Follow-up of @system macro

[mforets]

Originally posted by @ueliwechsler in https://github.com/JuliaReach/MathematicalSystems.jl/pull/125#issuecomment-575240459

The following things can be done in a later PR (in approximately increasing order of work needed):

• [x] add template f_(x_,u_) for allowing to write @system(x' = f(x,u_12) (add one line) (#154)
• [ ] ensure that state≠input≠noise (~#155~ #164)
• [ ] add docstring to all functions
• [ ] make f_(x_,u_) and f_(x_,w_) distinguishable
• [x] correct handling for arbitrary order of input set definition + unit_tests (#176)
• [ ] make @system(x' = x + Bu) work by extracting state_dim from B
• [ ] add LazySets.Universe(n) or nothing as default cases for undefined sets (a straightforward option would be to handle every set as a Constrained set with all set as Universe(n) by default and if a set definition is provided specify the corresponding set)
• [x] Make @system macro work without parantheses, e.g. @system x' = A*x (#156 )
• [ ] Add support for remaining types, e.g. polynomial
• [ ] Add support for SystemWithOutput, e.g. @system(x' = Ax + Bu, y=Bx + Du)
• [ ] Allow arbitrary input for dynamic equations without *, e.g. @system(x' = Ax + Bu1)
• [ ] Major revision by using type specific "templates" for each AbstractSystem type

[ueliwechsler]

from #137:

• [x] allow property initial_state as state(0) ∈ set to create IVP

[ueliwechsler]

from #139

• [x] consistent type conversion in x' = -x and x' = 2x

[ueliwechsler]

Additionally, we should try to remove the explicit dependency on MathematicalSystems in these lines https://github.com/JuliaReach/MathematicalSystems.jl/blob/641d4f5a067f2bea6fd0dd0de7c2f193a9a8632c/src/macros.jl#L535 https://github.com/JuliaReach/MathematicalSystems.jl/blob/641d4f5a067f2bea6fd0dd0de7c2f193a9a8632c/src/macros.jl#L538

by writing push!(params, I($state_dim), :B) or something similar.