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/*! @file Defines `boost::hana::demux`. Copyright Louis Dionne 2013-2022 Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt) */ #ifndef BOOST_HANA_FUNCTIONAL_DEMUX_HPP #define BOOST_HANA_FUNCTIONAL_DEMUX_HPP #include #include #include #include #include namespace boost { namespace hana { //! @ingroup group-functional //! Invoke a function with the results of invoking other functions //! on its arguments. //! //! Specifically, `demux(f)(g...)` is a function such that //! @code //! demux(f)(g...)(x...) == f(g(x...)...) //! @endcode //! //! Each `g` is called with all the arguments, and then `f` is called //! with the result of each `g`. Hence, the arity of `f` must match //! the number of `g`s. //! //! This is called `demux` because of a vague similarity between this //! device and a demultiplexer in signal processing. `demux` takes what //! can be seen as a continuation (`f`), a bunch of functions to split a //! signal (`g...`) and zero or more arguments representing the signal //! (`x...`). Then, it calls the continuation with the result of //! splitting the signal with whatever functions where given. //! //! @note //! When used with two functions only, `demux` is associative. In other //! words (and noting `demux(f, g) = demux(f)(g)` to ease the notation), //! it is true that `demux(demux(f, g), h) == demux(f, demux(g, h))`. //! //! //! Signature //! --------- //! The signature of `demux` is //! \f[ //! \mathtt{demux} : //! (B_1 \times \dotsb \times B_n \to C) //! \to ((A_1 \times \dotsb \times A_n \to B_1) //! \times \dotsb //! \times (A_1 \times \dotsb \times A_n \to B_n)) //! \to (A_1 \times \dotsb \times A_n \to C) //! \f] //! //! This can be rewritten more tersely as //! \f[ //! \mathtt{demux} : //! \left(\prod_{i=1}^n B_i \to C \right) //! \to \prod_{j=1}^n \left(\prod_{i=1}^n A_i \to B_j \right) //! \to \left(\prod_{i=1}^n A_i \to C \right) //! \f] //! //! //! Link with normal composition //! ---------------------------- //! The signature of `compose` is //! \f[ //! \mathtt{compose} : (B \to C) \times (A \to B) \to (A \to C) //! \f] //! //! A valid observation is that this coincides exactly with the type //! of `demux` when used with a single unary function. Actually, both //! functions are equivalent: //! @code //! demux(f)(g)(x) == compose(f, g)(x) //! @endcode //! //! However, let's now consider the curried version of `compose`, //! `curry<2>(compose)`: //! \f[ //! \mathtt{curry_2(compose)} : (B \to C) \to ((A \to B) \to (A \to C)) //! \f] //! //! For the rest of this explanation, we'll just consider the curried //! version of `compose` and so we'll use `compose` instead of //! `curry<2>(compose)` to lighten the notation. With currying, we can //! now consider `compose` applied to itself: //! \f[ //! \mathtt{compose(compose, compose)} : //! (B \to C) \to (A_1 \to A_2 \to B) \to (A_1 \to A_2 \to C) //! \f] //! //! If we uncurry deeply the above expression, we obtain //! \f[ //! \mathtt{compose(compose, compose)} : //! (B \to C) \times (A_1 \times A_2 \to B) \to (A_1 \times A_2 \to C) //! \f] //! //! This signature is exactly the same as that of `demux` when given a //! single binary function, and indeed they are equivalent definitions. //! We can also generalize this further by considering //! `compose(compose(compose, compose), compose)`: //! \f[ //! \mathtt{compose(compose(compose, compose), compose)} : //! (B \to C) \to (A_1 \to A_2 \to A_3 \to B) //! \to (A_1 \to A_2 \to A_3 \to C) //! \f] //! //! which uncurries to //! \f[ //! \mathtt{compose(compose(compose, compose), compose)} : //! (B \to C) \times (A_1 \times A_2 \times A_3 \to B) //! \to (A_1 \times A_2 \times A_3 \to C) //! \f] //! //! This signature is exactly the same as that of `demux` when given a //! single ternary function. Hence, for a single n-ary function `g`, //! `demux(f)(g)` is equivalent to the n-times composition of `compose` //! with itself, applied to `g` and `f`: //! @code //! demux(f)(g) == fold_left([compose, ..., compose], id, compose)(g, f) //! // ^^^^^^^^^^^^^^^^^^^^^ n times //! @endcode //! //! More information on this insight can be seen [here][1]. Also, I'm //! not sure how this insight could be generalized to more than one //! function `g`, or if that is even possible. //! //! //! Proof of associativity in the binary case //! ----------------------------------------- //! As explained above, `demux` is associative when it is used with //! two functions only. Indeed, given functions `f`, `g` and `h` with //! suitable signatures, we have //! @code //! demux(f)(demux(g)(h))(x...) == f(demux(g)(h)(x...)) //! == f(g(h(x...))) //! @endcode //! //! On the other hand, we have //! @code //! demux(demux(f)(g))(h)(x...) == demux(f)(g)(h(x...)) //! == f(g(h(x...))) //! @endcode //! //! and hence `demux` is associative in the binary case. //! //! //! Example //! ------- //! @include example/functional/demux.cpp //! //! [1]: http://stackoverflow.com/q/5821089/627587 #ifdef BOOST_HANA_DOXYGEN_INVOKED constexpr auto demux = [](auto&& f) { return [perfect-capture](auto&& ...g) { return [perfect-capture](auto&& ...x) -> decltype(auto) { // x... can't be forwarded unless there is a single g // function, or that could cause double-moves. return forwarded(f)(forwarded(g)(x...)...); }; }; }; #else template struct pre_demux_t; struct make_pre_demux_t { struct secret { }; template constexpr pre_demux_t::type> operator()(F&& f) const { return {static_cast(f)}; } }; template struct demux_t; template struct pre_demux_t { F f; template constexpr demux_t, F, typename detail::decay::type...> operator()(G&& ...g) const& { return {make_pre_demux_t::secret{}, this->f, static_cast(g)...}; } template constexpr demux_t, F, typename detail::decay::type...> operator()(G&& ...g) && { return {make_pre_demux_t::secret{}, static_cast(this->f), static_cast(g)...}; } }; template struct demux_t, F, G...> { template constexpr demux_t(make_pre_demux_t::secret, T&& ...t) : storage_{static_cast(t)...} { } basic_tuple storage_; template constexpr decltype(auto) operator()(X&& ...x) const& { return hana::at_c<0>(storage_)( hana::at_c(storage_)(x...)... ); } template constexpr decltype(auto) operator()(X&& ...x) & { return hana::at_c<0>(storage_)( hana::at_c(storage_)(x...)... ); } template constexpr decltype(auto) operator()(X&& ...x) && { return static_cast(hana::at_c<0>(storage_))( static_cast(hana::at_c(storage_))(x...)... ); } }; template struct demux_t, F, G> { template constexpr demux_t(make_pre_demux_t::secret, T&& ...t) : storage_{static_cast(t)...} { } basic_tuple storage_; template constexpr decltype(auto) operator()(X&& ...x) const& { return hana::at_c<0>(storage_)( hana::at_c<1>(storage_)(static_cast(x)...) ); } template constexpr decltype(auto) operator()(X&& ...x) & { return hana::at_c<0>(storage_)( hana::at_c<1>(storage_)(static_cast(x)...) ); } template constexpr decltype(auto) operator()(X&& ...x) && { return static_cast(hana::at_c<0>(storage_))( static_cast(hana::at_c<1>(storage_))(static_cast(x)...) ); } }; BOOST_HANA_INLINE_VARIABLE constexpr make_pre_demux_t demux{}; #endif }} // end namespace boost::hana #endif // !BOOST_HANA_FUNCTIONAL_DEMUX_HPP