From 83d8006a7cc3d5429bb5e2d0abcb3da7ea4a9035 Mon Sep 17 00:00:00 2001 From: zihang Date: Wed, 4 Dec 2024 11:20:42 +0800 Subject: [PATCH] doc(example): extract code for lambda --- next/locales/zh_CN/LC_MESSAGES/tutorial.po | 136 ++++++------ next/sources/lambda-expression/.gitignore | 2 + next/sources/lambda-expression/LICENSE | 202 ++++++++++++++++++ next/sources/lambda-expression/README.md | 3 + next/sources/lambda-expression/moon.mod.json | 10 + .../lambda-expression/src/moon.pkg.json | 1 + next/sources/lambda-expression/src/top.mbt | 120 +++++++++++ next/tutorial/example/lambda/index.md | 162 +++++--------- 8 files changed, 453 insertions(+), 183 deletions(-) create mode 100644 next/sources/lambda-expression/.gitignore create mode 100644 next/sources/lambda-expression/LICENSE create mode 100644 next/sources/lambda-expression/README.md create mode 100644 next/sources/lambda-expression/moon.mod.json create mode 100644 next/sources/lambda-expression/src/moon.pkg.json create mode 100644 next/sources/lambda-expression/src/top.mbt diff --git a/next/locales/zh_CN/LC_MESSAGES/tutorial.po b/next/locales/zh_CN/LC_MESSAGES/tutorial.po index 7244da76..a074b75b 100644 --- a/next/locales/zh_CN/LC_MESSAGES/tutorial.po +++ b/next/locales/zh_CN/LC_MESSAGES/tutorial.po @@ -9,7 +9,7 @@ msgid "" msgstr "" "Project-Id-Version: MoonBit Document \n" "Report-Msgid-Bugs-To: \n" -"POT-Creation-Date: 2024-12-03 16:21+0800\n" +"POT-Creation-Date: 2024-12-04 11:18+0800\n" "PO-Revision-Date: YEAR-MO-DA HO:MI+ZONE\n" "Last-Translator: FULL NAME \n" "Language: zh_CN\n" @@ -2614,18 +2614,18 @@ msgid "" "}\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:27 +#: ../../tutorial/example/lambda/index.md:25 msgid "" "Concepts encountered in daily programming such as boolean values, if " "expressions, natural number arithmetic, and even recursion can all be " "implemented using Lambda expressions. However, this is not the focus of " -"this article. Interested readers can refer to:" +"this article." msgstr "" -#: ../../tutorial/example/lambda/index.md:29 +#: ../../tutorial/example/lambda/index.md:28 msgid "" -"Programming with Nothing：[https://tomstu.art/programming-with-" -"nothing](https://tomstu.art/programming-with-nothing)" +"Interested readers can refer to: [Programming with " +"Nothing](https://tomstu.art/programming-with-nothing)" msgstr "" #: ../../tutorial/example/lambda/index.md:31 @@ -2711,7 +2711,7 @@ msgid "" "variables:" msgstr "" -#: ../../tutorial/example/lambda/index.md:67 +#: ../../tutorial/example/lambda/index.md:69 msgid "" "// (λx.E) T => E.subst(x, T)\n" "fn subst(self : Term, var : String, term : Term) -> Term {\n" @@ -2731,31 +2731,31 @@ msgid "" "}\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:86 +#: ../../tutorial/example/lambda/index.md:88 msgid "" "Next, I'll introduce a less popular but somewhat convenient method: de " "Bruijn index." msgstr "" -#: ../../tutorial/example/lambda/index.md:88 +#: ../../tutorial/example/lambda/index.md:90 msgid "De Bruijn Index" msgstr "" -#: ../../tutorial/example/lambda/index.md:90 +#: ../../tutorial/example/lambda/index.md:92 msgid "" "De Bruijn index is a technique for representing variables in Lambda terms" " using integers. Specifically, it replaces specific variables with " "Lambdas between the variable and its original imported position." msgstr "" -#: ../../tutorial/example/lambda/index.md:92 +#: ../../tutorial/example/lambda/index.md:94 msgid "" "λx.(λy.x (λz.z z))\n" "\n" "λ.(λ.1 (λ.0 0))\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:98 +#: ../../tutorial/example/lambda/index.md:100 msgid "" "In the example above, there is one Lambda `λy` between the variable `x` " "and its introduction position `λx`, so `x` is replaced with `1`. For " @@ -2766,19 +2766,19 @@ msgid "" " by the number of nested Lambdas." msgstr "" -#: ../../tutorial/example/lambda/index.md:100 +#: ../../tutorial/example/lambda/index.md:102 msgid "" "The same variable may be replaced with different integers in different " "positions." msgstr "" -#: ../../tutorial/example/lambda/index.md:102 +#: ../../tutorial/example/lambda/index.md:104 msgid "" "We define a new type `TermDBI` to represent Lambda terms using de Bruijn " "indices:" msgstr "" -#: ../../tutorial/example/lambda/index.md:104 +#: ../../tutorial/example/lambda/index.md:106 msgid "" "enum TermDBI {\n" " Var(String, Int)\n" @@ -2793,22 +2793,22 @@ msgid "" "form is painful, so we need to write a function `bruijn()` to convert " "`Term` to `TermDBI`. This is also why there is still a `String` in the " "definition of the `TermDBI` type, so that the original variable name can " -"be used for its `to_string` method, making it easy to print and view the " -"evaluation results with `println`." +"be used for its `Show` implementation, making it easy to print and view " +"the evaluation results with `println`." msgstr "" #: ../../tutorial/example/lambda/index.md:114 msgid "" -"fn to_string(self : TermDBI) -> String {\n" +"impl Show for TermDBI with output(self : TermDBI, logger) -> Unit {\n" " match self {\n" -" Var(name, _) => name\n" -" Abs(name, body) => \"(\\\\\\{name}.\\{body})\"\n" -" App(t1, t2) => \"\\{t1} \\{t2}\"\n" +" Var(name, _) => logger.write_string(name)\n" +" Abs(name, body) => logger.write_string(\"(\\\\\\{name}.\\{body})\")\n" +" App(t1, t2) => logger.write_string(\"\\{t1} \\{t2}\")\n" " }\n" "}\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:124 +#: ../../tutorial/example/lambda/index.md:121 msgid "" "To simplify implementation, if the input `Term` contains free variables, " "the `bruijn()` function will report an error directly. MoonBit provides a" @@ -2817,21 +2817,21 @@ msgid "" "respectively." msgstr "" -#: ../../tutorial/example/lambda/index.md:126 +#: ../../tutorial/example/lambda/index.md:123 msgid "Readers familiar with Rust should find this familiar." msgstr "" -#: ../../tutorial/example/lambda/index.md:128 +#: ../../tutorial/example/lambda/index.md:126 msgid "fn bruijn(self : Term) -> Result[TermDBI, String]\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:132 +#: ../../tutorial/example/lambda/index.md:130 msgid "" "We take a clumsy approach to save variable names and their associated " "nesting depth. First, we define the `Index` type:" msgstr "" -#: ../../tutorial/example/lambda/index.md:134 +#: ../../tutorial/example/lambda/index.md:132 msgid "" "struct Index {\n" " name : String\n" @@ -2839,41 +2839,41 @@ msgid "" "}\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:141 +#: ../../tutorial/example/lambda/index.md:138 msgid "" "Then we write a helper function to find the corresponding `depth` based " "on a specific `name` from `List[Index]`:" msgstr "" -#: ../../tutorial/example/lambda/index.md:143 +#: ../../tutorial/example/lambda/index.md:140 msgid "" -"// Find the first depth corresponding to varname in the environment\n" +"// Find the depth corresponding to the first varname in the environment\n" "fn find(map : @immut/list.T[Index], varname : String) -> Result[Int, " "String] {\n" " match map {\n" -" Nil => Err(varname) // abort(\"variable '(varname)' not found\")\n" -" Cons(i, rest) => {\n" +" Nil => Err(varname)\n" +" Cons(i, rest) =>\n" " if i.name == varname {\n" " Ok(i.depth)\n" " } else {\n" " find(rest, varname)\n" " }\n" -" }\n" " }\n" "}\n" +"\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:159 +#: ../../tutorial/example/lambda/index.md:147 msgid "Now we can complete the `bruijn()` function." msgstr "" -#: ../../tutorial/example/lambda/index.md:161 +#: ../../tutorial/example/lambda/index.md:149 msgid "" "Handling `Var` is the simplest, just look up the table to find the " "corresponding `depth`." msgstr "" -#: ../../tutorial/example/lambda/index.md:162 +#: ../../tutorial/example/lambda/index.md:150 msgid "" "`Abs` is a bit more complicated. First, add one to the `depth` of all " "`index` in the list (because the Lambda nesting depth has increased by " @@ -2881,13 +2881,13 @@ msgid "" "the list." msgstr "" -#: ../../tutorial/example/lambda/index.md:163 +#: ../../tutorial/example/lambda/index.md:151 msgid "" "`App` succeeds when both sub-items can be converted; otherwise, it " "returns an `Err`." msgstr "" -#: ../../tutorial/example/lambda/index.md:165 +#: ../../tutorial/example/lambda/index.md:153 msgid "" "fn go(m : @immut/list.T[Index], t : Term) -> Result[TermDBI, String] {\n" " match t {\n" @@ -2899,53 +2899,55 @@ msgid "" " }\n" " }\n" " Abs(varname, body) => {\n" -" let m = m.map(fn (index){\n" -" { name : index.name, depth : index.depth + 1 }\n" -" }).cons({ name : varname, depth : 0 })\n" +" let m = @immut/list.Cons(\n" +" { name: varname, depth: 0 },\n" +" m.map(fn(index) { { name: index.name, depth: index.depth + 1 } " +"}),\n" +" )\n" " let res = go(m, body)\n" " match res {\n" " Err(name) => Err(name)\n" " Ok(term) => Ok(Abs(varname, term))\n" " }\n" " }\n" -" App(e1, e2) => {\n" +" App(e1, e2) =>\n" " match (go(m, e1), go(m, e2)) {\n" " (Ok(e1), Ok(e2)) => Ok(App(e1, e2))\n" " (Err(name), _) => Err(name)\n" " (_, Err(name)) => Err(name)\n" " }\n" -" }\n" " }\n" "}\n" +"\n" "go(Nil, self)\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:197 +#: ../../tutorial/example/lambda/index.md:160 msgid "Reduce on TermDBI" msgstr "" -#: ../../tutorial/example/lambda/index.md:199 +#: ../../tutorial/example/lambda/index.md:162 msgid "Reduction mainly deals with App, i.e., calls:" msgstr "" -#: ../../tutorial/example/lambda/index.md:201 +#: ../../tutorial/example/lambda/index.md:164 msgid "" "fn eval(self : TermDBI) -> TermDBI {\n" " match self {\n" -" App(t1, t2) => {\n" +" App(t1, t2) =>\n" " match (eval(t1), eval(t2)) {\n" " (Abs(_, t1), t2) => eval(subst(t1, t2))\n" " (t1, t2) => App(t1, t2)\n" " }\n" -" }\n" " Abs(_) => self\n" -" otherwise => abort(\"eval(): \\{otherwise} \")\n" -" // eval should not encounter free variables\n" +" // Eval should not encounter any free variables\n" +" _ => panic()\n" " }\n" "}\n" +"\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:217 +#: ../../tutorial/example/lambda/index.md:170 msgid "" "First, attempt reduction on both sub-items, then see if `eval(t1)` " "results in a Lambda. If so, perform one step of variable substitution " @@ -2953,7 +2955,7 @@ msgid "" "(`Abs`), simply return them as they are." msgstr "" -#: ../../tutorial/example/lambda/index.md:219 +#: ../../tutorial/example/lambda/index.md:172 msgid "" "The implementation of the subst function becomes much simpler when we " "don't need to consider free variables. We just need to keep track of the " @@ -2961,41 +2963,35 @@ msgid "" "If they match, it's the variable to be replaced." msgstr "" -#: ../../tutorial/example/lambda/index.md:221 +#: ../../tutorial/example/lambda/index.md:174 msgid "" "fn subst(t1 : TermDBI, t2 : TermDBI) -> TermDBI {\n" " fn go(t1 : TermDBI, t2 : TermDBI, depth : Int) -> TermDBI {\n" " match t1 {\n" -" Var(name, d) => {\n" -" if d == depth {\n" -" t2\n" -" } else {\n" -" t1\n" -" }\n" -" }\n" -" Abs(name, t) => {\n" -" Abs(name, go(t, t2, depth + 1))\n" -" }\n" -" App(tl, tr) => {\n" -" App(go(tl, t2, depth), go(tr, t2, depth))\n" -" }\n" +" Var(_, d) => if d == depth { t2 } else { t1 }\n" +" Abs(name, t) => Abs(name, go(t, t2, depth + 1))\n" +" App(tl, tr) => App(go(tl, t2, depth), go(tr, t2, depth))\n" " }\n" " }\n" +"\n" " go(t1, t2, 0)\n" "}\n" +"\n" msgstr "" -#: ../../tutorial/example/lambda/index.md:244 +#: ../../tutorial/example/lambda/index.md:180 msgid "" -"The full code: " -"[try.moonbitlang.com/#b52b528b](https://try.moonbitlang.com/#b52b528b)" +"The full code: [GitHub repository](https://github.com/moonbitlang" +"/moonbit-docs/tree/main/next/sources/lambda-expression/src/top.mbt)" msgstr "" +"完整代码请查看 [GitHub 代码仓库](https://github.com/moonbitlang/moonbit-" +"docs/tree/main/next/sources/lambda-expression/src/top.mbt)。" -#: ../../tutorial/example/lambda/index.md:246 +#: ../../tutorial/example/lambda/index.md:182 msgid "Improvement" msgstr "" -#: ../../tutorial/example/lambda/index.md:248 +#: ../../tutorial/example/lambda/index.md:184 msgid "" "When mapping variable names to indices, we used the " "`@immut/list.T[Index]` type and updated the entire list every time we " diff --git a/next/sources/lambda-expression/.gitignore b/next/sources/lambda-expression/.gitignore new file mode 100644 index 00000000..b1283a74 --- /dev/null +++ b/next/sources/lambda-expression/.gitignore @@ -0,0 +1,2 @@ +target/ +.mooncakes/ diff --git a/next/sources/lambda-expression/LICENSE b/next/sources/lambda-expression/LICENSE new file mode 100644 index 00000000..d6456956 --- /dev/null +++ b/next/sources/lambda-expression/LICENSE @@ -0,0 +1,202 @@ + + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. 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@@ +# moonbit-community/lambda + +Check `tutorial/example/lambda`. \ No newline at end of file diff --git a/next/sources/lambda-expression/moon.mod.json b/next/sources/lambda-expression/moon.mod.json new file mode 100644 index 00000000..5f140707 --- /dev/null +++ b/next/sources/lambda-expression/moon.mod.json @@ -0,0 +1,10 @@ +{ + "name": "moonbit-community/lambda", + "version": "0.1.0", + "readme": "README.md", + "repository": "", + "license": "Apache-2.0", + "keywords": [], + "description": "", + "source": "src" +} \ No newline at end of file diff --git a/next/sources/lambda-expression/src/moon.pkg.json b/next/sources/lambda-expression/src/moon.pkg.json new file mode 100644 index 00000000..9e26dfee --- /dev/null +++ b/next/sources/lambda-expression/src/moon.pkg.json @@ -0,0 +1 @@ +{} \ No newline at end of file diff --git a/next/sources/lambda-expression/src/top.mbt b/next/sources/lambda-expression/src/top.mbt new file mode 100644 index 00000000..45f285e4 --- /dev/null +++ b/next/sources/lambda-expression/src/top.mbt @@ -0,0 +1,120 @@ +enum Term { + Var(String) // Variable + Abs(String, Term) // Define lambda, variables represented by strings + App(Term, Term) // Call lambda +} + +struct Index { + name : String + depth : Int +} + +enum TermDBI { + Var(String, Int) + Abs(String, TermDBI) + App(TermDBI, TermDBI) +} + +impl Show for TermDBI with output(self : TermDBI, logger) -> Unit { + match self { + Var(name, _) => logger.write_string(name) + Abs(name, body) => logger.write_string("(\\\{name}.\{body})") + App(t1, t2) => logger.write_string("\{t1} \{t2}") + } +} + +/// https://bor0.wordpress.com/2019/03/19/writing-a-lambda-calculus-evaluator-in-haskell/ +fn bruijn(self : Term) -> Result[TermDBI, String] { + // Find the depth corresponding to the first varname in the environment + fn find(map : @immut/list.T[Index], varname : String) -> Result[Int, String] { + match map { + Nil => Err(varname) + Cons(i, rest) => + if i.name == varname { + Ok(i.depth) + } else { + find(rest, varname) + } + } + } + + fn go(m : @immut/list.T[Index], t : Term) -> Result[TermDBI, String] { + match t { + Var(name) => { + let idx = find(m, name) + match idx { + Err(name) => Err(name) + Ok(idx) => Ok(Var(name, idx)) + } + } + Abs(varname, body) => { + let m = @immut/list.Cons( + { name: varname, depth: 0 }, + m.map(fn(index) { { name: index.name, depth: index.depth + 1 } }), + ) + let res = go(m, body) + match res { + Err(name) => Err(name) + Ok(term) => Ok(Abs(varname, term)) + } + } + App(e1, e2) => + match (go(m, e1), go(m, e2)) { + (Ok(e1), Ok(e2)) => Ok(App(e1, e2)) + (Err(name), _) => Err(name) + (_, Err(name)) => Err(name) + } + } + } + + go(Nil, self) +} + +fn subst(t1 : TermDBI, t2 : TermDBI) -> TermDBI { + fn go(t1 : TermDBI, t2 : TermDBI, depth : Int) -> TermDBI { + match t1 { + Var(_, d) => if d == depth { t2 } else { t1 } + Abs(name, t) => Abs(name, go(t, t2, depth + 1)) + App(tl, tr) => App(go(tl, t2, depth), go(tr, t2, depth)) + } + } + + go(t1, t2, 0) +} + +fn eval(self : TermDBI) -> TermDBI { + match self { + App(t1, t2) => + match (eval(t1), eval(t2)) { + (Abs(_, t1), t2) => eval(subst(t1, t2)) + (t1, t2) => App(t1, t2) + } + Abs(_) => self + // Eval should not encounter any free variables + _ => panic() + } +} + +test { + // (\x.x x) (\y.y) + let exp : Term = App(Abs("x", App(Var("x"), Var("x"))), Abs("x", Var("x"))) + let exp = exp.bruijn() + match exp { + Err(name) => fail!("err on \{name}\n") + Ok(t) => { + inspect!( + t, + content= + #|(\x.x x) (\x.x) + , + ) + let t = eval(t) + inspect!( + t, + content= + #|(\x.x) + , + ) + } + } +} diff --git a/next/tutorial/example/lambda/index.md b/next/tutorial/example/lambda/index.md index f68f9923..e9d0fddd 100644 --- a/next/tutorial/example/lambda/index.md +++ b/next/tutorial/example/lambda/index.md @@ -16,17 +16,17 @@ If we replace `x x` with `x(x)`, it might be more in line with the function call The combination of the above two expressions and the variables introduced when defining Lambdas are collectively referred to as the Lambda term. In MoonBit, we use an enum type to represent it: -```rust -enum Term { - Var(String) // Variable - Abs(String, Term) // Define lambda, variables represented by strings - App(Term, Term) // Call lambda -} +```{literalinclude} /sources/lambda-expression/src/top.mbt +:language: moonbit +:start-at: enum Term +:end-at: } ``` -Concepts encountered in daily programming such as boolean values, if expressions, natural number arithmetic, and even recursion can all be implemented using Lambda expressions. However, this is not the focus of this article. Interested readers can refer to: +Concepts encountered in daily programming such as boolean values, if expressions, natural number arithmetic, and even recursion can all be implemented using Lambda expressions. However, this is not the focus of this article. -- Programming with Nothing：[https://tomstu.art/programming-with-nothing](https://tomstu.art/programming-with-nothing) +```{seealso} +Interested readers can refer to: [Programming with Nothing](https://tomstu.art/programming-with-nothing) +``` To implement an interpreter for untyped Lambda calculus, the basic things we need to understand are just two rules: Alpha conversion and Beta reduction. @@ -34,13 +34,13 @@ To implement an interpreter for untyped Lambda calculus, the basic things we nee **Beta reduction** focuses on handling Lambda calls. Let's take an example: -```racket +``` (λx.(λy.x)) (λs.(λz.z)) ``` In untyped Lambda calculus, all that needs to be done after calling a Lambda is to substitute the parameter. In the example above, we need to replace the variable `x` with `(λs.(λz.z))`, resulting in: -```racket +``` (λy.(λs.(λz.z))) ``` @@ -50,13 +50,13 @@ If a variable in a Lambda term is not defined in its context, we call it a free During Beta reduction, if the Lambda term used for variable substitution contains free variables, it may lead to a behavior called "variable capture": -```racket +``` (λx.(λy.x)) (λz.y) ``` After substitution: -```racket +``` λy.λz.y ``` @@ -64,6 +64,8 @@ The free variable in `λz.y` is treated as a parameter of some Lambda, which is A common solution to the variable capture problem when writing interpreters is to traverse the expression before substitution to obtain a set of free variables. When encountering an inner Lambda during substitution, check if the variable name is in this set of free variables: + + ```moonbit // (λx.E) T => E.subst(x, T) fn subst(self : Term, var : String, term : Term) -> Term { @@ -89,7 +91,7 @@ Next, I'll introduce a less popular but somewhat convenient method: de Bruijn in De Bruijn index is a technique for representing variables in Lambda terms using integers. Specifically, it replaces specific variables with Lambdas between the variable and its original imported position. -```racket +``` λx.(λy.x (λz.z z)) λ.(λ.1 (λ.0 0)) @@ -101,59 +103,45 @@ In the example above, there is one Lambda `λy` between the variable `x` and its We define a new type `TermDBI` to represent Lambda terms using de Bruijn indices: -```moonbit -enum TermDBI { - Var(String, Int) - Abs(String, TermDBI) - App(TermDBI, TermDBI) -} +```{literalinclude} /sources/lambda-expression/src/top.mbt +:language: moonbit +:start-at: enum TermDBI +:end-at: } ``` -However, directly writing and reading Lambda terms in de Bruijn index form is painful, so we need to write a function `bruijn()` to convert `Term` to `TermDBI`. This is also why there is still a `String` in the definition of the `TermDBI` type, so that the original variable name can be used for its `to_string` method, making it easy to print and view the evaluation results with `println`. +However, directly writing and reading Lambda terms in de Bruijn index form is painful, so we need to write a function `bruijn()` to convert `Term` to `TermDBI`. This is also why there is still a `String` in the definition of the `TermDBI` type, so that the original variable name can be used for its `Show` implementation, making it easy to print and view the evaluation results with `println`. -```moonbit -fn to_string(self : TermDBI) -> String { - match self { - Var(name, _) => name - Abs(name, body) => "(\\\{name}.\{body})" - App(t1, t2) => "\{t1} \{t2}" - } -} +```{literalinclude} /sources/lambda-expression/src/top.mbt +:language: moonbit +:start-at: impl Show for TermDBI +:end-at: App(t1, t2) +:append: " }\n}" ``` To simplify implementation, if the input `Term` contains free variables, the `bruijn()` function will report an error directly. MoonBit provides a `Result[V, E]` type in the standard library, which has two constructors, `Ok(V)` and `Err(E)`, representing success and failure in computation, respectively. > Readers familiar with Rust should find this familiar. + ```moonbit fn bruijn(self : Term) -> Result[TermDBI, String] ``` We take a clumsy approach to save variable names and their associated nesting depth. First, we define the `Index` type: -```moonbit -struct Index { - name : String - depth : Int -} +```{literalinclude} /sources/lambda-expression/src/top.mbt +:language: moonbit +:start-at: struct Index +:end-at: } ``` Then we write a helper function to find the corresponding `depth` based on a specific `name` from `List[Index]`: -```moonbit -// Find the first depth corresponding to varname in the environment -fn find(map : @immut/list.T[Index], varname : String) -> Result[Int, String] { - match map { - Nil => Err(varname) // abort("variable '(varname)' not found") - Cons(i, rest) => { - if i.name == varname { - Ok(i.depth) - } else { - find(rest, varname) - } - } - } -} +```{literalinclude} /sources/lambda-expression/src/top.mbt +:language: moonbit +:dedent: +:start-at: // Find +:end-before: fn go ``` Now we can complete the `bruijn()` function. @@ -162,86 +150,34 @@ Now we can complete the `bruijn()` function. - `Abs` is a bit more complicated. First, add one to the `depth` of all `index` in the list (because the Lambda nesting depth has increased by one), and then add `{ name : varname, depth : 0 }` to the beginning of the list. - `App` succeeds when both sub-items can be converted; otherwise, it returns an `Err`. -```moonbit -fn go(m : @immut/list.T[Index], t : Term) -> Result[TermDBI, String] { - match t { - Var(name) => { - let idx = find(m, name) - match idx { - Err(name) => Err(name) - Ok(idx) => Ok(Var(name, idx)) - } - } - Abs(varname, body) => { - let m = m.map(fn (index){ - { name : index.name, depth : index.depth + 1 } - }).cons({ name : varname, depth : 0 }) - let res = go(m, body) - match res { - Err(name) => Err(name) - Ok(term) => Ok(Abs(varname, term)) - } - } - App(e1, e2) => { - match (go(m, e1), go(m, e2)) { - (Ok(e1), Ok(e2)) => Ok(App(e1, e2)) - (Err(name), _) => Err(name) - (_, Err(name)) => Err(name) - } - } - } -} -go(Nil, self) +```{literalinclude} /sources/lambda-expression/src/top.mbt +:language: moonbit +:start-at: fn go +:end-at: go(Nil, self) +:dedent: ``` ## Reduce on TermDBI Reduction mainly deals with App, i.e., calls: -```moonbit -fn eval(self : TermDBI) -> TermDBI { - match self { - App(t1, t2) => { - match (eval(t1), eval(t2)) { - (Abs(_, t1), t2) => eval(subst(t1, t2)) - (t1, t2) => App(t1, t2) - } - } - Abs(_) => self - otherwise => abort("eval(): \{otherwise} ") - // eval should not encounter free variables - } -} +```{literalinclude} /sources/lambda-expression/src/top.mbt +:language: moonbit +:start-at: fn eval +:end-before: test ``` First, attempt reduction on both sub-items, then see if `eval(t1)` results in a Lambda. If so, perform one step of variable substitution (via the subst function) and then continue simplifying. For Lambdas (`Abs`), simply return them as they are. The implementation of the subst function becomes much simpler when we don't need to consider free variables. We just need to keep track of the current depth recursively and compare it with the encountered variables. If they match, it's the variable to be replaced. -```moonbit -fn subst(t1 : TermDBI, t2 : TermDBI) -> TermDBI { - fn go(t1 : TermDBI, t2 : TermDBI, depth : Int) -> TermDBI { - match t1 { - Var(name, d) => { - if d == depth { - t2 - } else { - t1 - } - } - Abs(name, t) => { - Abs(name, go(t, t2, depth + 1)) - } - App(tl, tr) => { - App(go(tl, t2, depth), go(tr, t2, depth)) - } - } - } - go(t1, t2, 0) -} +```{literalinclude} /sources/lambda-expression/src/top.mbt +:language: moonbit +:start-at: fn subst +:end-before: fn eval ``` -The full code: [try.moonbitlang.com/#b52b528b](https://try.moonbitlang.com/#b52b528b) +The full code: [GitHub repository](https://github.com/moonbitlang/moonbit-docs/tree/main/next/sources/lambda-expression/src/top.mbt) ## Improvement