doc(examples/lean): add a new example with even/odd numbers

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

examples/lean/even.lean

@@ -0,0 +1,83 @@
+Import macros.
+
+(*
+In this example, we prove two simple theorems about even/odd numbers.
+First, we define the predicates even and odd.
+*)
+Definition even (a : Nat) := ∃ b, a = 2*b.
+Definition odd  (a : Nat) := ∃ b, a = 2*b + 1.
+
+(*
+   Prove that the sum of two even numbers is even.
+
+   Notes: we use the macro
+      Obtain [bindings] ',' 'from' [expr]_1 ',' [expr]_2
+
+   It is syntax sugar for existential elimination.
+   It expands to
+
+      ExistsElim [expr]_1 (fun [binding], [expr]_2)
+
+   It is defined in the file macros.lua.
+
+   We also use the calculational proof style.
+   See doc/lean/calc.md for more information.
+
+  We use the first two Obtain-expressions to extract the
+  witnesses w1 and w2 s.t. a = 2*w1 and b = 2*w2.
+  We can do that because H1 and H2 are evidence/proof for the
+  fact that 'a' and 'b' are even.
+
+  We use a calculational proof 'calc' expression to derive
+  the witness w1+w2 for the fact that a+b is also even.
+  That is, we provide a derivation showing that a+b = 2*(w1 + w2)
+*)
+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,
+     ExistsIntro (w1 + w2)
+                 (calc a + b  =  2*w1 + b      : { Hw1 }
+                         ...  =  2*w1 + 2*w2   : { Hw2 }
+                         ...  =  2*(w1 + w2)   : Symm (Nat::Distribute 2 w1 w2)).
+
+(*
+   In the following example, we omit the arguments for Nat::PlusAssoc, Refl and Nat::Distribute.
+   Lean can infer them automatically.
+
+   Refl is the reflexivity proof. (Refl a) is a proof that two
+   definitionally equal terms are indeed equal.
+   "Definitionally equal" means that they have the same normal form.
+   We can also view it as "Proof by computation".
+   The normal form of (1+1), and 2*1 is 2.
+
+   Another remark: '2*w + 1 + 1' is not definitionally equal to '2*w + 2*1'.
+   The gotcha is that '2*w + 1 + 1' is actually '(2*w + 1) + 1' since +
+   is left associative. Moreover, Lean normalizer does not use
+   any theorems such as + associativity.
+*)
+Theorem OddPlusOne {a : Nat} (H : odd a) : even (a + 1)
+:= Obtain (w : Nat) (Hw : a = 2*w + 1), from H,
+   ExistsIntro (w + 1)
+               (calc a + 1 = 2*w + 1 + 1   : { Hw }
+                       ... = 2*w + (1 + 1) : Symm (Nat::PlusAssoc _ _ _)
+                       ... = 2*w + 2*1     : Refl _
+                       ... = 2*(w + 1)     : Symm (Nat::Distribute _ _ _)).
+
+(*
+  The following command displays the proof object produced by Lean after
+  expanding macros, and infering implicit/missing arguments.
+*)
+Show Environment 2.
+
+(*
+  By default, Lean does not display implicit arguments.
+  The following command will force it to display them.
+*)
+SetOption pp::implicit true.
+
+Show Environment 2.
+
+(*
+  As an exercise, prove that the sum of two odd numbers is even,
+  and other similar theorems.
+*)