Sketch of linear relations in julia

linrel.jl

#A*x = 0
using LinearAlgebra
struct HRep{T}
   n :: Int64 #input size
   A :: AbstractArray{T}
end
struct VRep{T}
   n :: Int64 #input size
   A :: AbstractArray{T}
end

function vrep(x::HRep) 
    VRep(x.n, nullspace(x.A))
end
function hrep(x::VRep)
    HRep(x.n, nullspace(x.A')')
end


function meet(x::HRep, y::HRep)
    @assert x.n == y.n
    HRep(x.n, [x.A ; y.A])
end
function meet(x::VRep, y::VRep)
    @assert x.n == y.n
    xh = hrep(x)
    yh = hrep(y)
    HRep(x.n, [xh.A ; yh.A])
end
function meet(x::HRep, y::VRep)
    @assert x.n == y.n
    yh = hrep(y)
    HRep(x.n, [x.A ; yh.A])
end
function meet(x::VRep, y::HRep)
    @assert x.n == y.n
    xh = hrep(x)
    HRep(x.n, [xh.A ; y.A])
end


function join(x::VRep, y::VRep)
    @assert x.n == y.n
    VRep(x.n, [x.A y.A])
end
function join(x::HRep, y::HRep)
    yv = vrep(y)
    join(x,yv)
end
function join(x::VRep, y::HRep)
        xv = vrep(x)
        join(xv,y)
end
function join(x::HRep, y::HRep)
        xv = vrep(x)
        join(xv,y)
end
    
function rid(n)
    VRep(n, [ Matrix(I,n,n) ; Matrix(I,n,n) ]  )
end


function rsub(r :: VRep,p :: HRep)
    all(isapprox.(p.A * r.A , 0; atol=eps(Float64), rtol=0))
end

function rsub(r :: VRep, p :: VRep)
    p = hrep(p)
    all(isapprox.(p.A  * r.A , 0; atol=eps(Float64), rtol=0))
end

function heq(p,q)
   rsub(p,q) && rsub(q,p) 
end


function converse(p :: VRep)
   n, g = size(p.A)
    VRep( n - p.n  , [p.A[p.n + 1 : end, :] ;  p.A[1 : p.n, :]  ]) 
end

function converse(p :: HRep)
   c, n = size(p.A)
   HRep( n - p.n  , [p.A[:, p.n + 1 : end]  p.A[:, 1 : p.n]]) 
end

function top(n)
   VRep(n, Matrix(I, n, n)) 
end 
function bottom(n)
    HRep(n, Matrix(I, n, n)) 
end


function compose(x::HRep, y::HRep)
    cx, nx = size(x.A)
    cy, ny = size(y.A)    
    na = x.n
    nb = nx - x.n # which should equal y.n
    nc = ny - y.n
    B = [                x.A          zeros( cx, ny - y.n) ; 
         zeros( cy, x.n  )      y.A                        ]
    C = nullspace(B)
    return VRep([  C[1 : n.x, :]          ; 
                   C[ nx+y.n + 1:end, :] ])
end

function compose(x::VRep, y::VRep)
   x1 = hrep(x)
   y1 = hrep(y)
   compose(x1,y1)
end