feat (hott/init): move nat.of_num to init.num and make it reducible outside namespace nat

This is so that init.trunc can already use nat.of_num.
Also make nat.of_num reducible in the standard library
Also make gt and ge abbreviations

hott/init/nat.hlean

@@ -1,6 +1,8 @@
 /-
 Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
+
+Module init.nat
 Authors: Floris van Doorn, Leonardo de Moura
 -/
 
@@ -274,18 +276,15 @@ namespace nat
         have aux : a = max a b, from max.left_eq (lt.asymm h),
         eq.rec_on aux (le.of_lt h)))
 
-  definition gt a b := lt b a
+  abbreviation gt a b := lt b a
 
   notation a > b := gt a b
 
-  definition ge a b := le b a
+  abbreviation ge a b := le b a
 
   notation a ≥ b := ge a b
 
-  definition add (a b : nat) : nat :=
-  nat.rec_on b a (λ b₁ r, succ r)
-
-  notation a + b := add a b
+  -- add is defined in init.num
 
   definition sub (a b : nat) : nat :=
   nat.rec_on b a (λ b₁ r, pred r)
@@ -343,7 +342,4 @@ namespace nat
     (le.refl a)
     (λ b₁ ih, le.trans !pred_le ih)
 
-  definition of_num [coercion] [reducible] (n : num) : ℕ :=
-  num.rec zero
-    (λ n, pos_num.rec (succ zero) (λ n r, r + r + (succ zero)) (λ n r, r + r) n) n
 end nat

hott/init/num.hlean

@@ -2,6 +2,7 @@
 Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 
+Module init.num
 Authors: Leonardo de Moura
 -/
 prelude
@@ -72,3 +73,17 @@ namespace num
   notation a + b := add a b
   notation a * b := mul a b
 end num
+
+-- the coercion from num to nat is defined here, so that it can already be used in init.trunc
+namespace nat
+
+  definition add (a b : nat) : nat :=
+  nat.rec_on b a (λ b₁ r, succ r)
+
+  notation a + b := add a b
+
+  definition of_num [coercion] (n : num) : nat :=
+  num.rec zero
+    (λ n, pos_num.rec (succ zero) (λ n r, r + r + (succ zero)) (λ n r, r + r) n) n
+end nat
+attribute nat.of_num [reducible] -- of_num is also reducible if namespace "nat" is not opened

hott/init/trunc.hlean

@@ -6,20 +6,18 @@ Module: init.trunc
 Authors: Jeremy Avigad, Floris van Doorn
 
 Ported from Coq HoTT.
+
+TODO: can we replace some definitions with a hprop as codomain by theorems?
 -/
 
 prelude
-import .path .logic .datatypes .equiv .types.empty .types.sigma
+import .logic .equiv .types.empty .types.sigma
 open eq nat sigma unit
 
-/- Truncation levels -/
-
--- TODO: can we replace some definitions with a hprop as codomain by theorems?
-
-/- truncation indices -/
-
 namespace is_trunc
 
+ /- Truncation levels -/
+
   inductive trunc_index : Type₁ :=
   | minus_two : trunc_index
   | succ : trunc_index → trunc_index
@@ -32,7 +30,7 @@ namespace is_trunc
   postfix `.+2`:(max+1) := λn, (n .+1 .+1)
   notation `-2` := trunc_index.minus_two
   notation `-1` := -2.+1
-  export [coercions] nat -- does this export
+  export [coercions] nat
 
   namespace trunc_index
   definition add (n m : trunc_index) : trunc_index :=
@@ -81,7 +79,7 @@ namespace is_trunc
 
   abbreviation is_contr := is_trunc -2
   abbreviation is_hprop := is_trunc -1
-  abbreviation is_hset  := is_trunc nat.zero
+  abbreviation is_hset  := is_trunc 0
 
   variables {A B : Type}
 

hott/types/trunc.hlean

@@ -11,7 +11,6 @@ Properties of is_trunc
 import types.pi types.path
 
 open sigma sigma.ops pi function eq equiv path funext
-
 namespace is_trunc
 
   definition is_contr.sigma_char (A : Type) :
@@ -108,8 +107,9 @@ namespace is_trunc
 
   section
   open decidable
+  --this is proven differently in init.hedberg
   definition is_hset_of_decidable_eq (A : Type)
-    [H : Π(a b : A), decidable (a = b)] : is_hset A :=
+    [H : decidable_eq A] : is_hset A :=
   is_hset_of_double_neg_elim (λa b, by_contradiction)
   end
 

library/init/nat.lean

@@ -334,7 +334,8 @@ namespace nat
     (le.refl a)
     (λ b₁ ih, le.trans !pred_le ih)
 
-  definition of_num [coercion] [reducible] (n : num) : ℕ :=
+  definition of_num [coercion] (n : num) : ℕ :=
   num.rec zero
     (λ n, pos_num.rec (succ zero) (λ n r, r + r + (succ zero)) (λ n r, r + r) n) n
 end nat
+attribute nat.of_num [reducible] -- of_num is also reducible if namespace "nat" is not opened