* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
+ Considered Problem:
- Strict TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
goal(xs,ys) -> mul0(xs,ys)
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2} / {Cons/2,Nil/0,S/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0,goal,mul0} and constructors {Cons,Nil,S}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
goal(xs,ys) -> mul0(xs,ys)
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2} / {Cons/2,Nil/0,S/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0,goal,mul0} and constructors {Cons,Nil,S}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
add0(y,z){y -> Cons(x,y)} =
add0(Cons(x,y),z) ->^+ add0(y,Cons(S(),z))
= C[add0(y,Cons(S(),z)) = add0(y,z){z -> Cons(S(),z)}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
goal(xs,ys) -> mul0(xs,ys)
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2} / {Cons/2,Nil/0,S/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0,goal,mul0} and constructors {Cons,Nil,S}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
add0#(Nil(),y) -> c_2()
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
mul0#(Nil(),y) -> c_5()
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
add0#(Nil(),y) -> c_2()
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
mul0#(Nil(),y) -> c_5()
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
goal(xs,ys) -> mul0(xs,ys)
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
add0#(Nil(),y) -> c_2()
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
mul0#(Nil(),y) -> c_5()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
add0#(Nil(),y) -> c_2()
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
mul0#(Nil(),y) -> c_5()
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,5}
by application of
Pre({2,5}) = {1,3,4}.
Here rules are labelled as follows:
1: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
2: add0#(Nil(),y) -> c_2()
3: goal#(xs,ys) -> c_3(mul0#(xs,ys))
4: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
5: mul0#(Nil(),y) -> c_5()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak DPs:
add0#(Nil(),y) -> c_2()
mul0#(Nil(),y) -> c_5()
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
-->_1 add0#(Nil(),y) -> c_2():4
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
2:S:goal#(xs,ys) -> c_3(mul0#(xs,ys))
-->_1 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3
-->_1 mul0#(Nil(),y) -> c_5():5
3:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_2 mul0#(Nil(),y) -> c_5():5
-->_1 add0#(Nil(),y) -> c_2():4
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
4:W:add0#(Nil(),y) -> c_2()
5:W:mul0#(Nil(),y) -> c_5()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: mul0#(Nil(),y) -> c_5()
4: add0#(Nil(),y) -> c_2()
** Step 1.b:5: RemoveHeads WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
goal#(xs,ys) -> c_3(mul0#(xs,ys))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
2:S:goal#(xs,ys) -> c_3(mul0#(xs,ys))
-->_1 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3
3:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):3
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
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).
[(2,goal#(xs,ys) -> c_3(mul0#(xs,ys)))]
** Step 1.b:6: Decompose WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ 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:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
- Weak DPs:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
Problem (S)
- Strict DPs:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
*** Step 1.b:6.a:1: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
- Weak DPs:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ 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
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
and a lower component
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
Further, following extension rules are added to the lower component.
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
**** Step 1.b:6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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_4) = {1,2}
Following symbols are considered usable:
{add0#,goal#,mul0#}
TcT has computed the following interpretation:
p(Cons) = [1] x2 + [4]
p(Nil) = [2]
p(S) = [8]
p(add0) = [1] x2 + [9]
p(goal) = [4] x1 + [0]
p(mul0) = [1] x1 + [14]
p(add0#) = [1]
p(goal#) = [1] x1 + [0]
p(mul0#) = [4] x1 + [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [1]
p(c_3) = [8] x1 + [0]
p(c_4) = [1] x1 + [1] x2 + [11]
p(c_5) = [0]
Following rules are strictly oriented:
mul0#(Cons(x,xs),y) = [4] xs + [16]
> [4] xs + [12]
= c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
Following rules are (at-least) weakly oriented:
***** Step 1.b:6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
***** Step 1.b:6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 1.b:6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
- Weak DPs:
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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:
1: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
- Weak DPs:
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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_1) = {1}
Following symbols are considered usable:
{add0,mul0,add0#,goal#,mul0#}
TcT has computed the following interpretation:
p(Cons) = 1 + x2
p(Nil) = 0
p(S) = 1
p(add0) = x1 + x2
p(goal) = 1 + 4*x1 + x1*x2 + x1^2 + x2^2
p(mul0) = x1*x2 + 2*x1^2
p(add0#) = 3 + 2*x1
p(goal#) = 1
p(mul0#) = 2*x1 + 3*x1*x2 + 6*x1^2 + x2
p(c_1) = x1
p(c_2) = 0
p(c_3) = 0
p(c_4) = 1
p(c_5) = 1
Following rules are strictly oriented:
add0#(Cons(x,xs),y) = 5 + 2*xs
> 3 + 2*xs
= c_1(add0#(xs,Cons(S(),y)))
Following rules are (at-least) weakly oriented:
mul0#(Cons(x,xs),y) = 8 + 14*xs + 3*xs*y + 6*xs^2 + 4*y
>= 3 + 2*xs*y + 4*xs^2
= add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) = 8 + 14*xs + 3*xs*y + 6*xs^2 + 4*y
>= 2*xs + 3*xs*y + 6*xs^2 + y
= mul0#(xs,y)
add0(Cons(x,xs),y) = 1 + xs + y
>= 1 + xs + y
= add0(xs,Cons(S(),y))
add0(Nil(),y) = y
>= y
= y
mul0(Cons(x,xs),y) = 2 + 4*xs + xs*y + 2*xs^2 + y
>= xs*y + 2*xs^2 + y
= add0(mul0(xs,y),y)
mul0(Nil(),y) = 0
>= 0
= Nil()
***** Step 1.b:6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
mul0#(Cons(x,xs),y) -> mul0#(xs,y)
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
2:W:mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):1
3:W:mul0#(Cons(x,xs),y) -> mul0#(xs,y)
-->_1 mul0#(Cons(x,xs),y) -> mul0#(xs,y):3
-->_1 mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: mul0#(Cons(x,xs),y) -> mul0#(xs,y)
2: mul0#(Cons(x,xs),y) -> add0#(mul0(xs,y),y)
1: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
***** Step 1.b:6.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak DPs:
add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):2
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):1
2:W:add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
-->_1 add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y))):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: add0#(Cons(x,xs),y) -> c_1(add0#(xs,Cons(S(),y)))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y))
-->_2 mul0#(Cons(x,xs),y) -> c_4(add0#(mul0(xs,y),y),mul0#(xs,y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
*** Step 1.b:6.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
- Weak TRS:
add0(Cons(x,xs),y) -> add0(xs,Cons(S(),y))
add0(Nil(),y) -> y
mul0(Cons(x,xs),y) -> add0(mul0(xs,y),y)
mul0(Nil(),y) -> Nil()
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
*** Step 1.b:6.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,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_4) = {1}
Following symbols are considered usable:
{add0#,goal#,mul0#}
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [1] x2 + [8]
p(Nil) = [0]
p(S) = [0]
p(add0) = [1] x1 + [2] x2 + [0]
p(goal) = [1] x1 + [1] x2 + [0]
p(mul0) = [1] x1 + [1] x2 + [0]
p(add0#) = [0]
p(goal#) = [2]
p(mul0#) = [2] x1 + [2] x2 + [15]
p(c_1) = [2] x1 + [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [1]
p(c_4) = [1] x1 + [15]
p(c_5) = [1]
Following rules are strictly oriented:
mul0#(Cons(x,xs),y) = [2] x + [2] xs + [2] y + [31]
> [2] xs + [2] y + [30]
= c_4(mul0#(xs,y))
Following rules are (at-least) weakly oriented:
**** Step 1.b:6.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
-->_1 mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: mul0#(Cons(x,xs),y) -> c_4(mul0#(xs,y))
**** Step 1.b:6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{add0/2,goal/2,mul0/2,add0#/2,goal#/2,mul0#/2} / {Cons/2,Nil/0,S/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mul0#} and constructors {Cons,Nil,S}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^3))