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//! [Standard terms](https://en.wikipedia.org/wiki/Lambda_calculus#Standard_terms) and //! [combinators](https://en.wikipedia.org/wiki/Combinatory_logic#Combinatory_calculi) //! //! * [SKI](https://en.wikipedia.org/wiki/SKI_combinator_calculus) //! * [Iota](https://en.wikipedia.org/wiki/Iota_and_Jot) //! * [BCKW](https://en.wikipedia.org/wiki/B,_C,_K,_W_system) // //! * the recursion combinator U - needs more research //! * the looping combinator ω //! * the divergent combinator Ω //! * [the fixed-point combinator Y](https://en.wikipedia.org/wiki/Fixed-point_combinator) use term::*; use term::Term::*; /// I - the identity combinator. /// /// I := λx.x = λ 1 /// /// # Example /// ``` /// use lambda_calculus::combinators::i; /// use lambda_calculus::arithmetic::zero; /// use lambda_calculus::reduction::beta_full; /// /// assert_eq!(beta_full(i().app(zero())), zero()); /// ``` pub fn i() -> Term { abs(Var(1)) } /// K - the constant / discarding combinator. /// /// K := λxy.x = λ λ 2 = true /// /// # Example /// ``` /// use lambda_calculus::combinators::k; /// use lambda_calculus::arithmetic::{zero, one}; /// use lambda_calculus::reduction::beta_full; /// /// assert_eq!(beta_full(k().app(zero()).app(one())), zero()); /// ``` pub fn k() -> Term { abs(abs(Var(2))) } /// S - the substitution combinator. /// /// S := λxyz.x z (y z) = λ λ λ 3 1 (2 1) /// /// # Example /// ``` /// use lambda_calculus::term::Term; /// use lambda_calculus::combinators::s; /// use lambda_calculus::reduction::beta_full; /// /// assert_eq!(beta_full(s().app(0.into()).app(1.into()).app(2.into())), /// beta_full(Term::from(0).app(2.into()).app(Term::from(1).app(2.into())))); /// ``` pub fn s() -> Term { abs(abs(abs( Var(3) .app(Var(1)) .app(Var(2).app(Var(1))) ))) } /// Iota - the universal combinator. /// /// ι := λx.x S K = λ 1 S K /// /// # Example /// ``` /// use lambda_calculus::combinators::{iota, i, k, s}; /// use lambda_calculus::reduction::beta_full; /// /// assert_eq!(beta_full(iota().app(iota())), i()); /// assert_eq!(beta_full(iota().app(iota().app(iota().app(iota())))), k()); /// assert_eq!(beta_full(iota().app(iota().app(iota().app(iota().app(iota()))))), s()); /// ``` pub fn iota() -> Term { abs(Var(1).app(s()).app(k())) } /// B - the composition combinator. /// /// B := λxyz.x (y z) = λ λ λ 3 (2 1) /// /// # Example /// ``` /// use lambda_calculus::term::Term; /// use lambda_calculus::combinators::b; /// use lambda_calculus::reduction::beta_full; /// /// assert_eq!(beta_full(b().app(0.into()).app(1.into()).app(2.into())), /// beta_full(Term::from(0).app(Term::from(1).app(2.into())))); /// ``` pub fn b() -> Term { abs(abs(abs( Var(3) .app(Var(2).app(Var(1))) ))) } /// C - the swapping combinator. /// /// C := λxyz.x z y = λ λ λ 3 1 2 /// /// # Example /// ``` /// use lambda_calculus::term::Term; /// use lambda_calculus::combinators::c; /// use lambda_calculus::reduction::beta_full; /// /// assert_eq!(beta_full(c().app(0.into()).app(1.into()).app(2.into())), /// beta_full(Term::from(0).app(2.into()).app(1.into()))); /// ``` pub fn c() -> Term { abs(abs(abs( Var(3) .app(Var(1)) .app(Var(2)) ))) } /// W - the duplicating combinator. /// /// W := λxy.x y y = λ λ 2 1 1 /// /// # Example /// ``` /// use lambda_calculus::combinators::w; /// use lambda_calculus::arithmetic::{zero, one}; /// use lambda_calculus::reduction::beta_full; /// /// assert_eq!(beta_full(w().app(zero()).app(one())), /// beta_full(zero().app(one()).app(one()))); /// ``` pub fn w() -> Term { abs(abs( Var(2) .app(Var(1)) .app(Var(1)) )) } /* /// U - the recursion combinator. /// /// U := λxy.y (x x y) = λ λ 1 (2 2 1) pub fn u() -> Term { abs(abs(Var(1).app(Var(2).app(Var(2)).app(Var(1))))) } */ /// ω - the looping combinator. /// /// ω := λx.x x /// # Example /// /// ``` /// use lambda_calculus::combinators::om; /// use lambda_calculus::arithmetic::zero; /// use lambda_calculus::reduction::beta_full; /// /// assert_eq!(beta_full(om().app(zero())), beta_full(zero().app(zero()))); /// ``` pub fn om() -> Term { abs(Var(1).app(Var(1))) } /// Ω - the divergent combinator. /// /// Ω := ω ω /// /// # Example /// /// ``` /// use lambda_calculus::combinators::omm; /// /// let mut doesnt_reduce = omm(); /// /// doesnt_reduce.beta_once(); /// /// assert_eq!(doesnt_reduce, omm()); /// ``` pub fn omm() -> Term { om().app(om()) } /// Y - the fixed-point combinator. /// /// Y := λg.(λx.g (x x)) (λx.g (x x)) = λ (λ 2 (1 1)) (λ 2 (1 1)) /// # Example /// /// ``` /// use lambda_calculus::combinators::y; /// use lambda_calculus::arithmetic::zero; /// use lambda_calculus::reduction::beta_full; /// /// assert_eq!(beta_full(y().app(zero())), beta_full(zero().app(y().app(zero())))); /// ``` pub fn y() -> Term { abs(app( abs(Var(2).app(Var(1).app(Var(1)))), abs(Var(2).app(Var(1).app(Var(1)))) )) }