WORST_CASE(?,O(n^2))
* Step 1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
purge(nil()) -> nil()
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
- Signature:
{eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s
,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(add) = {2},
uargs(ifrm) = {1},
uargs(purge) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x1 + [1] x2 + [0]
p(eq) = [0]
p(false) = [5]
p(ifrm) = [1] x1 + [1] x3 + [0]
p(nil) = [0]
p(purge) = [1] x1 + [0]
p(rm) = [1] x2 + [1]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
ifrm(false(),N,add(M,X)) = [1] M + [1] X + [5]
> [1] M + [1] X + [1]
= add(M,rm(N,X))
rm(N,add(M,X)) = [1] M + [1] X + [1]
> [1] M + [1] X + [0]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [1]
> [0]
= nil()
Following rules are (at-least) weakly oriented:
eq(0(),0()) = [0]
>= [0]
= true()
eq(0(),s(X)) = [0]
>= [5]
= false()
eq(s(X),0()) = [0]
>= [5]
= false()
eq(s(X),s(Y)) = [0]
>= [0]
= eq(X,Y)
ifrm(true(),N,add(M,X)) = [1] M + [1] X + [0]
>= [1] X + [1]
= rm(N,X)
purge(add(N,X)) = [1] N + [1] X + [0]
>= [1] N + [1] X + [1]
= add(N,purge(rm(N,X)))
purge(nil()) = [0]
>= [0]
= nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(true(),N,add(M,X)) -> rm(N,X)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
purge(nil()) -> nil()
- Weak TRS:
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
- Signature:
{eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s
,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(add) = {2},
uargs(ifrm) = {1},
uargs(purge) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(add) = [1] x2 + [0]
p(eq) = [4]
p(false) = [8]
p(ifrm) = [1] x1 + [0]
p(nil) = [4]
p(purge) = [1] x1 + [10]
p(rm) = [4]
p(s) = [1] x1 + [8]
p(true) = [8]
Following rules are strictly oriented:
ifrm(true(),N,add(M,X)) = [8]
> [4]
= rm(N,X)
purge(nil()) = [14]
> [4]
= nil()
Following rules are (at-least) weakly oriented:
eq(0(),0()) = [4]
>= [8]
= true()
eq(0(),s(X)) = [4]
>= [8]
= false()
eq(s(X),0()) = [4]
>= [8]
= false()
eq(s(X),s(Y)) = [4]
>= [4]
= eq(X,Y)
ifrm(false(),N,add(M,X)) = [8]
>= [4]
= add(M,rm(N,X))
purge(add(N,X)) = [1] X + [10]
>= [14]
= add(N,purge(rm(N,X)))
rm(N,add(M,X)) = [4]
>= [4]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [4]
>= [4]
= nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
- Weak TRS:
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
purge(nil()) -> nil()
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
- Signature:
{eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s
,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(add) = {2},
uargs(ifrm) = {1},
uargs(purge) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x2 + [4]
p(eq) = [2]
p(false) = [2]
p(ifrm) = [1] x1 + [1] x3 + [0]
p(nil) = [0]
p(purge) = [1] x1 + [1]
p(rm) = [1] x2 + [2]
p(s) = [0]
p(true) = [0]
Following rules are strictly oriented:
eq(0(),0()) = [2]
> [0]
= true()
Following rules are (at-least) weakly oriented:
eq(0(),s(X)) = [2]
>= [2]
= false()
eq(s(X),0()) = [2]
>= [2]
= false()
eq(s(X),s(Y)) = [2]
>= [2]
= eq(X,Y)
ifrm(false(),N,add(M,X)) = [1] X + [6]
>= [1] X + [6]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [1] X + [4]
>= [1] X + [2]
= rm(N,X)
purge(add(N,X)) = [1] X + [5]
>= [1] X + [7]
= add(N,purge(rm(N,X)))
purge(nil()) = [1]
>= [0]
= nil()
rm(N,add(M,X)) = [1] X + [6]
>= [1] X + [6]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [2]
>= [0]
= nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
- Weak TRS:
eq(0(),0()) -> true()
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
purge(nil()) -> nil()
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
- Signature:
{eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s
,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(add) = {2},
uargs(ifrm) = {1},
uargs(purge) = {1}
Following symbols are considered usable:
{eq,ifrm,purge,rm}
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x2 + [8]
p(eq) = [0]
p(false) = [0]
p(ifrm) = [8] x1 + [1] x3 + [0]
p(nil) = [2]
p(purge) = [2] x1 + [0]
p(rm) = [1] x2 + [0]
p(s) = [0]
p(true) = [0]
Following rules are strictly oriented:
purge(add(N,X)) = [2] X + [16]
> [2] X + [8]
= add(N,purge(rm(N,X)))
Following rules are (at-least) weakly oriented:
eq(0(),0()) = [0]
>= [0]
= true()
eq(0(),s(X)) = [0]
>= [0]
= false()
eq(s(X),0()) = [0]
>= [0]
= false()
eq(s(X),s(Y)) = [0]
>= [0]
= eq(X,Y)
ifrm(false(),N,add(M,X)) = [1] X + [8]
>= [1] X + [8]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [1] X + [8]
>= [1] X + [0]
= rm(N,X)
purge(nil()) = [4]
>= [2]
= nil()
rm(N,add(M,X)) = [1] X + [8]
>= [1] X + [8]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [2]
>= [2]
= nil()
* Step 5: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
- Weak TRS:
eq(0(),0()) -> true()
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
purge(nil()) -> nil()
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
- Signature:
{eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s
,true}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(add) = {2},
uargs(ifrm) = {1},
uargs(purge) = {1}
Following symbols are considered usable:
{eq,ifrm,purge,rm}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(add) = [1 0 0] [1 0 2] [0]
[0 0 0] x1 + [0 0 2] x2 + [0]
[0 0 1] [0 0 1] [2]
p(eq) = [0 0 0] [1]
[0 0 0] x2 + [0]
[0 0 1] [1]
p(false) = [0]
[0]
[0]
p(ifrm) = [1 0 0] [0 0 0] [1 0 1] [0]
[2 0 2] x1 + [0 0 1] x2 + [0 1 0] x3 + [0]
[0 0 0] [0 0 0] [0 0 1] [0]
p(nil) = [0]
[0]
[0]
p(purge) = [2 1 1] [0]
[1 1 0] x1 + [1]
[0 0 1] [0]
p(rm) = [0 0 0] [1 0 1] [1]
[0 0 1] x1 + [0 0 2] x2 + [0]
[0 0 0] [0 0 1] [0]
p(s) = [0 0 0] [0]
[0 0 0] x1 + [0]
[0 0 1] [0]
p(true) = [0]
[0]
[1]
Following rules are strictly oriented:
eq(0(),s(X)) = [0 0 0] [1]
[0 0 0] X + [0]
[0 0 1] [1]
> [0]
[0]
[0]
= false()
eq(s(X),0()) = [1]
[0]
[1]
> [0]
[0]
[0]
= false()
Following rules are (at-least) weakly oriented:
eq(0(),0()) = [1]
[0]
[1]
>= [0]
[0]
[1]
= true()
eq(s(X),s(Y)) = [0 0 0] [1]
[0 0 0] Y + [0]
[0 0 1] [1]
>= [0 0 0] [1]
[0 0 0] Y + [0]
[0 0 1] [1]
= eq(X,Y)
ifrm(false(),N,add(M,X)) = [1 0 1] [0 0 0] [1 0 3] [2]
[0 0 0] M + [0 0 1] N + [0 0 2] X + [0]
[0 0 1] [0 0 0] [0 0 1] [2]
>= [1 0 0] [1 0 3] [1]
[0 0 0] M + [0 0 2] X + [0]
[0 0 1] [0 0 1] [2]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [1 0 1] [0 0 0] [1 0 3] [2]
[0 0 0] M + [0 0 1] N + [0 0 2] X + [2]
[0 0 1] [0 0 0] [0 0 1] [2]
>= [0 0 0] [1 0 1] [1]
[0 0 1] N + [0 0 2] X + [0]
[0 0 0] [0 0 1] [0]
= rm(N,X)
purge(add(N,X)) = [2 0 1] [2 0 7] [2]
[1 0 0] N + [1 0 4] X + [1]
[0 0 1] [0 0 1] [2]
>= [1 0 1] [2 0 7] [2]
[0 0 0] N + [0 0 2] X + [0]
[0 0 1] [0 0 1] [2]
= add(N,purge(rm(N,X)))
purge(nil()) = [0]
[1]
[0]
>= [0]
[0]
[0]
= nil()
rm(N,add(M,X)) = [1 0 1] [0 0 0] [1 0 3] [3]
[0 0 2] M + [0 0 1] N + [0 0 2] X + [4]
[0 0 1] [0 0 0] [0 0 1] [2]
>= [1 0 1] [0 0 0] [1 0 3] [3]
[0 0 2] M + [0 0 1] N + [0 0 2] X + [4]
[0 0 1] [0 0 0] [0 0 1] [2]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [0 0 0] [1]
[0 0 1] N + [0]
[0 0 0] [0]
>= [0]
[0]
[0]
= nil()
* Step 6: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
eq(s(X),s(Y)) -> eq(X,Y)
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
purge(nil()) -> nil()
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
- Signature:
{eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s
,true}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(add) = {2},
uargs(ifrm) = {1},
uargs(purge) = {1}
Following symbols are considered usable:
{eq,ifrm,purge,rm}
TcT has computed the following interpretation:
p(0) = [1]
[0]
[0]
p(add) = [0 0 0] [1 0 2] [0]
[0 0 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 0 1] [0]
p(eq) = [0 0 0] [0 0 1] [0]
[2 0 0] x1 + [2 3 0] x2 + [0]
[1 0 0] [2 1 0] [1]
p(false) = [0]
[2]
[0]
p(ifrm) = [1 0 0] [0 0 0] [1 1 0] [0]
[0 0 0] x1 + [0 2 0] x2 + [0 0 1] x3 + [2]
[0 0 0] [0 0 0] [0 0 1] [0]
p(nil) = [2]
[0]
[0]
p(purge) = [2 0 2] [0]
[0 0 2] x1 + [0]
[0 0 1] [0]
p(rm) = [0 0 0] [1 0 1] [0]
[0 2 0] x1 + [0 0 1] x2 + [2]
[0 0 0] [0 0 1] [0]
p(s) = [1 2 2] [0]
[0 0 0] x1 + [1]
[0 0 1] [2]
p(true) = [0]
[0]
[0]
Following rules are strictly oriented:
eq(s(X),s(Y)) = [0 0 0] [0 0 1] [2]
[2 4 4] X + [2 4 4] Y + [3]
[1 2 2] [2 4 4] [2]
> [0 0 0] [0 0 1] [0]
[2 0 0] X + [2 3 0] Y + [0]
[1 0 0] [2 1 0] [1]
= eq(X,Y)
Following rules are (at-least) weakly oriented:
eq(0(),0()) = [0]
[4]
[4]
>= [0]
[0]
[0]
= true()
eq(0(),s(X)) = [0 0 1] [2]
[2 4 4] X + [5]
[2 4 4] [3]
>= [0]
[2]
[0]
= false()
eq(s(X),0()) = [0 0 0] [0]
[2 4 4] X + [2]
[1 2 2] [3]
>= [0]
[2]
[0]
= false()
ifrm(false(),N,add(M,X)) = [0 0 0] [0 0 0] [1 0 3] [0]
[0 0 1] M + [0 2 0] N + [0 0 1] X + [2]
[0 0 1] [0 0 0] [0 0 1] [0]
>= [0 0 0] [1 0 3] [0]
[0 0 0] M + [0 0 1] X + [0]
[0 0 1] [0 0 1] [0]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [0 0 0] [0 0 0] [1 0 3] [0]
[0 0 1] M + [0 2 0] N + [0 0 1] X + [2]
[0 0 1] [0 0 0] [0 0 1] [0]
>= [0 0 0] [1 0 1] [0]
[0 2 0] N + [0 0 1] X + [2]
[0 0 0] [0 0 1] [0]
= rm(N,X)
purge(add(N,X)) = [0 0 2] [2 0 6] [0]
[0 0 2] N + [0 0 2] X + [0]
[0 0 1] [0 0 1] [0]
>= [0 0 0] [2 0 6] [0]
[0 0 0] N + [0 0 1] X + [0]
[0 0 1] [0 0 1] [0]
= add(N,purge(rm(N,X)))
purge(nil()) = [4]
[0]
[0]
>= [2]
[0]
[0]
= nil()
rm(N,add(M,X)) = [0 0 1] [0 0 0] [1 0 3] [0]
[0 0 1] M + [0 2 0] N + [0 0 1] X + [2]
[0 0 1] [0 0 0] [0 0 1] [0]
>= [0 0 1] [0 0 0] [1 0 3] [0]
[0 0 1] M + [0 2 0] N + [0 0 1] X + [2]
[0 0 1] [0 0 0] [0 0 1] [0]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [0 0 0] [2]
[0 2 0] N + [2]
[0 0 0] [0]
>= [2]
[0]
[0]
= nil()
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
eq(0(),0()) -> true()
eq(0(),s(X)) -> false()
eq(s(X),0()) -> false()
eq(s(X),s(Y)) -> eq(X,Y)
ifrm(false(),N,add(M,X)) -> add(M,rm(N,X))
ifrm(true(),N,add(M,X)) -> rm(N,X)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
purge(nil()) -> nil()
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) -> nil()
- Signature:
{eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s
,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))