112 lines
3.5 KiB
Markdown
112 lines
3.5 KiB
Markdown
Lean Tutorial
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=============
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Introduction
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------------
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Lean is an automatic and interactive theorem prover. It can be used to
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create specifications, build mathematical libraries, and solve
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constraints. In this tutorial, we introduce basic concepts, the logic
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used in Lean, and the main commands.
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Getting started
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---------------
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We can use Lean in interactive or batch mode.
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The following example just displays the message `hello world`.
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```lean
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print "hello world"
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```
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All we have to do to run your first example is to call the `lean` executable
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with the name of the text file that contains the command above.
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If you saved the above command in the file `hello.lean`, then you just have
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to execute
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lean hello.lean
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As a more complex example, the next example defines a function that doubles
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the input value, and then evaluates it on different values.
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```lean
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-- defines the double function
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definition double (x : Nat) := x + x
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eval double 10
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eval double 2
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eval double 3 > 4
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```
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Every expression has a unique type in Lean. The command `check` returns the
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type of a given expression.
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```lean
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check double 3
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check double
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```
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The last command returns `Nat → Nat`. That is, the type of double is a function
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from `Nat` to `Nat`, where `Nat` is the type of the natural numbers.
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The command `import` loads existing libraries and extensions. For example,
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the following command imports the command `find` that searches the Lean environment
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using regular expressions
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```lean
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import find
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find "Nat" -- find all object that start with the prefix Nat
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check Nat::ge -- display the signature of the Nat::ge definition
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```
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We say `Nat::ge` is a hierarchical name comprised of two parts: `Nat` and `ge`
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The command `using` creates aliases based on give prefix. For example, the following
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command creates aliases for all objects starting with `Nat`
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```lean
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using Nat
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check ge -- display the signature of the Nat::ge definition
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```
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In Lean, proofs are also expressions, and theorems are essentially definitions.
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In the following example we prove that `double x = 2 * x`
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```lean
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theorem double_x_eq_2x (x : Nat) : double x = 2 * x :=
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calc double x = x + x : refl (double x)
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... = 1*x + 1*x : { symm (mul_onel x) }
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... = (1 + 1)*x : symm (distributel 1 1 x)
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... = 2 * x : { refl (1 + 1) }
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```
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In the example above, we provided the proof manually using a calculational proof style.
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The terms after `:` are proof terms. They justify the equalities in the left-hand-side.
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The proof term `refl (double x)` produces a proof for `t = s` where `t` and `s` have the same
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normal form of `(double x)`. The proof term `{ symm (mul_onel x) }` is a justification for
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the equality `x = 1*x`. The curly braces instruct Lean to replace `x` with `1*x`.
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Similarly `{ symm (distributel 1 1 x) }` is a proof for `1*x + 1*x = (1 + 1)*x`.
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The exact semantics of these expressions is not important at this point.
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Objects
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-------
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In each Lean session, we create an enviroment, a sequence of named
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objects such as: definitions, axioms and theorems. Each object has
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a unique name. We use `hierarchical names` in Lean, i.e., a sequence
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of regular identifiers separated by `::`. Hierarchical names provide
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a cheap of simulating modules and namespaces in Lean.
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Expressions
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-----------
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Each expression has a unique type in Lean. The command `check` returns
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the type of an expression.
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```lean
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check 1+2.
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```
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