* Step 1: Sum WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
- Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
- Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
+ 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(U12) = {2,3},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{U11,U12,activate,plus}
TcT has computed the following interpretation:
p(0) = [4]
p(U11) = [8] x_1 + [1] x_2 + [2] x_3 + [0]
p(U12) = [1] x_2 + [2] x_3 + [0]
p(activate) = [1] x_1 + [0]
p(plus) = [2] x_1 + [1] x_2 + [0]
p(s) = [1] x_1 + [0]
p(tt) = [0]
Following rules are strictly oriented:
plus(N,0()) = [2] N + [4]
> [1] N + [0]
= N
Following rules are (at-least) weakly oriented:
U11(tt(),M,N) = [1] M + [2] N + [0]
>= [1] M + [2] N + [0]
= U12(tt(),activate(M),activate(N))
U12(tt(),M,N) = [1] M + [2] N + [0]
>= [1] M + [2] N + [0]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
plus(N,s(M)) = [1] M + [2] N + [0]
>= [1] M + [2] N + [0]
= U11(tt(),M,N)
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,s(M)) -> U11(tt(),M,N)
- Weak TRS:
plus(N,0()) -> N
- Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
+ 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(U12) = {2,3},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [1] x3 + [0]
p(U12) = [1] x2 + [1] x3 + [2]
p(activate) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [10]
p(tt) = [8]
Following rules are strictly oriented:
plus(N,s(M)) = [1] M + [1] N + [10]
> [1] M + [1] N + [0]
= U11(tt(),M,N)
Following rules are (at-least) weakly oriented:
U11(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [2]
= U12(tt(),activate(M),activate(N))
U12(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [10]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
plus(N,0()) = [1] N + [0]
>= [1] N + [0]
= N
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
- Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
+ 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(U12) = {2,3},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x2 + [1] x3 + [0]
p(U12) = [7] x1 + [1] x2 + [1] x3 + [0]
p(activate) = [1] x1 + [3]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [0]
p(tt) = [3]
Following rules are strictly oriented:
U12(tt(),M,N) = [1] M + [1] N + [21]
> [1] M + [1] N + [6]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [3]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
U11(tt(),M,N) = [1] M + [1] N + [0]
>= [1] M + [1] N + [27]
= U12(tt(),activate(M),activate(N))
plus(N,0()) = [1] N + [0]
>= [1] N + [0]
= N
plus(N,s(M)) = [1] M + [1] N + [0]
>= [1] M + [1] N + [0]
= U11(tt(),M,N)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
- Weak TRS:
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
- Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
+ 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(U12) = {2,3},
uargs(plus) = {1,2},
uargs(s) = {1}
Following symbols are considered usable:
{U11,U12,activate,plus}
TcT has computed the following interpretation:
p(0) = [4]
p(U11) = [4] x_2 + [2] x_3 + [7]
p(U12) = [1] x_1 + [4] x_2 + [2] x_3 + [4]
p(activate) = [1] x_1 + [0]
p(plus) = [2] x_1 + [4] x_2 + [1]
p(s) = [1] x_1 + [2]
p(tt) = [0]
Following rules are strictly oriented:
U11(tt(),M,N) = [4] M + [2] N + [7]
> [4] M + [2] N + [4]
= U12(tt(),activate(M),activate(N))
Following rules are (at-least) weakly oriented:
U12(tt(),M,N) = [4] M + [2] N + [4]
>= [4] M + [2] N + [3]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
plus(N,0()) = [2] N + [17]
>= [1] N + [0]
= N
plus(N,s(M)) = [4] M + [2] N + [9]
>= [4] M + [2] N + [7]
= U11(tt(),M,N)
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
- Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))