Explain in minute detail this Julia code for solving cryptic crossword clues and also output the same code with code explanations as comments added in the relevant places in the code: struct SolverState
    outputs::Dict{Arc{Rule}, Set{String}}
    derivations::Dict{Arc{Rule}, Dict{String, Vector{Vector{String}}}}
end

SolverState() = SolverState(Dict(), Dict())

@generated function _product(inputs, ::Val{N}) where {N}
    Expr(:call, :product, [:(inputs[$i]) for i in 1:N]...)
end

function apply(head::GrammaticalSymbol, args::Tuple{Vararg{GrammaticalSymbol, N}}, inputs) where {N}
    result = Set{String}()
    for input in _product(inputs, Val{N}())
        apply!(result, head, args, input)
    end
    result
end

function _apply(head::GrammaticalSymbol, args::Tuple{Vararg{GrammaticalSymbol, N}}, inputs::AbstractVector) where {N}
    apply(head, args, ntuple(i -> inputs[i], Val(N)))
end

function _apply(rule::Rule,
                constituents::AbstractVector)
    r = inner(rule)
    _apply(lhs(r), rhs(r), constituents)
end

solve!(state::SolverState, s::AbstractString) = Set([s])

function solve!(state::SolverState, arc::Arc{Rule})
    get!(state.outputs, arc) do
        inputs = [solve!(state, c) for c in constituents(arc)]
        _apply(rule(arc), inputs)
    end
end


struct SolvedArc
    arc::Arc{Rule}
    output::String
    similarity::Float64
end

function output_checker(len::Union{Integer, Nothing}, pattern::Regex)
    return word -> ((len === nothing || num_letters(word) == len) && is_word(word) && occursin(pattern, word))
end

function solve(clue;
        length::Union{Integer, Nothing} = nothing,
        pattern::Regex = r"",
        strategy = BottomUp(),
        min_grammar_score = 1e-6,
        should_continue = () -> true)
    state = SolverState()
    tokens = normalize.(split(clue))
    grammar = CrypticsGrammar()
    check = output_checker(length, pattern)

    function is_solvable(arc)
        if score(arc) < min_grammar_score
            return 0
        end
        outputs = solve!(state, arc)
        isempty(outputs)
        if isempty(outputs)
            return 0
        else
            return 1
        end
    end
    parser = ChartParser(tokens, grammar, BottomUp(),
                         is_solvable)
    solutions = SolvedArc[]
    lowest_score_seen = Inf
    for arc in parser
        if !should_continue()
            break
        end
        if !is_complete(arc, parser)
            continue
        end
        @assert score(arc) <= lowest_score_seen
        lowest_score_seen = score(arc)
        if score(arc) < min_grammar_score
            break
        end
        # TODO: probably don't need to call solve!() here
        outputs = solve!(state, arc)
        @assert !isempty(outputs)
        # for (output, inputs) in outputs
        for output in outputs
            if check(output)
                push!(solutions, SolvedArc(arc, output,
                                           solution_quality(arc, output)))
            end
        end
    end
    sort!(solutions, by=s -> s.similarity, rev=true)
    solutions, state
end

struct DerivedArc
    arc::Arc{Rule}
    output::String
    constituents::Vector{Union{DerivedArc, String}}
end

struct DerivedSolution
    derivation::DerivedArc
    output::String
    similarity::Float64
end

function derive(head::GrammaticalSymbol, args::Tuple{Vararg{GrammaticalSymbol, N}}, inputs, target) where {N}
    result = Vector{Vector{String}}()
    buffer = Vector{String}()
    for input in _product(inputs, Val{N}())
        empty!(buffer)
        apply!(buffer, head, args, input)
        input_vec = collect(input)
        for output in buffer
            if output == target
                push!(result, input_vec)
            end
        end
    end
    result
end

function _derive(head::GrammaticalSymbol, args::Tuple{Vararg{GrammaticalSymbol, N}}, inputs::AbstractVector, target::AbstractString) where {N}
    derive(head, args, ntuple(i -> inputs[i], Val(N)), target)
end

function _derive(rule::Rule,
                constituents::AbstractVector,
                target::AbstractString)
    r = inner(rule)
    _derive(lhs(r), rhs(r), constituents, target)
end

function find_derivations!(state::SolverState, arc::Arc{Rule}, target::AbstractString)
    arc_derivations = get!(state.derivations, arc) do
        Dict()
    end
    get!(arc_derivations, target) do
        inputs = [solve!(state, c) for c in constituents(arc)]
        _derive(rule(arc), inputs, target)
    end
end

derive!(state::SolverState, s::AbstractString, t::AbstractString) = [t]

function derive!(state::SolverState, arc::Arc{Rule}, target::AbstractString)
    result = DerivedArc[]
    input_lists = find_derivations!(state, arc, target)
    for inputs in input_lists
        for children in product(derive!.(Ref(state), constituents(arc), inputs)...)
            push!(result, DerivedArc(arc, target, collect(children)))
        end
    end
    result
end

function derive!(state::SolverState, solved::SolvedArc)
    derivations = derive!(state, solved.arc, solved.output)
    DerivedSolution.(derivations, Ref(solved.output), Ref(solved.similarity))
end

_show(io::IO, arc::DerivedArc) = print(io, arc)
_show(io::IO, s::AbstractString) = print(io, '"', s, '"')

function Base.show(io::IO, arc::DerivedArc)
    print(io, "($(lhs(rule(arc.arc))) -> ")
    for c in arc.constituents
        _show(io, c)
        print(io, " ")
    end
    print(io, "; $(score(arc.arc))) -> \"$(arc.output)\")")
end

function answer_similarity(word1::AbstractString, word2::AbstractString)
    if word2 in keys(CACHE.synonyms) && word1 in CACHE.synonyms[word2]
        1.0
    else
        SemanticSimilarity.similarity(word1, word2)
    end
end

function solution_quality(arc::Arc, output::AbstractString)
    @assert lhs(inner(rule(arc))) === Clue()
    answer_similarity(definition(arc), output)
end

tokens(arc::Arc) = join((tokens(x) for x in constituents(arc)), ' ')
tokens(s::AbstractString) = s

function definition(arc::Arc)
    @assert lhs(inner(rule(arc))) === Clue()
    tokens(first(x for x in constituents(arc) if lhs(inner(rule(x))) == Definition()))
end

definition(arc::SolvedArc) = definition(arc.arc)

num_letters(word::AbstractString) = count(!isequal(' '), word)