687 lines
27 KiB
Markdown
687 lines
27 KiB
Markdown
Lean Tutorial
|
||
=============
|
||
|
||
**WARNING: This tutoral is for Lean 0.1, before major modifications performed in version 0.2.
|
||
Version 0.2 is still under development. [Here you can find a snapshot of Lean 0.1](https://github.com/leanprover/lean0.1).**
|
||
|
||
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.
|
||
|
||
```lean
|
||
-- defines the double function
|
||
definition double (x : Nat) := x + x
|
||
```
|
||
|
||
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. 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 theorem.
|
||
|
||
```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 theorem.
|
||
|
||
```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 the _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 use `_` as a placeholder
|
||
that instructs Lean to synthesize this expression. The synthesis process is based on type inference, and it is
|
||
the most basic form of automation provided by Lean.
|
||
|
||
```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 theorems `refl`, `trans` and `symm` all have implicit arguments.
|
||
|
||
```lean
|
||
check @refl
|
||
check @trans
|
||
check @symm
|
||
```
|
||
|
||
We can also instruct Lean to display all implicit arguments when it prints expressions.
|
||
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
|
||
```
|
||
|
||
In the previous example, the `check` command stated that `nat_trans3i Hxy Hzy Hzw`
|
||
has type `@eq ℕ x w`. The expression `x = w` is just notational convenience.
|
||
|
||
We have seen many occurrences of `(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 theorem 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 users 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`.
|
||
In Lean, whenever we declare a new universe `V`, the system automatically adds the constraint `U ≥ V + 1`.
|
||
That is, `U` the _maximal_ universe in Lean.
|
||
Lean reports an universe inconsistency if the universe constraints are inconsistent. 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 automatically generate a Lean specification that is not based on this particular feature.
|
||
|
||
Propositional logic
|
||
-------------------
|
||
|
||
To manipulate formulas with a richer logical structure, it is important to master the notation Lean uses for building
|
||
composite logical expressions out of basic formulas using _logical connectives_. The logical connectives (`and`, `or`, `not`, etc)
|
||
are defined in the Lean [kernel](../../src/builtin/kernel.lean). The kernel also defines notational convention for rewriting formulas
|
||
in a natural way. Here is a table showing the notation for the so called propositional (or Boolean) connectives.
|
||
|
||
|
||
| Ascii | Ascii alt. | Unicode | Definition |
|
||
|-------|--------------|---------|--------------|
|
||
| true | | ⊤ | true |
|
||
| false | | ⊥ | false |
|
||
| not | | ¬ | not |
|
||
| /\ | && | ∧ | and |
|
||
| \/ | || | ∨ | or |
|
||
| -> | | → | implies |
|
||
| <-> | | ↔ | iff |
|
||
|
||
`true` and `false` are logical constants to denote the true and false propositions. Logical negation is a unary operator just like
|
||
arithmetical negation on numbers. The other connectives are all binary operators. The meaning of the operators is the usual one.
|
||
The table above makes clear that Lean supports unicode characters. We can use Ascii or/and unicode versions.
|
||
Here is a simple example using the connectives above.
|
||
|
||
```lean
|
||
variable q : Bool
|
||
check p → q → p ∧ q
|
||
check ¬ p → p ↔ false
|
||
check p ∨ q → q ∨ p
|
||
-- Ascii version
|
||
check p -> q -> p && q
|
||
check not p -> p <-> false
|
||
check p || q -> q \/ p
|
||
```
|
||
|
||
Depending on the platform, Lean uses unicode characters by default when printing expressions. The following commands can be used to
|
||
change this behavior.
|
||
|
||
```lean
|
||
set_option pp::unicode false
|
||
check p → q → p ∧ q
|
||
set_option pp::unicode true
|
||
check p → q → p ∧ q
|
||
```
|
||
|
||
Note that, it may seem that the symbols `->` and `→` are overloaded, and Lean uses them to represent Boolean implication and the type
|
||
of functions. Actually, they are not overloaded, they are the same symbols. In Lean, the Boolean `p → q` expression is also the type
|
||
of the functions that given a proof for `p`, returns a proof for `q`. This is very convenient for writing proofs.
|
||
|
||
```lean
|
||
-- Hpq is a function that takes a proof for p and returns a proof for q
|
||
axiom Hpq : p → q
|
||
-- Hq is a proof/certificate for p
|
||
axiom Hp : p
|
||
-- The expression Hpq Hp is a proof/certificate for q
|
||
check Hpq Hp
|
||
```
|
||
|
||
In composite expressions, the precedences of the various binary
|
||
connectives are in order of the above table, with `and` being the
|
||
strongest and `iff` the weakest. For example, `a ∧ b → c ∨ d ∧ e`
|
||
means `(a ∧ b) → (c ∨ (d ∧ e))`. All of them are right-associative.
|
||
So, `p ∧ q ∧ r` means `p ∧ (q ∧ r)`. The actual precedence and fixity of all
|
||
logical connectives is defined in the Lean [kernel definition file](../../src/builtin/kernel.lean).
|
||
|
||
Finally, `not`, `and`, `or` and `iff` are the actual names used when
|
||
defining the Boolean connectives. They can be used as any other function.
|
||
|
||
```lean
|
||
check and
|
||
check or
|
||
check not
|
||
```
|
||
|
||
Lean supports _currying_ `and true` is a function from `Bool` to `Bool`.
|
||
|
||
```lean
|
||
check and true
|
||
definition id := and true
|
||
```
|
||
|
||
Functions
|
||
---------
|
||
|
||
There are many variable-binding constructs in mathematics. Lean expresses
|
||
all of them using just one _abstraction_, which is a converse operation to
|
||
function application. Given a variable `x`, a type `A`, and a term `t` that
|
||
may or may not contain `x`, one can construct the so-called _lambda abstraction_
|
||
`fun x : A, t`, or using unicode notation `λ x : A, t`. Here is some simple
|
||
examples.
|
||
|
||
```lean
|
||
check fun x : Nat, x + 1
|
||
check fun x y : Nat, x + 2 * y
|
||
check fun x y : Bool, not (x ∧ y)
|
||
check λ x : Nat, x + 1
|
||
check λ (x : Nat) (p : Bool), x = 0 ∨ p
|
||
```
|
||
|
||
In many cases, Lean can automatically infer the type of the variable. Actually,
|
||
In all examples above, the type can be inferred automatically.
|
||
|
||
```lean
|
||
check fun x, x + 1
|
||
check fun x y, x + 2 * y
|
||
check fun x y, not (x ∧ y)
|
||
check λ x, x + 1
|
||
check λ x p, x = 0 ∨ p
|
||
```
|
||
|
||
However, Lean will complain that it cannot infer the type of the
|
||
variable `x` in `fun x, x` because any type would work in this example.
|
||
|
||
The following example shows how to use lambda abstractions in
|
||
function applications
|
||
|
||
```lean
|
||
check (fun x y, x + 2 * y) 1
|
||
check (fun x y, x + 2 * y) 1 2
|
||
check (fun x y, not (x ∧ y)) true false
|
||
```
|
||
|
||
Lambda abstractions are also used to create proofs for propositions of the form `A → B`.
|
||
This should be natural since we can "view" `A → B` as the type of functions that given
|
||
a proof for `A` returns a proof for `B`.
|
||
For example, a proof for `p → p` is just `fun H : p, H` (the identity function).
|
||
|
||
```lean
|
||
check fun H : p, H
|
||
```
|
||
|
||
Definitional equality
|
||
---------------------
|
||
|
||
The command `eval t` computes a normal form for the term `t`.
|
||
In Lean, we say two terms are _definitionally equal_ if the have the same
|
||
normal form. For example, the terms `(λ x : Nat, x + 1) a` and `a + 1`
|
||
are definitionally equal. The Lean type/proof checker uses the normalizer when
|
||
checking types/proofs. So, we can prove that two definitionally equal terms
|
||
are equal using just `refl`. Here is a simple example.
|
||
|
||
```lean
|
||
theorem def_eq_th (a : Nat) : ((λ x : Nat, x + 1) a) = a + 1
|
||
:= refl (a+1)
|
||
```
|
||
|
||
Provable equality
|
||
-----------------
|
||
|
||
In the previous examples, we have used `nat_trans3 x y z w Hxy Hzy Hzw`
|
||
to show that `x = w`. In this case, `x` and `w` are not definitionally equal,
|
||
but they are provably equal in the environment that contains `nat_trans3` and
|
||
axioms `Hxy`, `Hzy` and `Hzw`.
|
||
|
||
Proving
|
||
-------
|
||
|
||
The Lean kernel contains basic theorems for creating proof terms. The
|
||
basic theorems are useful for creating manual proofs. The are also the
|
||
basic building blocks used by all automated proof engines available in
|
||
Lean. The theorems can be broken into three different categories:
|
||
introduction, elimination, and rewriting. First, we cover the introduction
|
||
and elimination theorems for the basic Boolean connectives.
|
||
|
||
### And (conjuction)
|
||
|
||
The expression `and_intro H1 H2` creates a proof for `a ∧ b` using proofs
|
||
`H1 : a` and `H2 : b`. We say `and_intro` is the _and-introduction_ operation.
|
||
In the following example we use `and_intro` for creating a proof for
|
||
`p → q → p ∧ q`.
|
||
|
||
```lean
|
||
check fun (Hp : p) (Hq : q), and_intro Hp Hq
|
||
```
|
||
|
||
The expression `and_eliml H` creates a proof `a` from a proof `H : a ∧ b`.
|
||
Similarly `and_elimr H` is a proof for `b`. We say they are the _left/right and-elimination_.
|
||
|
||
```lean
|
||
-- Proof for p ∧ q → p
|
||
check fun H : p ∧ q, and_eliml H
|
||
-- Proof for p ∧ q → q
|
||
check fun H : p ∧ q, and_elimr H
|
||
```
|
||
|
||
Now, we prove `p ∧ q → q ∧ p` with the following simple proof term.
|
||
|
||
```lean
|
||
check fun H : p ∧ q, and_intro (and_elimr H) (and_eliml H)
|
||
```
|
||
|
||
Note that the proof term is very similar to a function that just swaps the
|
||
elements of a pair.
|
||
|
||
### Or (disjuction)
|
||
|
||
The expression `or_introl H1 b` creates a proof for `a ∨ b` using a proof `H1 : a`.
|
||
Similarly, `or_intror a H2` creates a proof for `a ∨ b` using a proof `H2 : b`.
|
||
We say they are the _left/right or-introduction_.
|
||
|
||
```lean
|
||
-- Proof for p → p ∨ q
|
||
check fun H : p, or_introl H q
|
||
-- Proof for q → p ∨ q
|
||
check fun H : q, or_intror p H
|
||
```
|
||
|
||
The or-elimination rule is slightly more complicated. The basic idea is the
|
||
following, we can prove `c` from `a ∨ b`, by showing we can prove `c`
|
||
by assuming `a` or by assuming `b`. It is essentially a proof by cases.
|
||
`or_elim Hab Hac Hbc` takes three arguments `Hab : a ∨ b`, `Hac : a → c` and `Hbc : b → c` and produces a proof for `c`.
|
||
In the following example, we use `or_elim` to prove that `p v q → q ∨ p`.
|
||
|
||
```lean
|
||
check fun H : p ∨ q,
|
||
or_elim H
|
||
(fun Hp : p, or_intror q Hp)
|
||
(fun Hq : q, or_introl Hq p)
|
||
```
|
||
|
||
### Not (negation)
|
||
|
||
`not_intro H` produces a proof for `¬ a` from `H : a → false`. That is,
|
||
we obtain `¬ a` if we can derive `false` from `a`. The expression
|
||
`absurd_elim b Ha Hna` produces a proof for `b` from `Ha : a` and `Hna : ¬ a`.
|
||
That is, we can deduce anything if we have `a` and `¬ a`.
|
||
We now use `not_intro` and `absurd_elim` to produce a proof term for
|
||
`(a → b) → ¬ b → ¬ a`
|
||
|
||
```lean
|
||
variables a b : Bool
|
||
check fun (Hab : a → b) (Hnb : ¬ b),
|
||
not_intro (fun Ha : a, absurd_elim false (Hab Ha) Hnb)
|
||
```
|
||
|
||
Here is the proof term for `¬ a → b → (b → a) → c`
|
||
|
||
```lean
|
||
variable c : Bool
|
||
check fun (Hna : ¬ a) (Hb : b) (Hba : b → a),
|
||
absurd_elim c (Hba Hb) Hna
|
||
```
|
||
|
||
### Iff (if-and-only-if)
|
||
|
||
The expression `iff_intro H1 H2` produces a proof for `a ↔ b` from `H1 : a → b` and `H2 : b → a`.
|
||
`iff_eliml H` produces a proof for `a → b` from `H : a ↔ b`. Similarly,
|
||
`iff_elimr H` produces a proof for `b → a` from `H : a ↔ b`.
|
||
Note that, in Lean, `a ↔ b` is definitionally equal to `a = b` when `a` and `b` have type `Bool`.
|
||
Here is the proof term for `a ∧ b ↔ b ∧ a`
|
||
|
||
```lean
|
||
check iff_intro (fun H : a ∧ b, and_intro (and_elimr H) (and_eliml H))
|
||
(fun H : b ∧ a, and_intro (and_elimr H) (and_eliml H))
|
||
```
|
||
|
||
### True and False
|
||
|
||
The expression `trivial` is a proof term for `true`, and `false_elim a H`
|
||
produces a proof for `a` from `H : false`.
|
||
|
||
Other basic operators used in proof construction are `eqt_intro`, `eqt_elim`, `eqf_intro` and `eqf_elim`.
|
||
`eqt_intro H` produces a proof for `a ↔ true` from `H : a`.
|
||
`eqt_elim H` produces a proof for `a` from `H : a ↔ true`.
|
||
`eqf_intro H` produces a proof for `a ↔ false` from `H : ¬ a`.
|
||
`eqf_elim H` produces a proof for `¬ a` from `H : a ↔ false`.
|
||
|
||
```lean
|
||
check @eqt_intro
|
||
check @eqt_elim
|
||
check @eqf_intro
|
||
check @eqf_elim
|
||
```
|
||
|
||
### Rewrite rules
|
||
|
||
The Lean kernel also contains many theorems that are meant to be used as rewriting/simplification rules.
|
||
The conclusion of these theorems is of the form `t = s` or `t ↔ s`. For example, `and_id a` is proof term for
|
||
`a ∧ a ↔ a`. The Lean simplifier can use these theorems to automatically create proof terms for us.
|
||
The expression `(by simp [rule-set])` is similar to `_`, but it tells Lean to synthesize the proof term using the simplifier
|
||
using the rewrite rule set named `[rule-set]`. In the following example, we create a simple rewrite rule set
|
||
and use it to prove a theorem that would be quite tedious to prove by hand.
|
||
|
||
```lean
|
||
-- import module that defines several tactics/strategies including "simp"
|
||
import tactic
|
||
-- create a rewrite rule set with name 'simple'
|
||
rewrite_set simple
|
||
-- add some theorems to the rewrite rule set 'simple'
|
||
add_rewrite and_id and_truer and_truel and_comm and_assoc and_left_comm iff_id : simple
|
||
theorem th1 (a b : Bool) : a ∧ b ∧ true ∧ b ∧ true ∧ b ↔ a ∧ b
|
||
:= (by simp simple)
|
||
```
|
||
|
||
In Lean, we can combine manual and automated proofs in a natural way. We can manually write the proof
|
||
skeleton and use the `by` construct to invoke automated proof engines like the simplifier for filling the
|
||
tedious steps. Here is a very simple example.
|
||
|
||
```lean
|
||
theorem th2 (a b : Bool) : a ∧ b ↔ b ∧ a
|
||
:= iff_intro
|
||
(fun H : a ∧ b, (by simp simple))
|
||
(fun H : b ∧ a, (by simp simple))
|
||
```
|
||
|
||
### Dependent functions and quantifiers
|
||
|
||
Lean supports _dependent functions_. In type theory, they are also called dependent product types or Pi-types.
|
||
The idea is quite simple, suppose we have a type `A` in some universe `(Type i)`, and a family of types `B : A → (Type j)` which assigns to each `a : A` a type `B a`. So a dependent function is a function whose range varies depending on its arguments.
|
||
In lean, the dependent functions is written as `forall a : A, B a`, or `∀ x : A, B a` using unicode.
|
||
The proposition as types paradigm is based on dependent functions. In the previous examples, we have seen many examples of dependent functions. The theorems `refl`, `trans` and `symm`, and the equality are all dependent functions,
|
||
|
||
```lean
|
||
check @refl
|
||
check @trans
|
||
check @symm
|
||
check @eq
|
||
```
|
||
|
||
The universal quantifier is also a dependent function. In Lean, if we have a family of types `B : A → Bool`, then `∀ x : A, B a` has type `Bool`. This features complicates the Lean set-theoretic model, but it improves usability. Several theorem provers have a `forall elimination` (aka instantiation) proof rule. In Lean (and other systems based on proposition as types), this rule is just function application. In the following example we add an axiom stating that `f x` is `0` forall `x`. Then we instantiate the axiom using function application.
|
||
|
||
```lean
|
||
variable f : Nat → Nat
|
||
axiom fzero : ∀ x, f x = 0
|
||
check fzero 1
|
||
check fzero x
|
||
```
|
||
|
||
Since we instantiate quantifiers using function application, it is
|
||
natural to create proof terms for universal quantifiers using lambda
|
||
abstraction. In the following example, we create a proof term showing that for all
|
||
`x` and `y`, `f x = f y`.
|
||
|
||
```lean
|
||
check λ x y, trans (fzero x) (symm (fzero y))
|
||
```
|
||
|
||
We can view the proof term above as a simple function or "recipe" for showing that
|
||
`f x = f y` for any `x` and `y`. The function "invokes" `fzero` for creating
|
||
proof terms for `f x = 0` and `f y = 0`. Then, it uses symmetry `symm` to create
|
||
a proof term for `0 = f y`. Finally, transitivity is used to combine the proofs
|
||
for `f x = 0` and `0 = f y`.
|
||
|
||
In Lean, the existential quantifier `exists x : A, B x` is defined as `¬ forall x : A, ¬ B x`.
|
||
We can also write existential quantifiers as `∃ x : A, B x`. Actually both versions are just
|
||
notational convenience for `Exists A (fun x : A, B x)`. That is, the existential quantifier
|
||
is actually a constant defined in the file `kernel.lean`. This file also defines the
|
||
`exists_intro` and `exists_elim` theorems. To build a proof for `∃ x : A, B x`, we should
|
||
provide a term `w : A` and a proof term `Hw : B w` to `exists_intro`.
|
||
We say `w` is the witness for the existential introduction. In previous examples,
|
||
`nat_trans3i Hxy Hzy Hzw` was a proof term for `x = w`. Then, we can create a proof term
|
||
for `∃ a : Nat, a = w` using
|
||
|
||
```lean
|
||
theorem ex_a_eq_w : exists a, a = w := exists_intro x (nat_trans3i Hxy Hzy Hzw)
|
||
check ex_a_eq_w
|
||
```
|
||
|
||
Note that `exists_intro` also has implicit arguments. For example, Lean has to infer the implicit argument
|
||
`P : A → Bool`, a predicate (aka function to Bool). This creates complications. For example, suppose
|
||
we have `Hg : g 0 0 = 0` and we invoke `exists_intro 0 Hg`. There are different possible values for `P`.
|
||
Each possible value corresponds to a different theorem: `∃ x, g x x = x`, `∃ x, g x x = 0`,
|
||
`∃ x, g x 0 = x`, etc. Lean uses the context where `exists_intro` occurs to infer the users intent.
|
||
In the example above, we were trying to prove the theorem `∃ a, a = w`. So, we are implicitly telling
|
||
Lean how to choose `P`. In the following example, we demonstrate this issue. We ask Lean to display
|
||
the implicit arguments using the option `pp::implicit`. We see that each instance of `exists_intro 0 Hg`
|
||
has different values for the implicit argument `P`.
|
||
|
||
```lean
|
||
check @exists_intro
|
||
variable g : Nat → Nat → Nat
|
||
axiom Hg : g 0 0 = 0
|
||
theorem gex1 : ∃ x, g x x = x := exists_intro 0 Hg
|
||
theorem gex2 : ∃ x, g x 0 = x := exists_intro 0 Hg
|
||
theorem gex3 : ∃ x, g 0 0 = x := exists_intro 0 Hg
|
||
theorem gex4 : ∃ x, g x x = 0 := exists_intro 0 Hg
|
||
set_option pp::implicit true -- display implicit arguments
|
||
print environment 4 -- print the last four theorems
|
||
set_option pp::implicit false -- hide implicit arguments
|
||
```
|
||
|
||
We can view `exists_intro` (aka existential introduction) as an information hiding procedure.
|
||
We are "hiding" what is the witness for some fact. The existential elimination performs the opposite
|
||
operation. The `exists_elim` theorem allows us to prove some proposition `B` from `∃ x : A, B x`
|
||
if we can derive `B` using an "abstract" witness `w` and a proof term `Hw : B w`.
|
||
|
||
```lean
|
||
check @exists_elim
|
||
```
|
||
|
||
In the following example, we define `even a` as `∃ b, a = 2*b`, and then we show that the sum
|
||
of two even numbers is an even number.
|
||
|
||
```lean
|
||
definition even (a : Nat) := ∃ b, a = 2*b
|
||
theorem EvenPlusEven {a b : Nat} (H1 : even a) (H2 : even b) : even (a + b)
|
||
:= exists_elim H1 (fun (w1 : Nat) (Hw1 : a = 2*w1),
|
||
exists_elim H2 (fun (w2 : Nat) (Hw2 : b = 2*w2),
|
||
exists_intro (w1 + w2)
|
||
(calc a + b = 2*w1 + b : { Hw1 }
|
||
... = 2*w1 + 2*w2 : { Hw2 }
|
||
... = 2*(w1 + w2) : symm (distributer 2 w1 w2))))
|
||
|
||
```
|
||
|
||
The example above also uses [_calculational proofs_](calc.md) to show that `a + b = 2*(w1 + w2)`.
|
||
The `calc` construct is just syntax sugar for creating proofs using transitivity and substitution.
|
||
|
||
The module `macros` provides notation for making proof terms more readable.
|
||
For example, it defines the `obtain _, from _, _` macro as syntax sugar for `exists_elim`.
|
||
With this macro we can write the example above as:
|
||
|
||
```lean
|
||
import macros
|
||
theorem EvenPlusEven2 {a b : Nat} (H1 : even a) (H2 : even b) : even (a + b)
|
||
:= obtain (w1 : Nat) (Hw1 : a = 2*w1), from H1,
|
||
obtain (w2 : Nat) (Hw2 : b = 2*w2), from H2,
|
||
exists_intro (w1 + w2)
|
||
(calc a + b = 2*w1 + b : { Hw1 }
|
||
... = 2*w1 + 2*w2 : { Hw2 }
|
||
... = 2*(w1 + w2) : symm (distributer 2 w1 w2))
|
||
|
||
```
|
||
|
||
The module `macros` also defines `take x : A, H` and `assume x : A, H`
|
||
as syntax sugar for `fun x : A, H`. This may been silly, but it allows us to simulate [Mizar](http://en.wikipedia.org/wiki/Mizar_system)-style declarative proofs in Lean. Using these macros, we can write
|
||
|
||
```lean
|
||
definition Set (A : Type) : Type := A → Bool
|
||
|
||
definition element {A : Type} (x : A) (s : Set A) := s x
|
||
infix 60 ∈ : element
|
||
|
||
definition subset {A : Type} (s1 : Set A) (s2 : Set A) := ∀ x, x ∈ s1 → x ∈ s2
|
||
infix 50 ⊆ : subset
|
||
|
||
theorem subset_trans {A : Type} {s1 s2 s3 : Set A} (H1 : s1 ⊆ s2) (H2 : s2 ⊆ s3) : s1 ⊆ s3
|
||
:= take x : A,
|
||
assume Hin : x ∈ s1,
|
||
show x ∈ s3, from
|
||
let L1 : x ∈ s2 := H1 x Hin
|
||
in H2 x L1
|
||
```
|
||
|
||
Finally, the construct `show A, from H` means "have" a proof for `A` using `H`. It is just syntax sugar for
|
||
`let H_show : A := H in H_show`. It is useful to document intermediate steps in manually constructed proofs.
|