104 lines
3.9 KiB
Markdown
104 lines
3.9 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 : Nat
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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 : Nat) : Nat := 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 `Nat -> Nat`.
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```lean
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check max 1
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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 : Nat) : Nat := 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 `set::option` 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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set::option pp::implicit true
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set::option pp::notation false
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check 1 = 2
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-- restore default configuration
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set::option pp::implicit false
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set::option pp::notation true
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check 1 = 2
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```
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