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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) x y is also a function application.
definition max (x y : Nat) : Nat := if (x >= y) x 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
Heterogeneous equality
For technical reasons, in Lean, we have heterogenous and homogeneous equality. In a nutshell, heterogenous is mainly used internally, and
homogeneous are used externally when writing programs and specifications in Lean.
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.
The expression t == s is a heterogenous equality that is true iff t and s have the same interpretation.
In the following example, we evaluate the expressions 1 == 2 and 2 == 2. The first evaluates to false and the second to true.
eval 1 == 2
eval 2 == 2
Since we can compare elements of different types, the following expression is type correct and evaluates to false.
eval 1 == true
This is consistent with the set theoretic semantics used in Lean, where the interpretation of all expressions are sets. The interpretation of heterogeneous equality is just set equality in the Lean seamtics.
We strongly discourage users from directly using heterogeneous equality. The main problem is that it is very easy to
write expressions that are false like the one above. The expression t = s is a homogeneous equality.
It expects t and s to have the same type. Thus, the expression 1 = true is type incorrect in Lean.
The symbol = is just notation. Internally, homogeneous equality is defined as:
definition eq {A : (Type U)} (x y : A) : Bool := x == y
infix 50 = : eq
The curly braces indicate that the first argument of eq is implicit. The symbol = is just a syntax sugar for eq.
In the following example, we use the set::option command to force Lean to display implicit arguments and
disable notation when pretty printing expressions.
set::option pp::implicit true
set::option pp::notation false
check 1 = 2
-- restore default configuration
set::option pp::implicit false
set::option pp::notation true
check 1 = 2