* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
- Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
- Obligation:
runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak dependency pairs:
Strict DPs
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3(y)
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3(y)
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3(y)
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3(y)
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ 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(.) = {2},
uargs(cond) = {2},
uargs(cond#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(.) = [1] x1 + [1] x2 + [0]
p(=) = [1]
p(admit) = [4] x1 + [4] x2 + [2]
p(carry) = [1] x1 + [0]
p(cond) = [2] x1 + [1] x2 + [0]
p(nil) = [2]
p(sum) = [1] x1 + [1] x2 + [0]
p(true) = [5]
p(w) = [2]
p(admit#) = [6] x1 + [4] x2 + [0]
p(cond#) = [1] x1 + [1] x2 + [3]
p(c_1) = [1] x1 + [4]
p(c_2) = [0]
p(c_3) = [1] x1 + [1]
Following rules are strictly oriented:
admit#(x,nil()) = [6] x + [8]
> [0]
= c_2()
cond#(true(),y) = [1] y + [8]
> [1] y + [1]
= c_3(y)
admit(x,.(u,.(v,.(w(),z)))) = [4] u + [4] v + [4] x + [4] z + [10]
> [1] u + [1] v + [4] x + [4] z + [6]
= cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) = [4] x + [10]
> [2]
= nil()
Following rules are (at-least) weakly oriented:
admit#(x,.(u,.(v,.(w(),z)))) = [4] u + [4] v + [6] x + [4] z + [8]
>= [1] u + [1] v + [4] x + [4] z + [12]
= c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
- Weak DPs:
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3(y)
- Weak TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
2:W:admit#(x,nil()) -> c_2()
3:W:cond#(true(),y) -> c_3(y)
-->_1 cond#(true(),y) -> c_3(y):3
-->_1 admit#(x,nil()) -> c_2():2
-->_1 admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w())
,.(u,.(v,.(w(),admit(carry(x,u,v),z)))))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: admit#(x,nil()) -> c_2()
* Step 5: SimplifyRHS WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
- Weak DPs:
cond#(true(),y) -> c_3(y)
- Weak TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
3:W:cond#(true(),y) -> c_3(y)
-->_1 cond#(true(),y) -> c_3(y):3
-->_1 admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w())
,.(u,.(v,.(w(),admit(carry(x,u,v),z)))))):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
* Step 6: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
- Weak DPs:
cond#(true(),y) -> c_3(y)
- Weak TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/0,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
cond#(true(),y) -> c_3(y)
* Step 7: PredecessorEstimationCP WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
- Weak DPs:
cond#(true(),y) -> c_3(y)
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/0,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
The strictly oriented rules are moved into the weak component.
** Step 7.a:1: NaturalMI WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
- Weak DPs:
cond#(true(),y) -> c_3(y)
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/0,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
none
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(.) = [0]
p(=) = [0]
p(admit) = [0]
p(carry) = [0]
p(cond) = [0]
p(nil) = [0]
p(sum) = [0]
p(true) = [0]
p(w) = [0]
p(admit#) = [2]
p(cond#) = [1] x2 + [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
Following rules are strictly oriented:
admit#(x,.(u,.(v,.(w(),z)))) = [2]
> [0]
= c_1()
Following rules are (at-least) weakly oriented:
cond#(true(),y) = [1] y + [0]
>= [1] y + [0]
= c_3(y)
** Step 7.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
cond#(true(),y) -> c_3(y)
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/0,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
cond#(true(),y) -> c_3(y)
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/0,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
2:W:cond#(true(),y) -> c_3(y)
-->_1 cond#(true(),y) -> c_3(y):2
-->_1 admit#(x,.(u,.(v,.(w(),z)))) -> c_1():1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: cond#(true(),y) -> c_3(y)
1: admit#(x,.(u,.(v,.(w(),z)))) -> c_1()
** Step 7.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/0,c_2/0,c_3/1}
- Obligation:
runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))