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Expressions
Lean is based on dependent type theory, and is very similar to the one used in the Boole and Coq systems. In contrast to Coq, Lean is classical.
In Lean, we have the following kind of expressions: constants, ,function applications, (heterogeneous) equality, local variables, lambdas, dependent function spaces (aka Pis), let expressions, and Types.
Constants
Constants are essentially references to variable declarations, definitions, axioms and theorems in the
environment. In the following example, we use the command variables to declare x and y as integers.
The check command displays the type of the given expression. The x and y in the check command
are constants. They reference the objects declared using the command variables.
variables x y : Nat
check x + y
In the following example, we define the constant s as the sum of x and y using the definition command.
The eval command evaluates (normalizes) the expression s + 1. In this example, eval will just expand
the definition of s, and return x + y + 1.
definition s := x + y
eval s + 1
Function applications
In Lean, the expression f t is a function application, where f is a function that is applied to t.
In the following example, we define the function max. The eval command evaluates the application max 1 2,
and returns the value 2. Note that, the expression if (x >= y) then x else y is also a function application.
It is notation for ite (x >= y) x y.
import if_then_else
definition max (x y : Nat) : Nat := if (x >= y) then x else y
eval max 1 2
The expression max 1 is also a valid expression in Lean, and it has type Nat -> Nat.
check max 1
In the following command, we define the function inc, and evaluate some expressions using inc and max.
definition inc (x : Nat) : Nat := x + 1
eval inc (inc (inc 2))
eval max (inc 2) 2 = 3