module MappingClassGroups

using AbstractAlgebra
using Groups

###############################################################################
#
#  Generating set
#
###############################################################################

Comm(x,y) = x*y*x^-1*y^-1

function generatingset(N::Int)
    if N == 2
        MCGroup = Groups.FPGroup(["a1","a2","a3","a4","a5"]);
        S = Nemo.gens(MCGroup)

        N = length(S)
        A = prod(reverse(S))*prod(S)

        relations = [
           [Comm(S[i], S[j]) for i in 1:N for j in 1:N if abs(i-j) > 1]...,
           [S[i]*S[i+1]*S[i]*inv(S[i+1]*S[i]*S[i+1]) for i in 1:N-1]...,
           (S[1]*S[2]*S[3])^4*inv(S[5])^2,
           Comm(A, S[1]),
           A^2
           ]

       relations = [relations; [inv(rel) for rel in relations]]
       Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))
       return MCGroup

    elseif N < 2
        throw("Genus must be at least 2!")
    end

    MCGroup = Groups.FPGroup(["a$i" for i in 0:2N])
    S = Nemo.gens(MCGroup)

    a0 = S[1]
    A = S[2:end]
    k = length(A)

    relations = [
        [Comm(A[i], A[j]) for i in 1:k for j in 1:k if abs(i-j) > 1]...,
        [Comm(a0, A[i]) for i in 1:k if i != 4]...,
        [A[i]*A[(i+1)]*A[i]*inv(A[i+1]*A[i]*A[i+1]) for i in 1:k-1]...,
        A[4]*a0*A[4]*inv(a0*A[4]*a0)
        ]

    # 3-chain relation
    c = prod(reverse(A[1:4]))*prod(A[1:4])
    b0 = c*a0*inv(c)
    push!(relations, (A[1]*A[2]*A[3])^4*inv(a0*b0))

    # Lantern relation
    b1 = inv(A[4]*A[5]*A[3]*A[4])*a0*(A[4]*A[5]*A[3]*A[4])
    b2 = inv(A[2]*A[3]*A[1]*A[2])*b1*(A[2]*A[3]*A[1]*A[2])
    u  = inv(A[6]*A[5])*b1*(A[6]*A[5])
    x  = prod(reverse(A[2:6]))*u*prod(inv.(A[1:4]))
    b3 = x*a0*inv(x)
    push!(relations, a0*b2*b1*inv(A[1]*A[3]*A[5]*b3))

    # Hyperelliptic relation
    X = prod(reverse(A))*prod(A)

    function n(i::Int, b=b0)
        if i == 1
            return A[1]
        elseif i == 2
            return b
        else
            return w(i-2)*n(i-2)*w(i-2)
        end
    end

    function w(i::Int)
       (A[2i+4]*A[2i+3]*A[2i+2]* n(i+1))*(A[2i+1]*A[2i]  *A[2i+2]*A[2i+1])*
       (A[2i+3]*A[2i+2]*A[2i+4]*A[2i+3])*( n(i+1)*A[2i+2]*A[2i+1]*A[2i]  )
    end

    # push!(relations, X*n(N)*inv(n(N)*X))

    relations = [relations; [inv(rel) for rel in relations]]
    Groups.add_rels!(MCGroup, Dict(rel => MCGroup() for rel in relations))

    return MCGroup, gens(MCGroup)
end

function generatingset(parsed_args)
    N = parsed_args["N"]
    return generatingset(N)
end

###############################################################################
#
#  Misc
#
###############################################################################

function groupname(parsed_args)
    N = parsed_args["N"]
    return groupname(N), N
end

groupname(N::Int) = "MCG$(N)"

end #of module MappingClassGroups