* Step 1: WeightGap WORST_CASE(?,O(n^3))
+ 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:
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) = {1,3},
uargs(purge) = {1},
uargs(rm) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x1 + [1] x2 + [0]
p(eq) = [5]
p(false) = [0]
p(ifrm) = [1] x1 + [1] x3 + [5]
p(nil) = [0]
p(purge) = [1] x1 + [5]
p(rm) = [1] x2 + [3]
p(s) = [1] x1 + [0]
p(true) = [0]
Following rules are strictly oriented:
eq(0(),0()) = [5]
> [0]
= true()
eq(0(),s(X)) = [5]
> [0]
= false()
eq(s(X),0()) = [5]
> [0]
= false()
ifrm(false(),N,add(M,X)) = [1] M + [1] X + [5]
> [1] M + [1] X + [3]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [1] M + [1] X + [5]
> [1] X + [3]
= rm(N,X)
purge(nil()) = [5]
> [0]
= nil()
rm(N,nil()) = [3]
> [0]
= nil()
Following rules are (at-least) weakly oriented:
eq(s(X),s(Y)) = [5]
>= [5]
= eq(X,Y)
purge(add(N,X)) = [1] N + [1] X + [5]
>= [1] N + [1] X + [8]
= add(N,purge(rm(N,X)))
rm(N,add(M,X)) = [1] M + [1] X + [3]
>= [1] M + [1] X + [10]
= ifrm(eq(N,M),N,add(M,X))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
eq(s(X),s(Y)) -> eq(X,Y)
purge(add(N,X)) -> add(N,purge(rm(N,X)))
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
- 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(nil()) -> nil()
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:
runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(add) = {2},
uargs(ifrm) = {1,3},
uargs(purge) = {1},
uargs(rm) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(add) = [1] x_1 + [1] x_2 + [3]
p(eq) = [0]
p(false) = [0]
p(ifrm) = [8] x_1 + [1] x_3 + [1]
p(nil) = [0]
p(purge) = [2] x_1 + [0]
p(rm) = [1] x_2 + [1]
p(s) = [4]
p(true) = [0]
Following rules are strictly oriented:
purge(add(N,X)) = [2] N + [2] X + [6]
> [1] N + [2] X + [5]
= 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] M + [1] X + [4]
>= [1] M + [1] X + [4]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [1] M + [1] X + [4]
>= [1] X + [1]
= rm(N,X)
purge(nil()) = [0]
>= [0]
= nil()
rm(N,add(M,X)) = [1] M + [1] X + [4]
>= [1] M + [1] X + [4]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [1]
>= [0]
= nil()
* Step 3: MI WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
eq(s(X),s(Y)) -> eq(X,Y)
rm(N,add(M,X)) -> ifrm(eq(N,M),N,add(M,X))
- 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,nil()) -> nil()
- Signature:
{eq/2,ifrm/3,purge/1,rm/2} / {0/0,add/2,false/0,nil/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(add) = {2},
uargs(ifrm) = {1,3},
uargs(purge) = {1},
uargs(rm) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
[0]
p(add) = [1 4] x_2 + [0]
[0 1] [2]
p(eq) = [0]
[2]
p(false) = [0]
[2]
p(ifrm) = [2 1] x_1 + [1 2] x_3 + [0]
[0 0] [0 1] [0]
p(nil) = [0]
[0]
p(purge) = [2 4] x_1 + [4]
[0 1] [0]
p(rm) = [1 2] x_2 + [3]
[0 1] [0]
p(s) = [1 0] x_1 + [0]
[0 1] [0]
p(true) = [0]
[2]
Following rules are strictly oriented:
rm(N,add(M,X)) = [1 6] X + [7]
[0 1] [2]
> [1 6] X + [6]
[0 1] [2]
= ifrm(eq(N,M),N,add(M,X))
Following rules are (at-least) weakly oriented:
eq(0(),0()) = [0]
[2]
>= [0]
[2]
= true()
eq(0(),s(X)) = [0]
[2]
>= [0]
[2]
= false()
eq(s(X),0()) = [0]
[2]
>= [0]
[2]
= false()
eq(s(X),s(Y)) = [0]
[2]
>= [0]
[2]
= eq(X,Y)
ifrm(false(),N,add(M,X)) = [1 6] X + [6]
[0 1] [2]
>= [1 6] X + [3]
[0 1] [2]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [1 6] X + [6]
[0 1] [2]
>= [1 2] X + [3]
[0 1] [0]
= rm(N,X)
purge(add(N,X)) = [2 12] X + [12]
[0 1] [2]
>= [2 12] X + [10]
[0 1] [2]
= add(N,purge(rm(N,X)))
purge(nil()) = [4]
[0]
>= [0]
[0]
= nil()
rm(N,nil()) = [3]
[0]
>= [0]
[0]
= nil()
* Step 4: MI WORST_CASE(?,O(n^3))
+ 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:
runtime complexity wrt. defined symbols {eq,ifrm,purge,rm} and constructors {0,add,false,nil,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 3))), miDimension = 4, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 3))):
The following argument positions are considered usable:
uargs(add) = {2},
uargs(ifrm) = {1,3},
uargs(purge) = {1},
uargs(rm) = {2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
[1]
[0]
[1]
p(add) = [0 0 0 0] [1 0 2 0] [0]
[0 0 0 0] x_1 + [0 0 1 0] x_2 + [0]
[0 0 0 1] [0 0 1 0] [1]
[0 0 0 0] [0 0 0 0] [0]
p(eq) = [0 0 0 0] [0 0 0 1] [1]
[1 1 1 1] x_1 + [0 0 0 1] x_2 + [0]
[0 0 0 0] [0 0 0 0] [0]
[1 1 0 2] [0 0 0 0] [2]
p(false) = [0]
[0]
[0]
[0]
p(ifrm) = [1 0 0 0] [0 0 0 0] [1 1 0 0] [0]
[0 0 0 0] x_1 + [0 0 2 0] x_2 + [1 0 0 0] x_3 + [2]
[0 0 0 0] [0 0 0 0] [0 0 1 0] [0]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [2]
p(nil) = [2]
[0]
[0]
[2]
p(purge) = [2 0 0 0] [0]
[0 1 0 0] x_1 + [0]
[0 0 1 0] [0]
[2 0 0 0] [0]
p(rm) = [0 0 0 0] [1 0 1 0] [0]
[0 0 2 0] x_1 + [1 0 0 0] x_2 + [2]
[0 0 0 0] [0 0 1 0] [0]
[0 0 0 0] [0 0 0 0] [2]
p(s) = [1 1 2 0] [0]
[0 0 0 0] x_1 + [2]
[0 0 0 0] [3]
[0 0 0 1] [1]
p(true) = [0]
[3]
[0]
[1]
Following rules are strictly oriented:
eq(s(X),s(Y)) = [0 0 0 0] [0 0 0 1] [2]
[1 1 2 1] X + [0 0 0 1] Y + [7]
[0 0 0 0] [0 0 0 0] [0]
[1 1 2 2] [0 0 0 0] [6]
> [0 0 0 0] [0 0 0 1] [1]
[1 1 1 1] X + [0 0 0 1] Y + [0]
[0 0 0 0] [0 0 0 0] [0]
[1 1 0 2] [0 0 0 0] [2]
= eq(X,Y)
Following rules are (at-least) weakly oriented:
eq(0(),0()) = [2]
[3]
[0]
[5]
>= [0]
[3]
[0]
[1]
= true()
eq(0(),s(X)) = [0 0 0 1] [2]
[0 0 0 1] X + [3]
[0 0 0 0] [0]
[0 0 0 0] [5]
>= [0]
[0]
[0]
[0]
= false()
eq(s(X),0()) = [0 0 0 0] [2]
[1 1 2 1] X + [7]
[0 0 0 0] [0]
[1 1 2 2] [6]
>= [0]
[0]
[0]
[0]
= false()
ifrm(false(),N,add(M,X)) = [0 0 0 0] [0 0 0 0] [1 0 3 0] [0]
[0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + [2]
[0 0 0 1] [0 0 0 0] [0 0 1 0] [1]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [2]
>= [0 0 0 0] [1 0 3 0] [0]
[0 0 0 0] M + [0 0 1 0] X + [0]
[0 0 0 1] [0 0 1 0] [1]
[0 0 0 0] [0 0 0 0] [0]
= add(M,rm(N,X))
ifrm(true(),N,add(M,X)) = [0 0 0 0] [0 0 0 0] [1 0 3 0] [0]
[0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + [2]
[0 0 0 1] [0 0 0 0] [0 0 1 0] [1]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [2]
>= [0 0 0 0] [1 0 1 0] [0]
[0 0 2 0] N + [1 0 0 0] X + [2]
[0 0 0 0] [0 0 1 0] [0]
[0 0 0 0] [0 0 0 0] [2]
= rm(N,X)
purge(add(N,X)) = [0 0 0 0] [2 0 4 0] [0]
[0 0 0 0] N + [0 0 1 0] X + [0]
[0 0 0 1] [0 0 1 0] [1]
[0 0 0 0] [2 0 4 0] [0]
>= [0 0 0 0] [2 0 4 0] [0]
[0 0 0 0] N + [0 0 1 0] X + [0]
[0 0 0 1] [0 0 1 0] [1]
[0 0 0 0] [0 0 0 0] [0]
= add(N,purge(rm(N,X)))
purge(nil()) = [4]
[0]
[0]
[4]
>= [2]
[0]
[0]
[2]
= nil()
rm(N,add(M,X)) = [0 0 0 1] [0 0 0 0] [1 0 3 0] [1]
[0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + [2]
[0 0 0 1] [0 0 0 0] [0 0 1 0] [1]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [2]
>= [0 0 0 1] [0 0 0 0] [1 0 3 0] [1]
[0 0 0 0] M + [0 0 2 0] N + [1 0 2 0] X + [2]
[0 0 0 1] [0 0 0 0] [0 0 1 0] [1]
[0 0 0 0] [0 0 0 0] [0 0 0 0] [2]
= ifrm(eq(N,M),N,add(M,X))
rm(N,nil()) = [0 0 0 0] [2]
[0 0 2 0] N + [4]
[0 0 0 0] [0]
[0 0 0 0] [2]
>= [2]
[0]
[0]
[2]
= nil()
* Step 5: 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:
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^3))