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`. ```lean 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. ```lean -- defines the double function definition double (x : Nat) := x + x eval double 10 eval double 2 eval double 3 > 4 ``` Basics ------ We can also view Lean as a suite of tools for evaluating and processing expressions representing terms, definitions, and theorems. Every expression has a unique type in Lean. The command `check` returns the type of a given expression. ```lean 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 ```lean 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 a given prefix. For example, the following command creates aliases for all objects starting with `Nat` ```lean using Nat check ge -- display the signature of the Nat::ge definition ``` The command `variable` assigns a type to an identifier. The following command postulates/assumes that `n`, `m` and `o` have type `Nat`. ```lean variable n : Nat variable m : Nat variable o : Nat ``` The command `variables n m o : Nat` can be used a shorthand for the three commands above. In Lean, proofs are also expressions, and all functionality provided for manipulating expressions is also available for manipulating proofs. For example, `refl n` is a proof for `n == n`. In Lean, `refl` is the reflexivity axiom. ```lean check refl n ``` The command `axiom` postulates that a given proposition (aka Boolean formula) is true. The following commands postulate two axioms `Ax1` and `Ax2` that state that `n = m` and `m = o`. ```lean axiom Ax1 : n = m axiom Ax2 : m = o ``` `Ax1` and `Ax2` are not just names. For example, `trans Ax1 Ax2` is a proof that `n == o`, where `trans` is the transitivity axiom. ```lean check trans Ax1 Ax2 ``` The expression `trans Ax1 Ax2` is just a function application like any other. Moreover, in Lean, _propositions are types_. Any Boolean expression `P` can be used as a type. The elements of type `P` can be viewed as the proofs of `P`. Moreover, in Lean, _proof checking is type checking_. For example, the Lean type checker will reject the type incorrect term `trans Ax2 Ax1`. Because we use _proposition as types_, we must support _empty types_. For example, the type `false` must be empty, since we don't have a proof for `false`. Most systems based on _propositions as types_ paradigm are based on constructive logic. Lean on the other hand is based on classical logic. The _excluded middle_ is a theorem in Lean, and `em p` is a proof for `p ∨ ¬ p`. ```lean variable p : Bool check em p ``` The commands `axiom` and `variable` are essentially the same command. We provide both just to make Lean files more readable. We encourage users to use `axiom` only for propostions, and `variable` for everything else. Similarly, a theorem is just a definition. The following command defines a new theorem called `nat_trans3` ```lean theorem nat_trans3 (a b c d : Nat) (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d := trans (trans H1 (symm H2)) H3 ``` The theorem `nat_trans3` has 7 parameters, it takes for natural numbers `a`, `b`, `c` and `d, and three proofs showing that `a = b`, `c = b` and `c = d`, and returns a proof that `a = d`. In the example above, `symm` is the symmetry theorem. Now, we use `nat_trans3` in a simple example. ```lean variables x y z w : Nat axiom Hxy : x = y axiom Hzy : z = y axiom Hzw : z = w check nat_trans3 x y z w Hxy Hzy Hzw ``` The theorem `nat_trans3` is somewhat inconvenient to use because it has 7 parameters. However, the first four parameters can be inferred from the last 3. We can `_` as placeholder that instructs Lean to synthesize this expression. ```lean check nat_trans3 _ _ _ _ Hxy Hzy Hzw ``` Lean also supports _implicit arguments_. We mark implicit arguments using curly braces instead of parenthesis. In the following example, we define the theorem `nat_trans3i` using implicit arguments. ```lean theorem nat_trans3i {a b c d : Nat} (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d := trans (trans H1 (symm H2)) H3 ``` It is identical to `nat_trans3`, the only difference is the use of curly braces. Lean will (try to) infer the implicit arguments. The idea behind implicit arguments is quite simple, we are just instructing Lean to automatically insert the placeholders `_` for us. ```lean check nat_trans3i Hxy Hzy Hzw ``` Sometimes, Lean will not be able to infer the parameters automatically. So, whenever we define a theorem/definition/axiom/variable containing implicit arguments, Lean will automatically create an _explicit_ version where all parameters are explicit. The explicit version uses the same name with a `@` prefix. ```lean check @nat_trans3i ``` The axiom `refl`, and the theorems `trans` and `symm` all have implicit arguments. ```lean check @refl check @trans check @symm ``` We can also instruct Lean to display all implicit arguments. This is useful when debugging non-trivial problems. ```lean set_option pp::implicit true -- show implicit arguments check nat_trans3i Hxy Hzy Hzw set_option pp::implicit false -- hide implicit arguments ``` Note that, in the examples above, we have seen two symbols for equality: `=` and `==`. Moreover, in the previous example, the `check` command stated that `nat_trans3i Hxy Hzy Hzw` has type `@eq ℕ x w`. 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. On the other hand `t = s` is a homogeneous equality, and is only type correct if `t` and `s` have the same type. The symbol `=` is actually notation for `eq t s`, where `eq` is defined (using heterogenous equality) as: ``` definition eq {A : TypeU} (a b : A) : Bool := a == b ``` We strongly discourage users from directly using heterogeneous equality. The main problem is that it is very easy to write nonsensical expressions such as `2 == true`. On the other hand, the expression `2 = true` is type incorrect, and Lean will report the mistake to the user. We have seen many occurrences of `TypeU`. It is just a definition: `(Type U)`, where `U` is a _universe variable_. In Lean, the type of `Nat` and `Bool` is `Type`. ```lean check Nat check Bool ``` We say `Type` is the type of all _small_ types, but what is the type of `Type`? ```lean check Type ``` Lean returns `(Type 1)`. Similarly, the type of `(Type 1)` is `(Type 2)`. In Lean, we also have _universe cumulativity_. That is, we can provide an element of type `(Type i)` where an element of type `(Type j)` is expected when `i ≤ j`. This makes the system more convenient to use. Otherwise, we would need a reflexivity axiom for `Type` (i.e., `(Type 0)`), `Type 1`, `Type 2`, etc. Universe cumulativity improves usability, but it is not enough because We would still have the question: how big should `i` be? Moreover, if we choose an `i` that is not big enough we have to go back and correct all libraries. This is not satisfactory and not modular. So, in Lean, we allow user to declare _universe variables_ and simple constraints between them. The Lean kernel defines one universe variable `U`, and states that `U ≥ 1` using the command `universe U ≥ 1`. The Lean type casting library defines another universe variable called `M` and states that `universe M ≥ 1` and `universe M ≥ U + 1`. Lean reports an universe inconsistency if the universe constraints are inconsistency. For example, it will return an error if execute the command `universe M ≥ U`. We can view universe variables as placeholders, and we can always solve the universe constraints and find and assignment for the universe variables used in our developments. This assignment allows us to produce a Lean specification that is not based on this particular feature.