PhysicalValues.hpp

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//                Copyright Oliver J. Rosten 2024.                //
// Distributed under the GNU GENERAL PUBLIC LICENSE, Version 3.0. //
//    (See accompanying file LICENSE.md or copy at                //
//          https://www.gnu.org/licenses/gpl-3.0.en.html)         //
 
#pragma once
 
#include "sequoia/Physics/PhysicalValuesDetails.hpp"
 
namespace sequoia::physics
{
  template<physical_unit T>
  struct is_invertible_unit : std::false_type {};
 
  template<physical_unit T>
    requires   std::is_same_v<typename T::validator_type, std::identity>
            || std::is_same_v<typename T::validator_type, maths::half_line_validator>
  struct is_invertible_unit<T> : std::true_type
  {};
 
  template<physical_unit T>
  using is_invertible_unit_t = is_invertible_unit<T>::type;
 
  template<physical_unit T>
  inline constexpr bool is_invertible_unit_v{is_invertible_unit<T>::value};
}
 
namespace sequoia::maths
{
  template<physics::physical_unit T>
    requires physics::is_invertible_unit_v<T>
  struct dual<T>
  {
    using validator_type = T::validator_type;
  };
 
  template<physics::physical_unit T>
    requires (!physics::is_invertible_unit_v<T>)
  struct dual<T>
  {
    using validator_type = void;
  };
}
 
namespace sequoia::physics
{
  using namespace maths;
 
  template<class... Ts>
  struct reduced_validator;
 
  template<class... Ts>
  using reduced_validator_t = reduced_validator<Ts...>::type;
 
  template<class T>
  struct reduced_validator<T>
  {
    using type = T;
  };
 
  template<class T, class... Us>
  struct reduced_validator<T, Us...>
  {
    using type = reduced_validator_t<T, reduced_validator_t<Us...>>;
  };
 
  struct no_unit_t
  {
    using validator_type = maths::half_line_validator;
  };
 
  inline constexpr no_unit_t no_unit{};
 
  template<physical_unit... Ts>
  struct composite_unit
  {
    using validator_type = reduced_validator_t<typename Ts::validator_type...>;
  };
 
  template<class T>
  inline constexpr bool has_arena_type_v{
    requires { typename T::arena_type;}
  };
 
  template<class T>
  struct arena_type_of;
 
  template<class T>
  using arena_type_of_t = arena_type_of<T>::type;
 
  template<class T>
    requires has_arena_type_v<T>
  struct arena_type_of<T>
  {
    using type = T::arena_type;
  };
 
  template<convex_space T>
    requires (!has_arena_type_v<dual<T>>)
  struct arena_type_of<dual<T>>
  {
    using type = arena_type_of_t<T>;
  };
 
  template<convex_space... Ts>
    requires (!has_arena_type_v<direct_product<Ts...>>)
  struct arena_type_of<direct_product<Ts...>>
  {
    using type = std::common_type_t<arena_type_of_t<Ts>...>;
  }; 
 
  template<class T>
  struct reduction;
 
  template<class T>
  using reduction_t = reduction<T>::type;
  
  template<class... Ts>
  struct composite_space;
 
  template<convex_space... Ts>
    requires (free_module<Ts> ||  ...)
  struct composite_space<Ts...>
  {    
    using direct_product_t      = direct_product<Ts...>;
    using set_type              = reduction<typename direct_product_t::set_type>;
    using commutative_ring_type = commutative_ring_type_of_t<direct_product_t>;
    using is_free_module        = std::true_type;
    using arena_type            = arena_type_of<direct_product<Ts...>>;
    constexpr static std::size_t dimension{std::ranges::max({dimension_of<Ts>...})};
  };
 
  template<convex_space... Ts>
    requires (!affine_space<Ts> && ...)
  struct composite_space<Ts...>
  {
    using direct_product_t = direct_product<Ts...>;
    using set_type         = reduction<typename direct_product_t::set_type>;
    using free_module_type = composite_space<free_module_type_of_t<Ts>...>;
    using is_convex_space  = std::true_type;
    using arena_type       = arena_type_of<direct_product<Ts...>>;
  };
 
  template<physical_unit T, physical_unit U>
  struct reduction<direct_product<T, U>>
  {
    using type = impl::simplify_t<direct_product<T>, direct_product<U>>;
  };
 
  template<physical_unit... Ts, physical_unit U>
  struct reduction<direct_product<composite_unit<Ts...>, U>>
  {
    using type = impl::simplify_t<direct_product<Ts...>, direct_product<U>>;
  };
 
  template<physical_unit T, physical_unit... Us>
  struct reduction<direct_product<T, composite_unit<Us...>>>
  {
    using type = impl::simplify_t<direct_product<Us...>, direct_product<T>>;
  };
 
  template<physical_unit... Ts, physical_unit... Us>
  struct reduction<direct_product<composite_unit<Ts...>, composite_unit<Us...>>>
  {
    using type = impl::simplify_t<direct_product<Ts...>, direct_product<Us...>>;
  };
 
  template<convex_space T, convex_space U>
  struct reduction<direct_product<T, U>>
  {    
    using type = impl::simplify_t<direct_product<T>, direct_product<U>>;
  };
 
  template<convex_space... Ts, convex_space U>
  struct reduction<direct_product<composite_space<Ts...>, U>>
  {
    using type = impl::simplify_t<direct_product<Ts...>, direct_product<U>>;
  };
 
  template<convex_space T, convex_space... Us>
  struct reduction<direct_product<T, composite_space<Us...>>>
  {
    using type = impl::simplify_t<direct_product<Us...>, direct_product<T>>;
  };  
 
  template<convex_space... Ts, convex_space... Us>
  struct reduction<direct_product<composite_space<Ts...>, composite_space<Us...>>>
  {
    using type = impl::simplify_t<direct_product<Ts...>, direct_product<Us...>>;
  };
 
  template<class T>
  struct reduced_validator<T, std::identity>
  {
    using type = std::identity;
  };
 
  template<class T>
    requires (!std::is_same_v<T, std::identity>)
  struct reduced_validator<std::identity, T>
  {
    using type = std::identity;
  };
 
  template<>
  struct reduced_validator<half_line_validator, half_line_validator>
  {
    using type = half_line_validator;
  };
 
  template<convex_space ValueSpace, physical_unit Unit>
  inline constexpr bool has_distinguished_origin{
       free_module<ValueSpace>
    || (!affine_space<ValueSpace> && defines_half_line_v<typename Unit::validator_type>)
  };
 
  template<convex_space ValueSpace, physical_unit Unit, class Origin>
  inline constexpr bool has_consistent_origin{
       ( has_distinguished_origin<ValueSpace, Unit> &&  std::is_same_v<Origin, distinguished_origin>)
    || (!has_distinguished_origin<ValueSpace, Unit> && !std::is_same_v<Origin, distinguished_origin>)
  };
 
  template<convex_space ValueSpace, validator_for<ValueSpace> Validator>
  inline constexpr bool has_consistent_validator{
    !affine_space<ValueSpace> || std::is_same_v<Validator, std::identity>
  };
 
  template<convex_space ValueSpace, physical_unit Unit>
  inline constexpr bool has_consistent_unit{
    !is_dual_v<Unit> || vector_space<ValueSpace> || (!affine_space<ValueSpace> && is_invertible_unit_v<dual_of_t<Unit>>)
  };
  
  template<convex_space ValueSpace, physical_unit Unit, basis_for<free_module_type_of_t<ValueSpace>> Basis, class Origin, validator_for<ValueSpace> Validator>
    requires    has_consistent_unit<ValueSpace, Unit>
             && has_consistent_validator<ValueSpace, Validator>
             && has_consistent_origin<ValueSpace, Unit, Origin>
  class physical_value;
 
  template<physical_unit Unit>
  struct unit_defined_origin{};
 
  template<affine_space T>
  struct implicit_affine_origin {};
 
  template<convex_space ValueSpace, physical_unit Unit>
  struct to_origin_type;
 
  template<convex_space ValueSpace, physical_unit Unit>
    requires (!has_distinguished_origin<ValueSpace, Unit>) && (!affine_space<ValueSpace>)
  struct to_origin_type<ValueSpace, Unit>
  {
    using type = unit_defined_origin<Unit>;
  };
 
  template<convex_space ValueSpace, physical_unit Unit>
  using to_origin_type_t = to_origin_type<ValueSpace, Unit>::type;
 
  template<convex_space ValueSpace, physical_unit Unit>
    requires has_distinguished_origin<ValueSpace, Unit>
  struct to_origin_type<ValueSpace, Unit>
  {
    using type = distinguished_origin;
  };
 
  template<convex_space ValueSpace, physical_unit Unit>
    requires (!has_distinguished_origin<ValueSpace, Unit> && affine_space<ValueSpace>)
  struct to_origin_type<ValueSpace, Unit>
  {
    using type = implicit_affine_origin<ValueSpace>;
  };
  
  template<convex_space ValueSpace, physical_unit Unit, basis_for<free_module_type_of_t<ValueSpace>> Basis, class Origin, class Validator>
  using to_coordinates_base_type
    = coordinates_base<
        ValueSpace,
        Basis,
        Origin,
        Validator,
        physical_value<free_module_type_of_t<ValueSpace>, Unit, Basis, distinguished_origin, std::identity>>;
 
  template<class T>
  inline constexpr bool has_base_space_v{
    requires { typename T::base_space; }
  };
 
  template<convex_space T>
  struct to_base_space
  {
    using type = T;
  };
 
  template<convex_space T>
  using to_base_space_t = to_base_space<T>::type;
 
  template<convex_space T>
    requires has_base_space_v<T>
  struct to_base_space<T>
  {
    using type = T::base_space;
  };
 
  template<convex_space T>
  struct to_base_space<dual<T>>
  {
    using type = dual<T>;
  };
 
  template<convex_space T>
    requires has_base_space_v<T>
  struct to_base_space<dual<T>>
  {
    using type = dual<typename T::base_space>;
  };
 
  template<convex_space... Ts>
  struct to_base_space<composite_space<Ts...>>
  {
    using sorted_direct_product_t = meta::stable_sort_t<direct_product<to_base_space_t<Ts>...>,  meta::type_comparator>;
    using type = impl::to_composite_space_t<reduction<impl::reduce_t<impl::count_and_combine_t<sorted_direct_product_t>>>>;
  };
 
  template<convex_space T, convex_space U>
  inline constexpr bool have_compatible_base_spaces_v{
    has_base_space_v<T> && has_base_space_v<U> && (free_module_type_of_t<T>::dimension == free_module_type_of_t<U>::dimension) && requires {
      typename std::common_type<typename T::base_space, typename U::base_space>::type;
    }
  };
 
  template<convex_space T, convex_space U>
  struct to_displacement_space;
 
  template<convex_space T, convex_space U>
  using to_displacement_space_t = to_displacement_space<T, U>::type;
 
  template<convex_space T>
  struct to_displacement_space<T, T>
  {
    using type = free_module_type_of_t<T>;
  };
 
  template<convex_space T, convex_space U>
    requires (!std::is_same_v<T, U>) && have_compatible_base_spaces_v<T, U>
  struct to_displacement_space<T, U>
  {
    using type = free_module_type_of_t<std::common_type_t<typename T::base_space, typename U::base_space>>;
  };
 
  template<class Basis1, class Basis2>
  struct consistent_bases : std::false_type {};
 
  template<class Basis1, class Basis2>
  inline constexpr bool consistent_bases_v{consistent_bases<Basis1, Basis2>::value};
 
  template<free_module M1, free_module M2>
  struct consistent_bases<canonical_right_handed_basis<M1>, canonical_right_handed_basis<M2>> : std::true_type
  {
    template<free_module M>
    using rebind_type = canonical_right_handed_basis<M>;
  };
  
  template<class T, class U>
  struct physical_value_product;
 
  template<class T, class U>
  using physical_value_product_t = physical_value_product<T, U>::type;
 
  template<convex_space C>
  inline constexpr bool is_1d_euclidean_v{
       std::is_same_v<euclidean_vector_space<commutative_ring_type_of_t<C>, 1, arena_type_of_t<C>>, C>
    || std::is_same_v<euclidean_half_space<commutative_ring_type_of_t<C>, arena_type_of_t<C>>, C>
  };
  
  template<
    convex_space LHSValueSpace, physical_unit LHSUnit, basis_for<free_module_type_of_t<LHSValueSpace>> LHSBasis, class LHSValidator,
    convex_space RHSValueSpace, physical_unit RHSUnit, basis_for<free_module_type_of_t<RHSValueSpace>> RHSBasis, class RHSValidator
  >
    requires consistent_bases_v<LHSBasis, RHSBasis> && (!is_1d_euclidean_v<LHSValueSpace> && !is_1d_euclidean_v<RHSValueSpace>)
  struct physical_value_product<physical_value<LHSValueSpace, LHSUnit, LHSBasis, distinguished_origin, LHSValidator>,
                                physical_value<RHSValueSpace, RHSUnit, RHSBasis, distinguished_origin, RHSValidator>>
  {
    using value_space_type = impl::to_composite_space_t<reduction_t<direct_product<LHSValueSpace, RHSValueSpace>>>;
    using type
      = physical_value<
          value_space_type,
          impl::to_composite_space_t<reduction_t<direct_product<LHSUnit, RHSUnit>>>,
          typename consistent_bases<LHSBasis, RHSBasis>::template rebind_type<free_module_type_of_t<value_space_type>>,
          distinguished_origin,
          reduced_validator_t<LHSValidator, RHSValidator>
        >;
  };
 
  template<
    convex_space LHSValueSpace, physical_unit LHSUnit, basis_for<free_module_type_of_t<LHSValueSpace>> LHSBasis, class LHSValidator,
    convex_space RHSValueSpace, physical_unit RHSUnit, basis_for<free_module_type_of_t<RHSValueSpace>> RHSBasis, class RHSValidator
  >
    requires  consistent_bases_v<LHSBasis, RHSBasis> && is_1d_euclidean_v<LHSValueSpace>
  struct physical_value_product<physical_value<LHSValueSpace, LHSUnit, LHSBasis, distinguished_origin, LHSValidator>,
                                physical_value<RHSValueSpace, RHSUnit, RHSBasis, distinguished_origin, RHSValidator>>
  {
    using type
      = physical_value<
          RHSValueSpace,
          RHSUnit,
          RHSBasis,
          distinguished_origin,
          reduced_validator_t<LHSValidator, RHSValidator>
        >;
  };
 
  template<
    convex_space LHSValueSpace, physical_unit LHSUnit, basis_for<free_module_type_of_t<LHSValueSpace>> LHSBasis, class LHSValidator,
    convex_space RHSValueSpace, physical_unit RHSUnit, basis_for<free_module_type_of_t<RHSValueSpace>> RHSBasis, class RHSValidator
  >
    requires  consistent_bases_v<LHSBasis, RHSBasis> && is_1d_euclidean_v<RHSValueSpace>
  struct physical_value_product<physical_value<LHSValueSpace, LHSUnit, LHSBasis, distinguished_origin, LHSValidator>,
                                physical_value<RHSValueSpace, RHSUnit, RHSBasis, distinguished_origin, RHSValidator>>
  {
    using type
      = physical_value<
          LHSValueSpace,
          LHSUnit,
          LHSBasis,
          distinguished_origin,
          reduced_validator_t<LHSValidator, RHSValidator>
        >;
  };
  
  template<
    convex_space ValueSpace,
    physical_unit Unit,
    basis_for<free_module_type_of_t<ValueSpace>> Basis = canonical_right_handed_basis<free_module_type_of_t<ValueSpace>>,
    class Origin                                       = to_origin_type_t<ValueSpace, Unit>,
    validator_for<ValueSpace> Validator                = typename Unit::validator_type
  >
    requires    has_consistent_unit<ValueSpace, Unit>
             && has_consistent_validator<ValueSpace, Validator>
             && has_consistent_origin<ValueSpace, Unit, Origin>
  class physical_value final : public to_coordinates_base_type<ValueSpace, Unit, Basis, Origin, Validator>
  {
  public:
    using coordinates_type         = to_coordinates_base_type<ValueSpace, Unit, Basis, Origin, Validator>;
    using space_type               = ValueSpace;
    using units_type               = Unit;
    using basis_type               = Basis;
    using origin_type              = Origin;
    using displacement_space_type  = free_module_type_of_t<ValueSpace>;
    using intrinsic_validator_type = Unit::validator_type;
    using validator_type           = Validator;
    using ring_type                = commutative_ring_type_of_t<ValueSpace>;
    using value_type               = ring_type;
    using displacement_type        = coordinates_type::displacement_coordinates_type;
 
    constexpr static std::size_t dimension{displacement_space_type::dimension};
    constexpr static std::size_t D{dimension};
 
    constexpr static bool is_intrinsically_absolute{
      (D == 1) && !affine_space<space_type> && defines_half_line_v<intrinsic_validator_type>
    };
 
    constexpr static bool is_effectively_absolute{is_intrinsically_absolute && std::is_same_v<Validator, intrinsic_validator_type>};
    constexpr static bool has_identity_validator{coordinates_type::has_identity_validator};
 
    template<convex_space RHSValueSpace, class RHSUnit, class RHSBasis, class RHSOrigin, class RHSValidator>
    constexpr static bool is_composable_with{
         consistent_bases_v<basis_type, RHSBasis>
      && (is_intrinsically_absolute || vector_space<space_type>)
      && (physical_value<RHSValueSpace, RHSUnit, RHSBasis, RHSOrigin, RHSValidator>::is_intrinsically_absolute || vector_space<RHSValueSpace>)
    };
 
    template<convex_space RHSValueSpace, class RHSUnit, class RHSBasis, class RHSOrigin, class RHSValidator>
    constexpr static bool is_multipicable_with{
         consistent_bases_v<basis_type, RHSBasis>
      && is_composable_with<RHSValueSpace, RHSUnit, RHSBasis, RHSOrigin, RHSValidator>
      && ((D == 1) || (physical_value<RHSValueSpace, RHSUnit, RHSBasis, RHSOrigin, RHSValidator>::D == 1))
    };
 
    template<convex_space RHSValueSpace, class RHSUnit, class RHSBasis, class RHSOrigin, class RHSValidator>
    constexpr static bool is_divisible_with{
         weak_field<ring_type>
      && weak_field<commutative_ring_type_of_t<RHSValueSpace>>
      && consistent_bases_v<basis_type, RHSBasis>
      && is_composable_with<RHSValueSpace, RHSUnit, RHSBasis, RHSOrigin, RHSValidator>
      && (physical_value<RHSValueSpace, RHSUnit, RHSBasis, RHSOrigin, RHSValidator>::D == 1)
    };
 
    constexpr physical_value() = default;
 
    constexpr physical_value(value_type val, units_type) requires (D == 1)
      : coordinates_type{val}
    {}
 
    constexpr physical_value(std::span<const value_type, D> val, units_type)
      : coordinates_type{val}
    {}
 
    [[nodiscard]]
    constexpr physical_value operator-() const noexcept(has_identity_validator)
      requires (coordinates_type::has_distinguished_origin) && (!std::is_unsigned_v<ring_type>) && (!is_effectively_absolute)
    {
      return physical_value{utilities::to_array(this->values(), [](value_type t) { return -t; }), units_type{}};
    }
 
    using coordinates_type::operator+=;
    
    template<convex_space OtherValueSpace, basis_for<free_module_type_of_t<OtherValueSpace>> OtherBasis>
      requires is_intrinsically_absolute && (std::is_base_of_v<ValueSpace, OtherValueSpace>) && consistent_bases_v<basis_type, OtherBasis>
    constexpr physical_value& operator+=(const physical_value<OtherValueSpace, Unit, OtherBasis, Origin, Validator>& other) noexcept(has_identity_validator)
    {
      this->apply_to_each_element(other.values(), [](value_type& lhs, value_type rhs){ lhs += rhs; });
      return *this;
    }
 
    using coordinates_type::operator-=;
    
    template<convex_space OtherValueSpace, basis_for<free_module_type_of_t<OtherValueSpace>> OtherBasis>
      requires is_intrinsically_absolute && (std::is_base_of_v<ValueSpace, OtherValueSpace>) && consistent_bases_v<basis_type, OtherBasis>
    constexpr physical_value& operator-=(const physical_value<OtherValueSpace, Unit, OtherBasis, Origin, Validator>& other) noexcept(has_identity_validator)
    {
      this->apply_to_each_element(other.values(), [](value_type& lhs, value_type rhs){ lhs -= rhs; });
      return *this;
    }
 
    template<convex_space OtherValueSpace, basis_for<free_module_type_of_t<OtherValueSpace>> OtherBasis>
      requires (!std::is_same_v<ValueSpace, OtherValueSpace>) && have_compatible_base_spaces_v<ValueSpace, OtherValueSpace> && consistent_bases_v<basis_type, OtherBasis>
    [[nodiscard]]
    friend constexpr auto operator+(const physical_value& lhs, const physical_value<OtherValueSpace, Unit, OtherBasis, Origin, Validator>& rhs)
    {
      using value_space_t = std::common_type_t<typename ValueSpace::base_space, typename OtherValueSpace::base_space>;
      using physical_value_t
        = physical_value<value_space_t, Unit, canonical_right_handed_basis<free_module_type_of_t<value_space_t>>, Origin, Validator>;
      return physical_value_t{lhs.values(), units_type{}} += rhs;
    }
 
    template<class OtherValueSpace, basis_for<free_module_type_of_t<OtherValueSpace>> OtherBasis>
    requires   (!std::is_same_v<OtherValueSpace, displacement_space_type>)
            && (std::is_same_v<ValueSpace, OtherValueSpace> || have_compatible_base_spaces_v<ValueSpace, OtherValueSpace>) && consistent_bases_v<basis_type, OtherBasis>
    [[nodiscard]]
    friend constexpr auto operator-(const physical_value& lhs, const physical_value<OtherValueSpace, Unit, OtherBasis, Origin, Validator>& rhs)
      noexcept(has_identity_validator)
    {
      using disp_space_t = to_displacement_space_t<ValueSpace, OtherValueSpace>;
      using basis_t = canonical_right_handed_basis<free_module_type_of_t<disp_space_t>>;
      using disp_t = to_coordinates_base_type<disp_space_t, Unit, basis_t, Origin, Validator>::displacement_coordinates_type;
      return[&] <std::size_t... Is>(std::index_sequence<Is...>) {
        return disp_t{std::array{(lhs.values()[Is] - rhs.values()[Is])...}, units_type{}};
      }(std::make_index_sequence<D>{});
    }
 
    template<convex_space RHSValueSpace, physical_unit RHSUnit, basis_for<free_module_type_of_t<RHSValueSpace>> RHSBasis, class RHSOrigin, class RHSValidator>
      requires is_multipicable_with<RHSValueSpace, RHSUnit, RHSBasis, RHSOrigin, RHSValidator>
    [[nodiscard]]
    friend constexpr auto operator*(const physical_value& lhs,
                                    const physical_value<RHSValueSpace, RHSUnit, RHSBasis, RHSOrigin, RHSValidator>& rhs)
    {
      using physical_value_t
        = physical_value_product_t<
            physical_value,
            physical_value<RHSValueSpace,RHSUnit, RHSBasis, RHSOrigin, RHSValidator>
          >;
 
      using derived_units_type = physical_value_t::units_type;
      return physical_value_t{lhs.value() * rhs.value(), derived_units_type{}};
    }
 
    template<convex_space RHSValueSpace, class RHSUnit,  basis_for<free_module_type_of_t<RHSValueSpace>> RHSBasis, class RHSOrigin, class RHSValidator>
      requires is_divisible_with<RHSValueSpace, RHSUnit, RHSBasis, RHSOrigin, RHSValidator>
    [[nodiscard]]
    friend constexpr auto operator/(const physical_value& lhs,
                                    const physical_value<RHSValueSpace, RHSUnit, RHSBasis, RHSOrigin, RHSValidator>& rhs)
    {
      using physical_value_t
        = physical_value_product_t<
            physical_value,
            physical_value<dual_of_t<RHSValueSpace>, dual_of_t<RHSUnit>, dual_of_t<RHSBasis>, RHSOrigin, RHSValidator>
          >;
      using derived_units_type = physical_value_t::units_type;
      if constexpr(dimension == 1)
      {
        return physical_value_t{lhs.value() / rhs.value(), derived_units_type{}};
      }
      else
      {
        return[&] <std::size_t... Is>(std::index_sequence<Is...>) {
          return physical_value_t{std::array{(lhs.values()[Is] / rhs.value())...}, derived_units_type{}};
        }(std::make_index_sequence<D>{});
      }
    }
 
    [[nodiscard]] friend constexpr auto operator/(value_type value, const physical_value& rhs)
      requires ((D == 1) && (is_intrinsically_absolute || vector_space<ValueSpace>))
    {
      using physical_value_t = physical_value<dual_of_t<ValueSpace>, dual_of_t<Unit>, dual_of_t<basis_type>, origin_type, validator_type>;
      using derived_units_type = physical_value_t::units_type;
      return physical_value_t{value / rhs.value(), derived_units_type{}};
    }
 
    // Reduce *everything* to its base space on both sides of the equation; if these are the same, allow the conversion 
    template<class LoweredValueSpace, class OtherUnit, basis_for<free_module_type_of_t<LoweredValueSpace>> OtherBasis>    
      requires std::same_as<to_base_space_t<space_type>, to_base_space_t<LoweredValueSpace>> && consistent_bases_v<basis_type, OtherBasis>
    [[nodiscard]]
    constexpr operator physical_value<LoweredValueSpace, OtherUnit, OtherBasis, origin_type, validator_type>() const noexcept
    {
      using value_space_t = to_base_space_t<LoweredValueSpace>;
      using basis_t = canonical_right_handed_basis<free_module_type_of_t<value_space_t>>;
      using physical_value_t = physical_value<to_base_space_t<LoweredValueSpace>, Unit, basis_t, origin_type, validator_type>;
      if constexpr(dimension == 1)
      {
        return physical_value_t{this->value(), OtherUnit{}};
      }
      else
      {
        return [this] <std::size_t... Is>(std::index_sequence<Is...>) {
          return physical_value_t{std::array{this->values()[Is]...}, OtherUnit{}};
        }(std::make_index_sequence<D>{});
      }
    }
  };
 
  namespace sets::classical
  {
    template<class Arena>
    struct masses
    {
      using arena_type = Arena;
    };
 
    template<class Arena>
    struct temperatures
    {
      using arena_type = Arena;
    };
 
    template<class Arena>
    struct electrical_currents
    {
      using arena_type = Arena;
    };
 
    template<class Arena>
    struct times
    {
      using arena_type = Arena;
    };
 
    template<class Arena>
    struct time_intervals
    {
      using arena_type = Arena;
    };
 
    template<class Arena>
    struct positions
    {
      using arena_type = Arena;
    };
 
    template<class Arena>
    struct lengths
    {
      using arena_type = Arena;
    };
 
    template<class Arena>
    struct angles
    {
      using arena_type = Arena;
    };
 
    template<class PhysicalValueSet>
    struct differences
    {
      using physical_value_set_type = PhysicalValueSet;
    };
  }
  
  template<class Space>
  struct associated_displacement_space
  {
    constexpr static std::size_t dimension{Space::dimension};
    using set_type              = sets::classical::differences<typename Space::set_type>;
    using commutative_ring_type = Space::representation_type;
    using is_free_module        = std::true_type;
    using arena_type            = Space::arena_type;
  };
 
  template<class Space>
      requires has_base_space_v<Space>
  struct associated_displacement_space<Space>
  {
    constexpr static std::size_t dimension{Space::dimension};
    using set_type              = sets::classical::differences<typename Space::set_type>;
    using commutative_ring_type = Space::representation_type;
    using is_free_module        = std::true_type;
    using base_space            = associated_displacement_space<typename Space::base_space>;
    using arena_type            = Space::arena_type;
  };
 
  template<class PhysicalValueSet, arithmetic Rep, std::size_t D, class Derived>
  struct physical_value_convex_space
  {
    constexpr static std::size_t dimension{D};
    using set_type            = PhysicalValueSet;
    using representation_type = Rep;
    using free_module_type    = associated_displacement_space<Derived>;
    using is_convex_space     = std::true_type;
    using arena_type          = PhysicalValueSet::arena_type;
  };
 
  template<class PhysicalValueSet, arithmetic Rep, std::size_t D, class Derived>
  struct physical_value_affine_space
  {
    constexpr static std::size_t dimension{D};
    using set_type            = PhysicalValueSet;
    using representation_type = Rep;
    using free_module_type    = associated_displacement_space<Derived>;
    using is_affine_space     = std::true_type;
    using arena_type          = PhysicalValueSet::arena_type;
  };
 
  template<class PhysicalValueSet, arithmetic Rep, std::size_t D, class Derived>
  struct physical_value_vector_space
  {
    constexpr static std::size_t dimension{D};
    using set_type            = PhysicalValueSet;
    using representation_type = Rep;
    using field_type          = Rep;
    using is_vector_space     = std::true_type;
  };
 
  template<std::floating_point Rep, class Arena>
  struct mass_space
    : physical_value_convex_space<sets::classical::masses<Arena>, Rep, 1, mass_space<Rep, Arena>>
  {
    using arena_type = Arena;
    using base_space = mass_space;
  };
 
  template<std::floating_point Rep, class Arena>
  struct temperature_space
    : physical_value_convex_space<sets::classical::temperatures<Arena>, Rep, 1, temperature_space<Rep, Arena>>
  {
    using arena_type = Arena;
    using base_space = temperature_space;
  };
 
  template<std::floating_point Rep, class Arena>
  struct electrical_current_space
    : physical_value_vector_space<sets::classical::electrical_currents<Arena>, Rep, 1, electrical_current_space<Rep, Arena>>
  {
    using arena_type = Arena;
    using base_space = electrical_current_space;
  };
 
  template<std::floating_point Rep, class Arena>
  struct angular_space : physical_value_vector_space<sets::classical::angles<Arena>, Rep, 1, angular_space<Rep, Arena>>
  {
    using arena_type = Arena;
    using base_space = angular_space;
  };
 
  template<arithmetic Rep, class Arena>
  struct length_space
    : physical_value_convex_space<sets::classical::lengths<Arena>, Rep, 1, length_space<Rep, Arena>>
  {
    using arena_type = Arena;
    using base_space = length_space;
  };
 
  template<arithmetic Rep, class Arena>
  struct width_space : length_space<Rep, Arena>
  {
    struct free_module_type : associated_displacement_space<width_space<Rep, Arena>> {};
  };
 
  template<arithmetic Rep, class Arena>
  struct height_space : length_space<Rep, Arena>
  {
    struct free_module_type : associated_displacement_space<height_space<Rep, Arena>> {};
  };
 
  template<arithmetic Rep, class Arena>
  struct time_interval_space
    : physical_value_convex_space<sets::classical::time_intervals<Arena>, Rep, 1, time_interval_space<Rep, Arena>>
  {
    using arena_type = Arena;
  };
  
  template<arithmetic Rep, class Arena>
  struct time_space : physical_value_affine_space<sets::classical::times<Arena>, Rep, 1, time_space<Rep, Arena>>
  {
    using arena_type = Arena;
  };
 
  template<arithmetic Rep, std::size_t D, class Arena>
  struct position_space : physical_value_affine_space<sets::classical::positions<Arena>, Rep, D, position_space<Rep, D, Arena>>
  {
    using arena_type = Arena;
  };
  
  struct implicit_common_arena {};
 
  template<physical_unit U>
  inline constexpr bool has_symbol_v{
    requires {
      { U::symbol } -> std::convertible_to<std::string_view>; }
  };
  
  namespace si
  {
    namespace units
    {
      struct ampere_t
      {
        using validator_type = std::identity;
        constexpr static std::string_view symbol{"A"};
      };
    
      struct kilogram_t
      {
        using validator_type = half_line_validator;
        constexpr static std::string_view symbol{"kg"};
      };
 
      struct metre_t
      {
        using validator_type = half_line_validator;
        constexpr static std::string_view symbol{"m"};
      };
 
      struct second_t
      {
        using validator_type = half_line_validator;
        constexpr static std::string_view symbol{"s"};
      };
 
      struct kelvin_t
      {
        using validator_type = half_line_validator;
        constexpr static std::string_view symbol{"K"};
      };
 
      struct coulomb_t
      {
        using validator_type = std::identity;
        constexpr static std::string_view symbol{"C"};
      };
 
      struct radian_t
      {
        using validator_type = std::identity;
        constexpr static std::string_view symbol{"rad"};
      };
 
 
      struct celsius_t
      {
        struct validator
        {
          template<std::floating_point T>
          [[nodiscard]]
          constexpr T operator()(const T val) const
          {
            if(val < T(-273.15)) throw std::domain_error{std::format("Value {} less than -273.15", val)};
 
            return val;
          }
        };
 
        using validator_type = validator;
        constexpr static std::string_view symbol{"degC"};
      };
 
      inline constexpr ampere_t   ampere{};
      inline constexpr kilogram_t kilogram{};
      inline constexpr metre_t    metre{};
      inline constexpr second_t   second{};
      inline constexpr kelvin_t   kelvin{};
      inline constexpr coulomb_t  coulomb{};
      inline constexpr radian_t   radian{};
 
      inline constexpr celsius_t celsius{};
    }
 
    template<std::floating_point T, class Arena=implicit_common_arena>
    using mass = physical_value<mass_space<T, Arena>, units::kilogram_t>;
 
    template<std::floating_point T, class Arena=implicit_common_arena>
    using length = physical_value<length_space<T, Arena>, units::metre_t>;
 
    template<std::floating_point T, class Arena=implicit_common_arena>
    using time_interval = physical_value<time_interval_space<T, Arena>, units::second_t>;
 
    template<std::floating_point T, class Arena=implicit_common_arena>
    using temperature = physical_value<temperature_space<T, Arena>, units::kelvin_t>;
 
    template<std::floating_point T, class Arena=implicit_common_arena>
    using electrical_current = physical_value<electrical_current_space<T, Arena>, units::ampere_t>;
 
    template<std::floating_point T, class Arena=implicit_common_arena>
    using angle = physical_value<angular_space<T, Arena>, units::radian_t>;
 
    template<std::floating_point T, class Arena=implicit_common_arena>
    using width = physical_value<width_space<T, Arena>, units::metre_t>;
 
    template<std::floating_point T, class Arena=implicit_common_arena>
    using height = physical_value<height_space<T, Arena>, units::metre_t>;
 
    template<
      std::floating_point T,
      class Arena=implicit_common_arena,
      class Origin=implicit_affine_origin<time_space<T, Arena>>
    >
    using time = physical_value<time_space<T, Arena>, units::second_t, canonical_right_handed_basis<free_module_type_of_t<time_space<T, Arena>>>, Origin, std::identity>;
 
    template<      
      std::floating_point T,
      std::size_t D,
      class Arena  = implicit_common_arena,      
      basis_for<free_module_type_of_t<position_space<T, D, Arena>>> Basis = canonical_right_handed_basis<free_module_type_of_t<position_space<T, D, Arena>>>,
      class Origin = implicit_affine_origin<position_space<T, D, Arena>>
    >
    using position = physical_value<position_space<T, D, Arena>, units::metre_t, Basis, Origin, std::identity>;
  }
 
  template<vector_space ValueSpace, physical_unit Unit, class Basis, class Origin, validator_for<ValueSpace> Validator>
    requires (dimension_of<ValueSpace> == 1)
  [[nodiscard]]
  constexpr physical_value<ValueSpace, Unit, Basis, Origin, Validator> abs(physical_value<ValueSpace, Unit, Basis, Origin, Validator> q)
  {
    return {std::abs(q.value()), Unit{}};
  }
 
  template<std::floating_point T, class Arena=implicit_common_arena>
  [[nodiscard]]
  constexpr T sin(physical_value<angular_space<T, Arena>, si::units::radian_t> theta)
  {
    return std::sin(theta.value());
  }
 
  template<std::floating_point T, class Arena=implicit_common_arena>
  [[nodiscard]]
  constexpr T cos(physical_value<angular_space<T, Arena>, si::units::radian_t> theta)
  {
    return std::cos(theta.value());
  }
 
  template<std::floating_point T, class Arena=implicit_common_arena>
  [[nodiscard]]
  constexpr T tan(physical_value<angular_space<T, Arena>, si::units::radian_t> theta)
  {
    return std::tan(theta.value());
  }
 
  template<class Arena=implicit_common_arena, std::floating_point T>
  [[nodiscard]]
  constexpr physical_value<angular_space<T, Arena>, si::units::radian_t> asin(T x)
  {
    return {std::asin(x), si::units::radian};
  }
 
  template<class Arena=implicit_common_arena, std::floating_point T>
  [[nodiscard]]
  constexpr physical_value<angular_space<T, Arena>, si::units::radian_t> acos(T x)
  {
    return {std::acos(x), si::units::radian};
  }
 
  template<class Arena=implicit_common_arena, std::floating_point T>
  [[nodiscard]]
  constexpr physical_value<angular_space<T, Arena>, si::units::radian_t> atan(T x)
  {
    return {std::atan(x), si::units::radian};
  }
 
  template<
    convex_space ValueSpace,
    physical_unit Unit,
    validator_for<ValueSpace> Validator=typename Unit::validator_type
  >
    requires has_consistent_validator<ValueSpace, Validator>
  using quantity = physical_value<ValueSpace, Unit, canonical_right_handed_basis<free_module_type_of_t<ValueSpace>>, to_origin_type_t<ValueSpace, Unit>, Validator>;
 
  
  template<std::floating_point Rep, class Arena=implicit_common_arena>
  using euclidean_1d_vector_quantity = quantity<euclidean_vector_space<Rep, 1, Arena>, no_unit_t, std::identity>;
 
  template<std::floating_point Rep, class Arena=implicit_common_arena>
  using euclidean_half_line_quantity = quantity<euclidean_half_space<Rep, Arena>, no_unit_t>;
}
 
// TO DO: extend this
template<
  sequoia::maths::convex_space ValueSpace,
  sequoia::physics::physical_unit Unit,
  sequoia::maths::validator_for<ValueSpace> Validator
>
  requires (sequoia::maths::dimension_of<ValueSpace> == 1)
struct std::formatter<sequoia::physics::quantity<ValueSpace, Unit, Validator>>
{
  constexpr auto parse(auto& ctx)
  {
    return ctx.begin();
  }
 
  auto format(const sequoia::physics::quantity<ValueSpace, Unit, Validator>& q, auto& ctx) const
  {
    if constexpr(sequoia::physics::has_symbol_v<Unit>)
      return std::format_to(ctx.out(), "{} {}", q.value(), Unit::symbol);
    else
      return std::format_to(ctx.out(), "{}", q.value());
  }
};