Leonardo de Moura d9c41e7282 doc(lean): add tutorial draft, and fix lexical documentation

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

2014-01-07 00:06:32 -08:00

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Lean Tutorial

Introduction

Lean is an automatic and interactive theorem prover. It can be used to create specifications, build mathematical libraries, and solve constraints. In this tutorial, we introduce basic concepts, the logic used in Lean, and the main commands.

Getting started

We can use Lean in interactive or batch mode. The following example just displays the message hello world.

        print "hello world"

All we have to do to run your first example is to call the lean executable with the name of the text file that contains the command above. If you saved the above command in the file hello.lean, then you just have to execute

   lean hello.lean

As a more complex example, the next example defines a function that doubles the input value, and then evaluates it on different values.

      -- defines the double function
      definition double (x : Nat) := x + x

      eval double 10
      eval double 2
      eval double 3 > 4

Every expression has a unique type in Lean. The command check returns the type of a given expression.

      check double 3
      check double

The last command returns Nat → Nat. That is, the type of double is a function from Nat to Nat, where Nat is the type of the natural numbers.

The command import loads existing libraries and extensions. For example, the following command imports the command find that searches the Lean environment using regular expressions

      import find

      find "Nat"      -- find all object that start with the prefix Nat
      check Nat::ge   -- display the signature of the Nat::ge definition

We say Nat::ge is a hierarchical name comprised of two parts: Nat and ge

The command using creates aliases based on give prefix. For example, the following command creates aliases for all objects starting with Nat

      using Nat
      check ge       -- display the signature of the Nat::ge definition

In Lean, proofs are also expressions, and theorems are essentially definitions. In the following example we prove that double x = 2 * x

      theorem double_x_eq_2x (x : Nat) : double x = 2 * x :=
        calc double x  =  x + x      :   refl (double x)
                   ... =  1*x + 1*x  :   { symm (mul::onel x) }
                   ... =  (1 + 1)*x  :   symm (distributel 1 1 x)
                   ... =  2 * x      :   { refl (1 + 1) }

In the example above, we provided the proof manually using a calculational proof style. The terms after : are proof terms. They justify the equalities in the left-hand-side. The proof term refl (double x) produces a proof for t = s where t and s have the same normal form of (double x). The proof term { symm (mul::onel x) } is a justification for the equality x = 1*x. The curly braces instruct Lean to replace x with 1*x. Similarly { symm (distributel 1 1 x) } is a proof for 1*x + 1*x = (1 + 1)*x. The exact semantics of these expressions is not important at this point.

Objects

In each Lean session, we create an enviroment, a sequence of named objects such as: definitions, axioms and theorems. Each object has a unique name. We use hierarchical names in Lean, i.e., a sequence of regular identifiers separated by ::. Hierarchical names provide a cheap of simulating modules and namespaces in Lean.

Expressions

Each expression has a unique type in Lean. The command check returns the type of an expression.

        check 1+2.

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