.
Here rules are labelled as follows:
1: f#(0(),y,0(),u) -> c_1()
2: f#(0(),y,s(z),u) -> c_2()
3: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
4: f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
5: perfectp#(0()) -> c_5()
6: perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
- Weak DPs:
f#(0(),y,0(),u) -> c_1()
f#(0(),y,s(z),u) -> c_2()
perfectp#(0()) -> c_5()
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
-->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
-->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
2:S:f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
-->_2 f#(0(),y,s(z),u) -> c_2():5
-->_2 f#(0(),y,0(),u) -> c_1():4
-->_2 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
-->_2 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
-->_1 f#(0(),y,s(z),u) -> c_2():5
-->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
4:W:f#(0(),y,0(),u) -> c_1()
5:W:f#(0(),y,s(z),u) -> c_2()
6:W:perfectp#(0()) -> c_5()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: perfectp#(0()) -> c_5()
4: f#(0(),y,0(),u) -> c_1()
5: f#(0(),y,s(z),u) -> c_2()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
-->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
-->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
2:S:f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
-->_2 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
-->_2 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
-->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
* Step 6: RemoveHeads WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
-->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2
-->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
2:S:f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
-->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2
-->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
-->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(3,perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))))]
* Step 7: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
The strictly oriented rules are moved into the weak component.
** Step 7.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,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_3) = {1},
uargs(c_4) = {1}
Following symbols are considered usable:
{f#,perfectp#}
TcT has computed the following interpretation:
p(0) = [4]
p(f) = [1] x1 + [1] x3 + [0]
p(false) = [1]
p(if) = [1] x1 + [1] x3 + [4]
p(le) = [1] x2 + [1]
p(minus) = [1]
p(perfectp) = [4]
p(s) = [1] x1 + [4]
p(true) = [4]
p(f#) = [2] x1 + [1] x3 + [2] x4 + [0]
p(perfectp#) = [1] x1 + [0]
p(c_1) = [0]
p(c_2) = [1]
p(c_3) = [1] x1 + [7]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [1]
Following rules are strictly oriented:
f#(s(x),s(y),z,u) = [2] u + [2] x + [1] z + [8]
> [2] u + [2] x + [1] z + [0]
= c_4(f#(x,u,z,u))
Following rules are (at-least) weakly oriented:
f#(s(x),0(),z,u) = [2] u + [2] x + [1] z + [8]
>= [2] u + [2] x + [8]
= c_3(f#(x,u,minus(z,s(x)),u))
** Step 7.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
- Weak DPs:
f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
** Step 7.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
- Weak DPs:
f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,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: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
The strictly oriented rules are moved into the weak component.
*** Step 7.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
- Weak DPs:
f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,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_3) = {1},
uargs(c_4) = {1}
Following symbols are considered usable:
{f#,perfectp#}
TcT has computed the following interpretation:
p(0) = [4]
p(f) = [1] x2 + [1] x3 + [1] x4 + [2]
p(false) = [2]
p(if) = [1] x1 + [1]
p(le) = [0]
p(minus) = [1] x1 + [3]
p(perfectp) = [1] x1 + [0]
p(s) = [1] x1 + [14]
p(true) = [0]
p(f#) = [1] x1 + [2] x3 + [4] x4 + [2]
p(perfectp#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [8]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
Following rules are strictly oriented:
f#(s(x),0(),z,u) = [4] u + [1] x + [2] z + [16]
> [4] u + [1] x + [2] z + [8]
= c_3(f#(x,u,minus(z,s(x)),u))
Following rules are (at-least) weakly oriented:
f#(s(x),s(y),z,u) = [4] u + [1] x + [2] z + [16]
>= [4] u + [1] x + [2] z + [10]
= c_4(f#(x,u,z,u))
*** Step 7.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 7.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
-->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2
-->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
2:W:f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
-->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2
-->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
2: f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
*** Step 7.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))