115 lines
4.3 KiB
Markdown
115 lines
4.3 KiB
Markdown
# Expressions
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Lean is based on dependent type theory, and is very similar to the one
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used in the [Boole](https://github.com/avigad/boole) and
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[Coq](http://coq.inria.fr/) systems. In contrast to Coq, Lean is
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classical.
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In Lean, we have the following kind of expressions: _constants_,
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,_function applications_, _(heterogeneous) equality_, _local variables_,
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_lambdas_, _dependent function spaces_ (aka _Pis_), _let expressions_,
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and _Types_.
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## Constants
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Constants are essentially references to variable declarations, definitions, axioms and theorems in the
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environment. In the following example, we use the command `Variables` to declare `x` and `y` as integers.
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The `Check` command displays the type of the given expression. The `x` and `y` in the `Check` command
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are constants. They reference the objects declared using the command `Variables`.
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```lean
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Variables x y : Int.
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Check x + y.
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```
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In the following example, we define the constant `s` as the sum of `x` and `y` using the `Definition` command.
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The `Eval` command evaluates (normalizes) the expression `s + 1`. In this example, `Eval` will just expand
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the definition of `s`, and return `x + y + 1`.
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```lean
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Definition s := x + y.
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Eval s + 1.
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```
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## Function applications
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In Lean, the expression `f t` is a function application, where `f` is a function that is applied to `t`.
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In the following example, we define the function `max`. The `Eval` command evaluates the application `max 1 2`,
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and returns the value `2`. Note that, the expression `if (x >= y) x y` is also a function application.
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```lean
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Definition max (x y : Int) : Int := if (x >= y) x y.
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Eval max 1 2.
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```
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The expression `max 1` is also a valid expression in Lean, and it has type `Int -> Int`.
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```lean
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Check max 1.
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```
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Remark: we can make the expression `if (x >= y) x y` more "user-friendly" by using custom "Notation".
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The following `Notation` command defines the usual `if-then-else` expression. The value `40` is the precedence
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of the new notation.
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```lean
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Notation 40 if _ then _ else _ : if
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Check if x >= y then x else y.
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```
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In the following command, we define the function `inc`, and evaluate some expressions using `inc` and `max`.
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```lean
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Definition inc (x : Int) : Int := x + 1.
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Eval inc (inc (inc 2)).
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Eval max (inc 2) 2 = 3.
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```
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## Heterogeneous equality
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For technical reasons, in Lean, we have heterogenous and homogeneous equality. In a nutshell, heterogenous is mainly used internally, and
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homogeneous are used externally when writing programs and specifications in Lean.
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Heterogenous equality allows us to compare elements of any type, and homogenous equality is just a definition on top of the heterogenous equality that expects arguments of the same type.
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The expression `t == s` is a heterogenous equality that is true iff `t` and `s` have the same interpretation.
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In the following example, we evaluate the expressions `1 == 2` and `2 == 2`. The first evaluates to `false` and the second to `true`.
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```lean
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Eval 1 == 2.
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Eval 2 == 2.
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```
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Since we can compare elements of different types, the following expression is type correct and evaluates to `false`.
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```lean
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Eval 1 == true.
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```
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This is consistent with the set theoretic semantics used in Lean, where the interpretation of all expressions are sets.
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The interpretation of heterogeneous equality is just set equality in the Lean seamtics.
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We strongly discourage users from directly using heterogeneous equality. The main problem is that it is very easy to
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write expressions that are false like the one above. The expression `t = s` is a homogeneous equality.
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It expects `t` and `s` to have the same type. Thus, the expression `1 = true` is type incorrect in Lean.
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The symbol `=` is just notation. Internally, homogeneous equality is defined as:
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```
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Definition eq {A : (Type U)} (x y : A) : Bool := x == y.
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Infix 50 = : eq.
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```
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The curly braces indicate that the first argument of `eq` is implicit. The symbol `=` is just a syntax sugar for `eq`.
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In the following example, we use the `SetOption` command to force Lean to display implicit arguments and
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disable notation when pretty printing expressions.
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```lean
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SetOption pp::implicit true.
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SetOption pp::notation false.
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Check 1 = 2.
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(* restore default configuration *)
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SetOption pp::implicit false.
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SetOption pp::notation true.
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Check 1 = 2.
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```
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Note that, like the SML programming language, `(* comment *)` is a comment.
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