feat(library/data/int/countable): show that int is encodable, isomorphic to nat, and countable

library/data/int/countable.lean

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+/-
+Copyright (c) 2015 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Leonardo de Moura
+-/
+import data.equiv data.int.basic data.encodable data.countable
+open equiv bool sum
+
+namespace int
+definition int_equiv_bool_nat : int ≃ (bool × nat) :=
+equiv.mk
+  (λ i, match i with of_nat a := (tt, a) | neg_succ_of_nat a := (ff, a) end)
+  (λ p, match p with (tt, a) := of_nat a | (ff, a) := neg_succ_of_nat a end)
+  (λ i, begin cases i, repeat reflexivity end)
+  (λ p, begin cases p with b a, cases b, repeat reflexivity end)
+
+definition int_equiv_nat : int ≃ nat :=
+calc int ≃ (bool × nat)                : int_equiv_bool_nat
+     ... ≃ ((unit + unit) × nat)       : prod_congr bool_equiv_unit_sum_unit !_root_.equiv.refl
+     ... ≃ (unit × nat) + (unit × nat) : sum_prod_distrib
+     ... ≃ nat + nat                   : sum_congr !prod_unit_left !prod_unit_left
+     ... ≃ nat                         : nat_sum_nat_equiv_nat
+
+definition encodable_int [instance] : encodable int :=
+encodable_of_equiv (_root_.equiv.symm int_equiv_nat)
+
+lemma countable_int : countable int :=
+countable_of_encodable encodable_int
+
+end int