* 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`.

#+BEGIN_SRC lean
  print "hello world"
#+END_SRC

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

#+BEGIN_SRC shell
  lean hello.lean
#+END_SRC

As a more complex example, the next example defines a function that doubles
the input value.

#+BEGIN_SRC lean
  import data.nat
  open nat
  -- defines the double function
  definition double (x : nat) := x + x
#+END_SRC

** 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.

#+BEGIN_SRC lean
  import logic
  check true
  check and
#+END_SRC

The last command returns =Prop → Prop → Prop=. That is, the type of
=and= is a function that takes two _propositions_ and return a
proposition, =Prop= is the type of propositions.

The command =import= loads existing libraries and extensions.

#+BEGIN_SRC lean
  import data.nat
  check nat.ge
#+END_SRC

We say =nat.ge= is a hierarchical name comprised of two parts: =nat= and =ge=.

The command =open= creates aliases based on a given prefix. The
command also imports notation, hints, and other features. We will
discuss its other applications later. Regarding aliases,
the following command creates aliases for all objects starting with
=nat=, and imports all notations defined in this namespace.

#+BEGIN_SRC lean
  import data.nat
  open nat
  check ge -- display the type of nat.ge
#+END_SRC

The command =constant= assigns a type to an identifier. The following command postulates/assumes
that =n=, =m= and =o= have type =nat=.

#+BEGIN_SRC lean
  import data.nat
  open nat
  constant n : nat
  constant m : nat
  constant o : nat
  -- The command 'open nat' also imported the notation defined at the namespace 'nat'
  check n + m
  check n ≤ m
#+END_SRC

The command =constants n m o : nat= can be used as 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, =eq.refl n= is a proof
for =n = n=. In Lean, =eq.refl= is the reflexivity theorem.

#+BEGIN_SRC lean
  import data.nat
  open nat
  constant n : nat
  check eq.refl n
#+END_SRC

The command =axiom= postulates that a given proposition holds.
The following commands postulate two axioms =Ax1= and =Ax2= that state that =n = m= and
=m = o=. =Ax1= and =Ax2= are not just names. For example, =eq.trans Ax1 Ax2= is a proof that
=n = o=, where =eq.trans= is the transitivity theorem.

#+BEGIN_SRC lean
  import data.nat
  open nat
  constants m n o : nat
  axiom Ax1 : n = m
  axiom Ax2 : m = o
  check eq.trans Ax1 Ax2
#+END_SRC

The expression =eq.trans Ax1 Ax2= is just a function application like any other.
Moreover, in Lean, _propositions are types_. Any proposition =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 =eq.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.
In Lean, we support classical and constructive logic. We can load
_classical choice axiom_ by using =import logic.choice= or =import classical=. When the classical
extensions are loaded, the _excluded middle_ is a theorem,
and =em p= is a proof for =p ∨ ¬ p=.

#+BEGIN_SRC lean
  import logic.choice
  constant p : Prop
  check em p
#+END_SRC

The commands =axiom= and =constant= are essentially the same command. We provide both
just to make Lean files more readable. We encourage users to use =axiom= only for
propositions, and =constant= for everything else.

Similarly, a theorem is just a definition. The following command defines a new theorem
called =nat_trans3=, and then use it to prove something else. In this
example, =eq.symm= is the symmetry theorem.

#+BEGIN_SRC lean
  import data.nat
  open nat

  theorem nat_trans3 (a b c d : nat) (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d :=
  eq.trans (eq.trans H1 (eq.symm H2)) H3

  -- Example using nat_trans3
  constants 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
#+END_SRC

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=.

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.
In the example above, we can use =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.

#+BEGIN_SRC lean
  import data.nat
  open nat

  theorem nat_trans3i {a b c d : nat} (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d :=
  eq.trans (eq.trans H1 (eq.symm H2)) H3

  -- Example using nat_trans3
  constants x y z w : nat
  axiom Hxy : x = y
  axiom Hzy : z = y
  axiom Hzw : z = w
  check nat_trans3i Hxy Hzy Hzw
#+END_SRC

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.

Sometimes, Lean will not be able to infer the parameters automatically.
The annotation =@f= instructs Lean that we want to provide the
implicit arguments for =f= explicitly.
The theorems =eq.refl=, =eq.trans= and =eq.symm= all have implicit arguments.

#+BEGIN_SRC lean
  import logic
  check @eq.refl
  check @eq.symm
  check @eq.trans
#+END_SRC

We can also instruct Lean to display all implicit arguments when it prints expressions.
This is useful when debugging non-trivial problems.

#+BEGIN_SRC lean
  import data.nat
  open nat

  constants a b c : nat
  axiom H1 : a = b
  axiom H2 : b = c
  check eq.trans H1 H2

  set_option pp.implicit true
  -- Now, Lean will display all implicit arguments
  check eq.trans H1 H2
#+END_SRC

In the previous example, the =check= command stated that =eq.trans H1 H2=
has type =@eq ℕ a c=. The expression =a = c= is just notational convenience.

We have seen many occurrences of =Type=.
In Lean, the type of =nat= and =Prop= is =Type=.
What is the type of =Type=?

#+BEGIN_SRC lean
  check Type
#+END_SRC

Lean reports =Type : Type=, is Lean inconsistent? No, it is not.
Internally, Lean maintains a hierarchy of Types. We say each one of
them _lives_ in a universe. Lean is universe polymorphic, and by
default all universes are hidden from the user. Like implicit
arguments, we can instruct Lean to display the universe levels
explicitly.

#+BEGIN_SRC lean
  set_option pp.universes true
  check Type
#+END_SRC

In the command above, Lean reports that =Type.{l_1}= that lives in
universe =l_1= has type =Type.{succ l_1}=. That is, its type lives in
the universe =l_1 + 1=.

Definitions such as =eq.refl=, =eq.symm= and =eq.trans= are all universe
polymorphic.

#+BEGIN_SRC lean
  import logic
  set_option pp.universes true
  check @eq.refl
  check @eq.symm
  check @eq.trans
#+END_SRC

Whenever we declare a new constant, Lean automatically infers the
universe parameters. We can also provide the universe levels
explicitly.

#+BEGIN_SRC lean
  import logic

  definition id.{l} {A : Type.{l}} (a : A) : A := a

  check id true
#+END_SRC

The universes can be explicitly provided for each constant and =Type=
by using the notation =.{ ... }=. Unlike other systems, Lean does not
have _universe cumulativity_. That is, the type =Type.{i}= is *not* an
element of =Type.{j}= for =j > i=.

** 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 file [[../../library/standard/logic.lean][logic.lean]]. This file also defines notational convention for writing formulas
in a natural way. Here is a table showing the notation for the so called propositional (or Boolean) connectives.


| Ascii |             Unicode   | Definition  |
|-------|-----------------------|--------------|
| true  |                       |   true       |
| false |                       |   false      |
| not   |               ¬       |   not        |
| /\    |               ∧       |   and        |
| ‌\/    |               ∨       |   or         |
| ->    |               →       |              |
| <->   |               ↔       |   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.

#+BEGIN_SRC lean
  import logic
  constants p q : Prop
  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
#+END_SRC

Depending on the platform, Lean uses unicode characters by default when printing expressions. The following commands can be used to
change this behavior.

#+BEGIN_SRC lean
  import logic
  set_option pp.unicode false
  constants p q : Prop
  check p → q → p ∧ q
  set_option pp.unicode true
  check p → q → p ∧ q
#+END_SRC

Note that, it may seem that the symbols =->= and =→= are overloaded, and Lean uses them to represent implication and the type
of functions. Actually, they are not overloaded, they are the same symbols. In Lean, the Proposition =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.

#+BEGIN_SRC lean
  import logic
  constants p q : Prop
  -- 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
#+END_SRC

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
[[../../library/standard/logic.lean][logic definition file]].
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 supports _currying_ =and true= is a function from =Prop= to =Prop=.

** 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.

#+BEGIN_SRC lean
  import data.nat
  open nat

  check fun x : nat, x + 1
  check fun x y : nat, x + 2 * y
  check fun x y : Prop, not (x ∧ y)
  check λ x : nat, x + 1
  check λ (x : nat) (p : Prop), x = 0 ∨ p
#+END_SRC

In many cases, Lean can automatically infer the type of the variable. Actually,
in all examples above, the type can be inferred automatically.

#+BEGIN_SRC lean
  import data.nat
  open nat

  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
#+END_SRC

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

#+BEGIN_SRC lean
  import data.nat
  open nat
  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
#+END_SRC

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).

#+BEGIN_SRC lean
  import logic
  constant p : Prop
  check fun H : p, H
#+END_SRC

** 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 =eq.refl=. Here is a simple example.

#+BEGIN_SRC lean
  import data.nat
  open nat

  theorem def_eq_th (a : nat) : ((λ x : nat, x + 1) a) = a + 1 := eq.refl (a+1)
#+END_SRC

** 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 standard library 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 (conjunction)

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=.

#+BEGIN_SRC lean
  import logic
  constants p q : Prop
  check fun (Hp : p) (Hq : q), and.intro Hp Hq
#+END_SRC

The expression =and.elim_left H= creates a proof =a= from a proof =H : a ∧ b=.
Similarly =and.elim_right H= is a proof for =b=. We say they are the _left/right and-eliminators_.

#+BEGIN_SRC lean
  import logic
  constants p q : Prop
  -- Proof for p ∧ q → p
  check fun H : p ∧ q, and.elim_left H
  -- Proof for p ∧ q → q
  check fun H : p ∧ q, and.elim_right H
#+END_SRC

Now, we prove =p ∧ q → q ∧ p= with the following simple proof term.

#+BEGIN_SRC lean
  import logic
  constants p q : Prop
  check fun H : p ∧ q, and.intro (and.elim_right H) (and.elim_left H)
#+END_SRC

Note that the proof term is very similar to a function that just swaps the
elements of a pair.

*** Or (disjunction)

The expression =or.intro_left b H1= creates a proof for =a ∨ b= using a proof =H1 : a=.
Similarly, =or.intro_right a H2= creates a proof for =a ∨ b= using a proof =H2 : b=.
We say they are the _left/right or-introduction_.

#+BEGIN_SRC lean
  import logic
  constants p q : Prop
  -- Proof for p → p ∨ q
  check fun H : p, or.intro_left q H
  -- Proof for q → p ∨ q
  check fun H : q, or.intro_right p H
#+END_SRC

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=.

#+BEGIN_SRC lean
  import logic
  constants p q : Prop
  check fun H : p ∨ q,
           or.elim H
              (fun Hp : p, or.intro_right q Hp)
              (fun Hq : q, or.intro_left  p Hq)

#+END_SRC

In most cases, the first argument of =or.intro_right= and
=or.intro_left= can be inferred automatically by Lean. Moreover, Lean
provides =or.inr= and =or.inl= as shorthands for =or.intro_right _=
and =or.intro_left _=. These two shorthands are extensively used in
the Lean standard library.

#+BEGIN_SRC lean
  import logic
  constants p q : Prop
  check fun H : p ∨ q,
           or.elim H
              (fun Hp : p, or.inr Hp)
              (fun Hq : q, or.inl Hq)

#+END_SRC

*** 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 Ha Hna= produces a proof for some =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= to produce a proof term for
=(a → b) → ¬b → ¬a=.

#+BEGIN_SRC lean
  import logic
  constants a b : Prop
  check fun (Hab : a → b) (Hnb : ¬ b),
            not.intro (fun Ha : a, absurd (Hab Ha) Hnb)

#+END_SRC

In the standard library, =not a= is actually just an _abbreviation_
for =a → false=. Thus, we don't really need to use =not_intro=
explicitly.

#+BEGIN_SRC lean
  import logic
  constants a b : Prop
  check fun (Hab : a → b) (Hnb : ¬ b),
            (fun Ha : a, Hnb (Hab Ha))

#+END_SRC

Now, here is the proof term for =¬a → b → (b → a) → c=

#+BEGIN_SRC lean
  import logic
  constants a b c : Prop
  check fun (Hna : ¬ a) (Hb : b) (Hba : b → a),
            absurd (Hba Hb) Hna
#+END_SRC

*** 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.elim_left H= produces a proof for =a → b= from =H : a ↔ b=. Similarly,
=iff.elim_right H= produces a proof for =b → a= from =H : a ↔ b=.
Here is the proof term for =a ∧ b ↔ b ∧ a=

#+BEGIN_SRC lean
  import logic
  constants a b : Prop
  check iff.intro
          (fun H : a ∧ b, and.intro (and.elim_right H) (and.elim_left H))
          (fun H : b ∧ a, and.intro (and.elim_right H) (and.elim_left H))
#+END_SRC

In Lean, we can use =assume= instead of =fun= to make proof terms look
more like proofs found in text books.

#+BEGIN_SRC lean
  import logic
  constants a b : Prop
  check iff.intro
          (assume H : a ∧ b, and.intro (and.elim_right H) (and.elim_left H))
          (assume H : b ∧ a, and.intro (and.elim_right H) (and.elim_left H))
#+END_SRC

*** 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=.

*** Rewrite rules

*WARNING: We did not port this section to Lean 0.2 yet*

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.

#+BEGIN_SRC
  -- 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)
#+END_SRC

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.

#+BEGIN_SRC
   theorem th2 (a b : Prop) : a ∧ b ↔ b ∧ a
   := iff.intro
        (fun H : a ∧ b, (by simp simple))
        (fun H : b ∧ a, (by simp simple))
#+END_SRC

** 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 : Type=, and a family of types =B : A → Type= 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 type of a dependent function is written as =forall a : A, B a=,
=Pi a : A, B a=, =∀ x : A, B a=, or =Π x : A, B a=. We usually use
=forall= and =∀= for propositions, and =Pi= and =Π= for everything
else. In the previous examples, we have seen many examples of
dependent functions. The theorems =eq.refl=, =eq.trans= and =eq.symm=, and the
equality are all dependent functions.

The universal quantifier is just a dependent function.
In Lean, if we have a family of types =B : A → Prop=,
then =∀ x : A, B a= has type =Prop=.
This feature 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.

#+BEGIN_SRC lean
  import data.nat
  open nat

  constant f : nat → nat
  axiom fzero : ∀ x, f x = 0
  check fzero 1
  constant a : nat
  check fzero a
#+END_SRC

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=.

#+BEGIN_SRC lean
  import data.nat
  open nat

  constant f : nat → nat
  axiom fzero : ∀ x, f x = 0
  check λ x y, eq.trans (fzero x) (eq.symm (fzero y))
#+END_SRC

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 =eq.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 can be written as =exists x : A, B
x= or =∃ x : A, B x=. Actually both versions are just
notational convenience for =Exists (fun x : A, B x)=. That is, the existential quantifier
is actually a constant defined in the file =logic.lean=.
This file also defines the =exists.intro= and =exists.elim=.
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

#+BEGIN_SRC lean
  import data.nat
  open nat

  theorem nat_trans3i {a b c d : nat} (H1 : a = b) (H2 : c = b) (H3 : c = d) : a = d :=
  eq.trans (eq.trans H1 (eq.symm H2)) H3

  constants x y z w : nat
  axiom Hxy : x = y
  axiom Hzy : z = y
  axiom Hzw : z = w

  theorem ex_a_eq_w : exists a, a = w  := exists.intro x (nat_trans3i Hxy Hzy Hzw)
  check ex_a_eq_w
#+END_SRC


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 Prop). 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=.

#+BEGIN_SRC lean
  import data.nat
  open nat

  check @exists.intro
  constant 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
  check gex1
  check gex2
  check gex3
  check gex4
#+END_SRC

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=.

#+BEGIN_SRC lean
  import logic
   check @exists.elim
#+END_SRC

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.

#+BEGIN_SRC lean
  import data.nat
  open nat

  namespace hide
  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)   : eq.symm !mul.left_distrib)))
  end hide
#+END_SRC

The example above also uses [[./calc.org][calculational proofs]] to show that =a + b = 2*(w1 + w2)=.
The =calc= construct is just syntax sugar for creating proofs using transitivity and substitution.

In Lean, we can use =obtain _, from _, _= as syntax sugar for =exists.elim=.
With this macro we can write the example above in a more natural way

#+BEGIN_SRC lean
  import data.nat
  open nat

  namespace hide
  definition even (a : nat) := ∃ b, a = 2*b
  theorem EvenPlusEven {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)   : eq.symm !mul.left_distrib)
  end hide
#+END_SRC