feat/fix(library/data/nat,int): add power to int, add trans attributes, power notation

library/data/int/default.lean

@@ -5,4 +5,4 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Module: data.int.default
 Author: Jeremy Avigad
 -/
-import .basic .order .div
+import .basic .order .div .power

library/data/int/int.md

@@ -6,3 +6,4 @@ The integers.
 * [basic](basic.lean) : the integers, with basic operations
 * [order](order.lean) : the order relations and the sign function
 * [div](div.lean)     : div, mod, gcd, lcm
+* [power](power.lean) 

library/data/int/order.lean

@@ -259,6 +259,9 @@ section migrate_algebra
 
   migrate from algebra with int
   replacing has_le.ge → ge, has_lt.gt → gt, sign → sign, abs → abs, dvd → dvd, sub → sub
+
+  attribute le.trans ge.trans lt.trans gt.trans [trans]
+  attribute lt_of_lt_of_le lt_of_le_of_lt gt_of_gt_of_ge gt_of_ge_of_gt [trans]
 end migrate_algebra
 
 /- more facts specific to int -/

library/data/int/power.lean

@@ -0,0 +1,30 @@
+/-
+Copyright (c) 2015 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Jeremy Avigad
+
+The power function on the integers.
+-/
+import data.int.basic data.int.order data.int.div algebra.group_power
+
+namespace int
+
+section migrate_algebra
+  open [classes] algebra
+
+  local attribute int.integral_domain [instance]
+  local attribute int.linear_ordered_comm_ring [instance]
+  local attribute int.decidable_linear_ordered_comm_ring [instance]
+
+  definition pow (a : ℤ) (n : ℕ) : ℤ := algebra.pow a n
+  infix ^ := pow
+
+  migrate from algebra with int
+    replacing dvd → dvd, has_le.ge → ge, has_lt.gt → gt, pow → pow
+    hiding add_pos_of_pos_of_nonneg,  add_pos_of_nonneg_of_pos,
+      add_eq_zero_iff_eq_zero_and_eq_zero_of_nonneg_of_nonneg, le_add_of_nonneg_of_le,
+      le_add_of_le_of_nonneg, lt_add_of_nonneg_of_lt, lt_add_of_lt_of_nonneg,
+      lt_of_mul_lt_mul_left, lt_of_mul_lt_mul_right, pos_of_mul_pos_left, pos_of_mul_pos_right
+end migrate_algebra
+
+end int

library/data/nat/order.lean

@@ -175,6 +175,9 @@ section migrate_algebra
       le_add_of_le_of_nonneg, lt_add_of_nonneg_of_lt, lt_add_of_lt_of_nonneg,
       lt_of_mul_lt_mul_left, lt_of_mul_lt_mul_right, pos_of_mul_pos_left, pos_of_mul_pos_right
 
+  attribute le.trans ge.trans lt.trans gt.trans [trans]
+  attribute lt_of_lt_of_le lt_of_le_of_lt gt_of_gt_of_ge gt_of_ge_of_gt [trans]
+
   theorem add_pos_left : ∀{a : ℕ}, 0 < a → ∀b : ℕ, 0 < a + b :=
     take a H b, @algebra.add_pos_of_pos_of_nonneg _ _ a b H !zero_le
   theorem add_pos_right : ∀{a : ℕ}, 0 < a → ∀b : ℕ, 0 < b + a :=

library/data/nat/power.lean

@@ -15,6 +15,8 @@ section migrate_algebra
   local attribute nat.linear_ordered_semiring [instance]
 
   definition pow (a : ℕ) (n : ℕ) : ℕ := algebra.pow a n
+  infix ^ := pow
+
   migrate from algebra with nat
     replacing dvd → dvd, has_le.ge → ge, has_lt.gt → gt, pow → pow
     hiding add_pos_of_pos_of_nonneg,  add_pos_of_nonneg_of_pos,

library/data/rat/default.lean

@@ -1,8 +1,6 @@
 /-
 Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-
-Module: data.rat.default
 Author: Jeremy Avigad
 -/
-import .basic
+import .basic .order