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#![deny(missing_docs)] //! # Poi //! a pragmatic point-free theorem prover assistant //! //! ```text //! === Poi Reduce 0.2 === //! Type `help` for more information. //! > and[not] //! and[not] //! or ( and[not] => or ) //! ``` //! //! To run Poi Reduce from your Terminal, type: //! //! ```text //! cargo install --example poireduce poi //! ``` //! //! Then, to run: //! //! ```text //! poireduce //! ``` //! //! ### Example //! //! When computing the length of two concatenated lists, //! there is a faster way, which is to compute the length of each list and add them together: //! //! ```text //! > (len . concat)(a, b) //! (len · concat)(a, b) //! (len · concat)(a)(b) //! (concat[len] · (len · fst, len · snd))(a)(b) //! (add · (len · fst, len · snd))(a)(b) //! <=> add((len · fst)(a)(b), (len · snd)(a)(b)) //! > add((len · fst)(a)(b), (len · snd)(a)(b)) //! add((len · fst)(a)(b), (len · snd)(a)(b)) //! add((len · fst)(a)(b))((len · snd)(a)(b)) //! add(len(a))((len · snd)(a)(b)) //! add(len(a))(len(b)) //! ``` //! //! ### Introduction to Poi and Path Semantics //! //! In "point-free" or "tacit" programming, functions do not identify the arguments //! (or "points") on which they operate. See [Wikipedia article](https://en.wikipedia.org/wiki/Tacit_programming). //! //! Poi is an implementation of a small subset of [Path Semantics](https://github.com/advancedresearch/path_semantics). //! In order to explain how Poi works, one needs to explain a bit about Path Semantics. //! //! [Path Semantics](https://github.com/advancedresearch/path_semantics) is an extremely expressive language for mathematical programming, //! which has a "path-space" in addition to normal computation. //! If normal programming is 2D, then Path Semantics is 3D. //! Path Semantics is often used in combination with [Category Theory](https://en.wikipedia.org/wiki/Category_theory), [Logic](https://en.wikipedia.org/wiki/Logic), etc. //! //! A "path" (or "normal path") is a way of navigating between functions, for example: //! //! ```text //! and[not] <=> or //! ``` //! //! Translated into words, this sentence means: //! //! ```text //! If you flip the input and output bits of an `and` function, //! then you can predict the output directly from the input bits //! using the function `or`. //! ``` //! //! In normal programming, there is no way to express this idea directly, //! but you can represent the logical relationship as an equation: //! //! ```text //! not(and(a, b)) = or(not(a), not(b)) //! ``` //! //! This is known as one of [De Morgan's laws](https://en.wikipedia.org/wiki/De_Morgan's_laws). //! //! When represented as a commutative diagram, one can visualize the dimensions: //! //! ```text //! not x not //! o ---------> o o -------> path-space //! | | | //! and | | or | //! V V | //! o ---------> o V //! not computation //! ``` //! //! This is written in asymmetric path notation: //! //! ```text //! and[not x not -> not] <=> or //! ``` //! //! In symmetric path notation: //! //! ```text //! and[not] <=> or //! ``` //! //! Both computation and path-space are directional, //! meaning that one can not always find the inverse. //! Composition in path-space is just function composition: //! //! ```text //! f[g][h] <=> f[h . g] //! ``` //! //! If one imagines `computation = 2D`, then `computation + path-space = 3D`. //! //! Path Semantics can be thought of as "point-free style" sub-set of equations. //! This sub-set of equations is particularly helpful in programming. //! //! ### The Problem of Complexity //! //! Efficient mathematical knowledge useful for programming depends on knowing //! the identity of functions. This means that the more knowledge you want to build, //! the more functions you need to name and refer to symbolically. //! //! This means that mathematical theories using a "birds-eye view" are not as //! useful to solve specific problems, except as a guide to find the solution. //! The more general and expressive a theory is, the harder it is to do proof search. //! //! As a consequence, theorem proving along both `computation + path-space` is //! much harder than just theorem proving for `computation`. //! //! For example, [Type Theory](https://en.wikipedia.org/wiki/Type_theory) is useful //! to check that programs are correct, but for higher categories, it becomes //! increasingly hard to ground the semantics while staying efficient and usable. //! //! [Path Semantics](https://github.com/advancedresearch/path_semantics) uses a //! different approach, which is based on symbols. //! When a symbol is created, the theory "commits" to preserving the "paths" //! from the symbol, which is known in [Homotopy Type Theory](https://homotopytypetheory.org/) to correspond to "proofs". //! Since the symbols themselves encode this relationship to proofs, //! it means that proofs can be arbitrary complex without affecting complexity. //! //! This is different from a pure axiomatic system. //! In a pure axiomatic system, the symbols do not have meaning except //! the relationship to each other (the axioms). //! As a result, you get non-standard interpretations of the Peano axioms. //! In Path Semantics, if you say "the natural numbers", you *mean* the natural numbers, not the natural numbers as described by the Peano axioms. //! The symbol "the natural numbers" *is the proof* of what you mean, //! using the background of path semantical knowledge to interpret it. //! This is what the "semantics" in Path Semantics means. //! //! It is possible to express ideas in Path Semantics which are believed to be true, //! yet can not be proven to be true in any formal language. Someday, a formal //! language might be invented to prove the sentence true, but programmers do not //! wait for this to happen. Instead, they default to pragmatic strategies, such as //! testing extensively. For example, the Goldebach conjecture has been tested up //! to some limit, so it holds for all natural numbers below that limit. //! A pragmatic strategy is what you do when you can not idealize the problem away. //! //! Poi uses a pragmatic approach in its design because a lot of proofs in //! Path Semantics requires no or little type checking in the "point-free style". //! //! ### Design of Poi //! //! Poi is designed to be used as a Rust library. //! //! It means that anybody can create their own tools on top of Poi, //! without needing a lot of dependencies. //! //! Poi uses primarily rewriting-rules for theorem proving. //! This means that the core design is "stupid" and will do dumb things like running //! in infinite loops when given the wrong rules. //! //! However, this design makes also Poi very flexible, because it can pattern match //! in any way, independent of computational direction. //! It is relatively easy to define such rules in Rust code. //! //! #### Syntax //! //! Poi uses [Piston-Meta](https://github.com/pistondevelopers/meta) to describe its syntax. Piston-Meta is a meta parsing language for human readable text documents. //! It makes it possible to easily make changes to Poi's grammar, //! and also preserve backward compatibility. //! //! Since Piston-Meta can describe its own grammar rules, it means that future //! versions of Piston-Meta can parse grammars of old versions of Poi. //! The old documents can then be transformed into new versions of Poi using synthesis. //! //! #### Core Design //! //! At the core of Poi, there is the `Expr` structure: //! //! ```rust(ignore) //! /// Function expression. //! #[derive(Clone, PartialEq, Debug)] //! pub enum Expr { //! /// A symbol that is used together with symbolic knowledge. //! Sym(Symbol), //! /// Some function that returns a value, ignoring the argument. //! /// //! /// This can also be used to store values, since zero arguments is a value. //! Ret(Value), //! /// A binary operation on functions. //! Op(Op, Box<Expr>, Box<Expr>), //! /// A tuple for more than one argument. //! Tup(Vec<Expr>), //! /// A list. //! List(Vec<Expr>), //! } //! ``` //! //! The simplicity of the `Expr` structure is important and heavily based on //! advanced path semantical knowledge. //! //! A symbol contains every domain-specific symbol and "avatar extensions" //! of symbols. An "avatar extension" is a technique of integrating information //! processing from building blocks that have no relations for introspection. //! This means, that even some variants of `Symbol` are not symbols in a direct sense, //! they are put there because they "integrate information" of symbols. //! For example, a variable is classified as a `1-avatar` since it "integrates information" of a single symbol or expression. "Avatar extensions" occur //! frequently in Path Semantics for very sophisticated mathematical relations, but usually do not need to be represented explicitly. //! Instead, they are used as a "guide" to design. //! See the paper [Avatar Graphs](https://github.com/advancedresearch/path_semantics/blob/master/papers-wip/avatar-graphs.pdf) for more information. //! //! The `Ret` variant comes from the notation used in [Higher Order Operator Overloading](https://github.com/advancedresearch/path_semantics/blob/master/sequences.md#higher-order-operator-overloading). Instead of describing a value as value, //! it is thought of as a function of some unknown input type, which returns a known value. For example, if a function returns `2` for all inputs, this is written `\2`. //! This means that point-free transformations on functions sometimes can compute stuff, without explicitly needing to reference the concrete value directly. //! See paper [Higher Order Operator Overloading and Existential Path Equations](https://github.com/advancedresearch/path_semantics/blob/master/papers-wip/higher-order-operator-overloading-and-existential-path-equations.pdf) for more information. //! //! The `Op` variant generalizes binary operators on functions, //! such as `Composition`, `Path` (normal path), //! `Apply` (call a function) and `Constrain` (partial functions). //! //! The `Tup` variant represents tuples of expressions, where a singleton (a tuple of one element) is //! "lifted up" one level. This is used e.g. to transition from `and[not x not -> not]` to `and[not]` without having to write rules for asymmetric cases. //! //! The `List` variant represents lists of expressions, e.g. `[1, 2, 3]`. //! This differs from `Tup` by the property that singletons are not "lifted up". //! //! #### Representing Knowledge //! //! In higher dimensions of functional programming, the definition of "normalization" //! depends on the domain specific use of a theory. Intuitively, since there are //! more directions, what counts as progression toward an answer is somewhat //! chosen arbitrarily. Therefore, the subjectivity of this choice must be //! reflected in the representation of knowledge. //! //! Poi's representation of knowledge is designed for multi-purposes. //! Unlike in normal programming, you do not want to always do e.g. evaluation. //! Instead, you design different tools for different purposes, using the same //! knowledge. //! //! The `Knowledge` struct represents mathematical knowledge in form of rules: //! //! ```rust(ignore) //! /// Represents knowledge about symbols. //! pub enum Knowledge { //! /// A symbol has some definition. //! Def(Symbol, Expr), //! /// A reduction from a more complex expression into another by normalization. //! Red(Expr, Expr), //! /// Two expressions that are equivalent but neither normalizes the other. //! Eqv(Expr, Expr), //! } //! ``` //! //! The `Def` variant represents a definition. //! A definition is inlined when evaluating an expression. //! //! The `Red` variant represents what counts as "normalization" in a domain specific theory. //! It can use computation in the sense of normal evaluation, or use path-space. //! This rule is directional, which means it pattern matches on the first expression //! and binds variables, which are synthesized using the second expression. //! //! The `Eqv` variant represents choices that one can make when traveling along a path. //! Going in one direction might be as good as another. //! This is used when it is not clear which direction one should go. //! This rule is bi-directional, which means one can treat it as a reduction both ways. //! use std::sync::Arc; use self::Expr::*; use self::Op::*; use self::Value::*; use self::Knowledge::*; use self::Symbol::*; pub use val::*; pub use expr::*; pub use sym::*; pub use standard_library::*; pub use parsing::*; pub use knowledge::*; mod val; mod expr; mod sym; mod knowledge; mod standard_library; mod parsing; /// Used to global import enum variants. pub mod prelude { pub use super::*; pub use super::Expr::*; pub use super::Op::*; pub use super::Value::*; pub use super::Knowledge::*; pub use super::Symbol::*; } impl Into<Expr> for bool { fn into(self) -> Expr {Ret(Bool(self))} } impl Into<Expr> for f64 { fn into(self) -> Expr {Ret(F64(self))} } impl<T, U> Into<Expr> for (T, U) where T: Into<Expr>, U: Into<Expr> { fn into(self) -> Expr {Tup(vec![self.0.into(), self.1.into()])} } impl<T0, T1, T2> Into<Expr> for (T0, T1, T2) where T0: Into<Expr>, T1: Into<Expr>, T2: Into<Expr> { fn into(self) -> Expr {Tup(vec![self.0.into(), self.1.into(), self.2.into()])} } impl Expr { /// Returns available equivalences of the expression, using a knowledge base. pub fn equivalences(&self, knowledge: &[Knowledge]) -> Vec<(Expr, usize)> { let mut ctx = Context {vars: vec![]}; let mut res = vec![]; for i in 0..knowledge.len() { if let Eqv(a, b) = &knowledge[i] { if ctx.bind(a, self) { let expr = ctx.substitute(b).unwrap(); res.push((expr, i)); ctx.vars.clear(); } else if ctx.bind(b, self) { let expr = ctx.substitute(a).unwrap(); res.push((expr, i)); ctx.vars.clear(); } } } res } /// Evaluate an expression using a knowledge base. /// /// This combines reductions and inlining of all symbols. pub fn eval(&self, knowledge: &[Knowledge]) -> Result<Expr, Error> { let mut me = self.clone(); loop { let expr = me.reduce_all(knowledge).inline_all(knowledge)?; if expr == me {break}; me = expr; } Ok(me) } /// Reduces an expression using a knowledge base, until it can not be reduces further. pub fn reduce_all(&self, knowledge: &[Knowledge]) -> Expr { let mut me = self.clone(); while let Ok((expr, _)) = me.reduce(knowledge) {me = expr} me } /// Reduces expression one step using a knowledge base. pub fn reduce(&self, knowledge: &[Knowledge]) -> Result<(Expr, usize), Error> { let mut ctx = Context {vars: vec![]}; let mut me: Option<(Expr, usize)> = None; for i in 0..knowledge.len() { if let Red(a, b) = &knowledge[i] { if ctx.bind(a, self) { let expr = ctx.substitute(b)?; me = Some((expr, i)); break; } } } match self { Op(op, a, b) => { if let Ok((a, i)) = a.reduce(knowledge) { // Prefer the reduction that matches the first rule. if let Some((expr, j)) = me {if j < i {return Ok((expr, j))}}; return Ok((Op(*op, Box::new(a), b.clone()), i)); } if let Ok((b, i)) = b.reduce(knowledge) { // Prefer the reduction that matches the first rule. if let Some((expr, j)) = me {if j < i {return Ok((expr, j))}}; return Ok((Op(*op, a.clone(), Box::new(b)), i)); } } Tup(a) => { let mut res = vec![]; for i in 0..a.len() { if let Ok((n, j)) = a[i].reduce(knowledge) { // Prefer the reduction that matches the first rule. if let Some((expr, k)) = me {if k < j {return Ok((expr, k))}}; res.push(n); res.extend(a[i+1..].iter().map(|n| n.clone())); return Ok((Tup(res), j)); } else { res.push(a[i].clone()); } } } _ => {} } if let Some((expr, i)) = me { Ok((expr, i)) } else { Err(Error::NoReductionRule) } } /// Inlines all symbols using a knowledge base. /// /// Ignores missing definitions in domain constraints. pub fn inline_all(&self, knowledge: &[Knowledge]) -> Result<Expr, Error> { match self { Sym(a) => { for i in 0..knowledge.len() { if let Def(b, c) = &knowledge[i] { if b == a { return Ok(c.clone()); } } } Err(Error::NoDefinition) } Ret(_) => Ok(self.clone()), Op(op, a, b) => { if let Constrain = op { let a = a.inline_all(knowledge)?; match b.inline_all(knowledge) { Err(Error::NoDefinition) => Ok(a), Err(err) => Err(err), Ok(b) => Ok(constr(a, b)), } } else { Ok(Op( *op, Box::new(a.inline_all(knowledge)?), Box::new(b.inline_all(knowledge)?) )) } } Tup(a) => { let mut res = vec![]; for i in 0..a.len() { res.push(a[i].inline_all(knowledge)?); } Ok(Tup(res)) } List(a) => { let mut res = vec![]; for i in 0..a.len() { res.push(a[i].inline_all(knowledge)?); } Ok(List(res)) } } } /// Inline a symbol using a knowledge base. pub fn inline(&self, sym: &Symbol, knowledge: &[Knowledge]) -> Result<Expr, Error> { match self { Sym(a) if a == sym => { for i in 0..knowledge.len() { if let Def(b, c) = &knowledge[i] { if b == a { return Ok(c.clone()); } } } Err(Error::NoDefinition) } Sym(_) | Ret(_) => Ok(self.clone()), Op(op, a, b) => { Ok(Op( *op, Box::new(a.inline(sym, knowledge)?), Box::new(b.inline(sym, knowledge)?) )) } Tup(a) => { let mut res = vec![]; for i in 0..a.len() { res.push(a[i].inline(sym, knowledge)?); } Ok(Tup(res)) } List(a) => { let mut res = vec![]; for i in 0..a.len() { res.push(a[i].inline(sym, knowledge)?); } Ok(List(res)) } } } } /// Stores variables bound by context. pub struct Context { /// Contains the variables in the context. pub vars: Vec<(Arc<String>, Expr)>, } impl Context { /// Binds patterns of a `name` expression to a `value` expression. pub fn bind(&mut self, name: &Expr, value: &Expr) -> bool { match (name, value) { (Sym(NoConstrVar(_)), Op(Constrain, _, _)) => { self.vars.clear(); false } (Sym(Var(_)), Tup(_)) | (Sym(NoConstrVar(_)), Tup(_)) => { self.vars.clear(); false } (Sym(Var(name)), x) | (Sym(NoConstrVar(name)), x) => { for i in (0..self.vars.len()).rev() { if &self.vars[i].0 == name { if &self.vars[i].1 == x { break } else { self.vars.clear(); return false; } } } self.vars.push((name.clone(), x.clone())); true } (Sym(RetVar(name)), Ret(_)) => { for i in (0..self.vars.len()).rev() { if &self.vars[i].0 == name { if &self.vars[i].1 == value { break } else { self.vars.clear(); return false; } } } self.vars.push((name.clone(), value.clone())); true } (Sym(HeadTail(head, tail)), Tup(list)) => { if list.len() < 2 {return false}; let r = self.bind(head, &list[0]); let b: Expr = if list[1..].len() == 1 { list[1].clone() } else { Tup(list[1..].into()) }; let r = r && self.bind(tail, &b); if !r {self.vars.clear()}; r } (Sym(Any), _) => true, (Sym(a), Sym(b)) if a == b => true, (Ret(a), Ret(b)) if a == b => true, (Op(op1, a1, b1), Op(op2, a2, b2)) if op1 == op2 => { let r = self.bind(a1, a2) && self.bind(b1, b2); if !r {self.vars.clear()}; r } (Tup(a), Tup(b)) if a.len() == b.len() => { let mut all = true; for i in 0..a.len() { let r = self.bind(&a[i], &b[i]); if !r { all = false; break; } } if !all {self.vars.clear()}; all } (List(a), List(b)) if a.len() == b.len() => { let mut all = true; for i in 0..a.len() { let r = self.bind(&a[i], &b[i]); if !r { all = false; break; } } if !all {self.vars.clear()}; all } _ => { self.vars.clear(); false } } } /// Substitute free occurences of variables in context. /// /// This is used on the right side in a reduction rule. pub fn substitute(&self, x: &Expr) -> Result<Expr, Error> { match x { Sym(Var(name)) => { for i in (0..self.vars.len()).rev() { if &self.vars[i].0 == name { return Ok(self.vars[i].1.clone()) } } Err(Error::CouldNotFind(name.clone())) } Sym(UnopRetVar(a, f)) => { let mut av: Option<Expr> = None; for i in (0..self.vars.len()).rev() { if &self.vars[i].0 == a { av = Some(self.vars[i].1.clone()); } } match av { Some(Ret(F64(a))) => { Ok(match **f { Neg => Ret(F64(-a)), _ => return Err(Error::InvalidComputation), }) } Some(List(a)) => { Ok(match **f { Len => Ret(F64(a.len() as f64)), _ => return Err(Error::InvalidComputation), }) } _ => Err(Error::CouldNotFind(a.clone())), } } Sym(BinopRetVar(a, b, f)) => { let mut av: Option<Expr> = None; let mut bv: Option<Expr> = None; for i in (0..self.vars.len()).rev() { if &self.vars[i].0 == a { av = Some(self.vars[i].1.clone()); } if &self.vars[i].0 == b { bv = Some(self.vars[i].1.clone()); } } match (av, bv) { (Some(Ret(a)), Some(Ret(b))) if **f == Eq => Ok(Ret(Bool(a == b))), (Some(Ret(F64(a))), Some(Ret(F64(b)))) => { Ok(Ret(F64(match **f { Lt => return Ok(Ret(Bool(a < b))), Le => return Ok(Ret(Bool(a <= b))), Gt => return Ok(Ret(Bool(a > b))), Ge => return Ok(Ret(Bool(a >= b))), Add => a + b, Sub => a - b, Mul => a * b, Div => if b == 0.0 { return Err(Error::InvalidComputation) } else { a / b } _ => return Err(Error::InvalidComputation), }))) } (Some(List(a)), Some(List(b))) => { Ok(match **f { Concat => { let mut a = a.clone(); a.extend(b.iter().map(|n| n.clone())); List(a) } _ => return Err(Error::InvalidComputation), }) } (av, _) => { if av.is_none() { Err(Error::CouldNotFind(a.clone())) } else { Err(Error::CouldNotFind(b.clone())) } } } } Sym(_) | Ret(_) => Ok(x.clone()), Op(op, a, b) => { Ok(Op(*op, Box::new(self.substitute(a)?), Box::new(self.substitute(b)?))) } Tup(a) => { let mut res = vec![]; for i in 0..a.len() { res.push(self.substitute(&a[i])?); } Ok(Tup(res)) } List(a) => { let mut res = vec![]; for i in 0..a.len() { res.push(self.substitute(&a[i])?); } Ok(List(res)) } } } } /// Represents an error. #[derive(Debug, PartialEq)] pub enum Error { /// Invalid function for computing something from left side of expression to right side. InvalidComputation, /// There was no defintion of the symbol. NoDefinition, /// There was no matching reduction rule. NoReductionRule, /// Could not find variable. CouldNotFind(Arc<String>), } /// Binary operation on functions. #[derive(Clone, Copy, PartialEq, Eq, Debug)] pub enum Op { /// Function composition `f . g` Compose, /// Path `f[g]` Path, /// Apply function to some argument. Apply, /// Constrain function input. Constrain, } impl Into<Expr> for Symbol { fn into(self) -> Expr {Sym(self)} } impl Into<Expr> for &'static str { fn into(self) -> Expr {Sym(Var(Arc::new(self.into())))} } impl Into<Symbol> for &'static str { fn into(self) -> Symbol {Var(Arc::new(self.into()))} } /// A function applied to some argument. pub fn app<A: Into<Expr>, B: Into<Expr>>(a: A, b: B) -> Expr { Op(Apply, Box::new(a.into()), Box::new(b.into())) } /// A function composition. pub fn comp<A: Into<Expr>, B: Into<Expr>>(a: A, b: B) -> Expr { Op(Compose, Box::new(a.into()), Box::new(b.into())) } /// A normal path expression. pub fn path<A: Into<Expr>, B: Into<Expr>>(a: A, b: B) -> Expr { Op(Path, Box::new(a.into()), Box::new(b.into())) } /// A function domain constraint. pub fn constr<A: Into<Expr>, B: Into<Expr>>(a: A, b: B) -> Expr { Op(Constrain, Box::new(a.into()), Box::new(b.into())) } /// An `if` expression. pub fn _if<A: Into<Expr>, B: Into<Expr>>(a: A, b: B) -> Expr {app(app(If, a), b)} /// A head-tail pattern match on a tuple. pub fn head_tail<A: Into<Expr>, B: Into<Expr>>(a: A, b: B) -> Expr { HeadTail(Box::new(a.into()), Box::new(b.into())).into() } /// A value variable. pub fn ret_var<A: Into<String>>(a: A) -> Expr {Sym(RetVar(Arc::new(a.into())))} /// Compute a binary function. pub fn binop_ret_var<A: Into<String>, B: Into<String>, F: Into<Symbol>>(a: A, b: B, f: F) -> Expr { Sym(BinopRetVar(Arc::new(a.into()), Arc::new(b.into()), Box::new(f.into()))) } /// Compute a unary function. pub fn unop_ret_var<A: Into<String>, F: Into<Symbol>>(a: A, f: F) -> Expr { Sym(UnopRetVar(Arc::new(a.into()), Box::new(f.into()))) } /// A function without domain constraints. pub fn no_constr<A: Into<String>>(a: A) -> Expr { Sym(NoConstrVar(Arc::new(a.into()))) } /// Knowledge about a commuative binary operator. pub fn commutative<S: Into<Symbol>>(s: S) -> Knowledge { let s = s.into(); let a: Expr = "a".into(); let b: Expr = "b".into(); Eqv(app(app(s.clone(), a.clone()), b.clone()), app(app(s, b), a)) } /// Knowledge about an associative binary operator. pub fn associative<S: Into<Symbol>>(s: S) -> Knowledge { let s = s.into(); let a: Expr = "a".into(); let b: Expr = "b".into(); let c: Expr = "c".into(); Eqv(app(app(s.clone(), a.clone()), app(app(s.clone(), b.clone()), c.clone())), app(app(s.clone(), app(app(s, a), b)), c)) } /// Knowledge about a distributive relationship. pub fn distributive<M: Into<Symbol>, A: Into<Symbol>>(mul: M, add: A) -> Knowledge { let mul = mul.into(); let add = add.into(); let a: Expr = "a".into(); let b: Expr = "b".into(); let c: Expr = "c".into(); Eqv(app(app(mul.clone(), a.clone()), app(app(add.clone(), b.clone()), c.clone())), app(app(add, app(app(mul.clone(), a.clone()), b)), app(app(mul, a), c))) } #[cfg(test)] mod tests { use super::*; #[test] fn apply_not() { let ref std = std(); let a = app(Not, true); let a = a.inline(&Not, std).unwrap(); let a = a.reduce(std).unwrap().0; assert_eq!(a, false.into()); let a = app(Not, false); let a = a.inline(&Not, std).unwrap(); let a = a.reduce(std).unwrap().0; assert_eq!(a, true.into()); } #[test] fn comp_not_not() { let ref std = std(); let a = comp(Not, Not); let a = a.reduce(std).unwrap().0; assert_eq!(a, Idb.into()); } #[test] fn path_not_not() { let ref std = std(); let a = path(Not, Not); let a = a.reduce(std).unwrap().0; assert_eq!(a, Not.into()); } #[test] fn comp_id() { let ref std = std(); let a = comp(Not, Id); let a = a.reduce(std).unwrap().0; assert_eq!(a, Not.into()); let a = comp(Id, Not); let a = a.reduce(std).unwrap().0; assert_eq!(a, Not.into()); } #[test] fn path_not_id() { let ref std = std(); let a = path(Not, Id); let a = a.reduce(std).unwrap().0; assert_eq!(a, Not.into()); } #[test] fn red_singleton() { let ref std = std(); let a = Tup(vec![true.into()]); let a = a.reduce(std).unwrap().0; assert_eq!(a, true.into()); } }