* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
copy(0(),y,z) -> f(z)
copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
copy(0(),y,z) -> f(z)
copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
copy(x,y,z){x -> s(x)} =
copy(s(x),y,z) ->^+ copy(x,y,cons(f(y),z))
= C[copy(x,y,cons(f(y),z)) = copy(x,y,z){z -> cons(f(y),z)}]
** Step 1.b:1: InnermostRuleRemoval WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
copy(0(),y,z) -> f(z)
copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
+ Applied Processor:
InnermostRuleRemoval
+ Details:
Arguments of following rules are not normal-forms.
f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
All above mentioned rules can be savely removed.
** Step 1.b:2: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
copy(0(),y,z) -> f(z)
copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
copy#(0(),y,z) -> c_1(f#(z))
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
f#(cons(nil(),y)) -> c_3()
Weak DPs
and mark the set of starting terms.
** Step 1.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
copy#(0(),y,z) -> c_1(f#(z))
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
f#(cons(nil(),y)) -> c_3()
- Weak TRS:
copy(0(),y,z) -> f(z)
copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/2,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f(cons(nil(),y)) -> y
copy#(0(),y,z) -> c_1(f#(z))
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
f#(cons(nil(),y)) -> c_3()
** Step 1.b:4: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
copy#(0(),y,z) -> c_1(f#(z))
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
f#(cons(nil(),y)) -> c_3()
- Weak TRS:
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/2,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{3}
by application of
Pre({3}) = {1,2}.
Here rules are labelled as follows:
1: copy#(0(),y,z) -> c_1(f#(z))
2: copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
3: f#(cons(nil(),y)) -> c_3()
** Step 1.b:5: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
copy#(0(),y,z) -> c_1(f#(z))
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
- Weak DPs:
f#(cons(nil(),y)) -> c_3()
- Weak TRS:
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/2,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {2}.
Here rules are labelled as follows:
1: copy#(0(),y,z) -> c_1(f#(z))
2: copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
3: f#(cons(nil(),y)) -> c_3()
** Step 1.b:6: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
- Weak DPs:
copy#(0(),y,z) -> c_1(f#(z))
f#(cons(nil(),y)) -> c_3()
- Weak TRS:
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/2,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
-->_1 copy#(0(),y,z) -> c_1(f#(z)):2
-->_2 f#(cons(nil(),y)) -> c_3():3
-->_1 copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y)):1
2:W:copy#(0(),y,z) -> c_1(f#(z))
-->_1 f#(cons(nil(),y)) -> c_3():3
3:W:f#(cons(nil(),y)) -> c_3()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: copy#(0(),y,z) -> c_1(f#(z))
3: f#(cons(nil(),y)) -> c_3()
** Step 1.b:7: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
- Weak TRS:
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/2,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
-->_1 copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
** Step 1.b:8: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
- Weak TRS:
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,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: copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
The strictly oriented rules are moved into the weak component.
*** Step 1.b:8.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
- Weak TRS:
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,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_2) = {1}
Following symbols are considered usable:
{copy#,f#}
TcT has computed the following interpretation:
p(0) = [1]
p(cons) = [1] x1 + [2]
p(copy) = [1] x1 + [1]
p(f) = [2] x1 + [0]
p(n) = [1]
p(nil) = [6]
p(s) = [1] x1 + [2]
p(copy#) = [8] x1 + [1] x2 + [0]
p(f#) = [1] x1 + [1]
p(c_1) = [2]
p(c_2) = [1] x1 + [12]
p(c_3) = [1]
Following rules are strictly oriented:
copy#(s(x),y,z) = [8] x + [1] y + [16]
> [8] x + [1] y + [12]
= c_2(copy#(x,y,cons(f(y),z)))
Following rules are (at-least) weakly oriented:
*** Step 1.b:8.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
- Weak TRS:
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 1.b:8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
- Weak TRS:
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
-->_1 copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
*** Step 1.b:8.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/1,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))