fix(library/data/real/{basic,division,order}: use notation for 0 and 1

These changes were needed because e.g. real.add_zero was "x + zero = x", and rewrite
would not match it to a goal "t + 0".

The fix was a lot harder than I expected. At first, migrate failed with resource
errors. In the end, what worked was this: I defined the coercion from num to real
directly (rather than infer num -> nat -> int -> rat -> real).

I still don't understand what the issues are, though. There are subtle issues with
numerals and coercions and migrate.

(We are not alone. Isabelle also suffers from the fact that there are too many
"zero"s and "one"s.)

library/data/rat/basic.lean

@@ -512,7 +512,7 @@ section migrate_algebra
     add_comm         := add.comm,
     mul              := mul,
     mul_assoc        := mul.assoc,
-    one              := (of_num 1),
+    one              := 1,
     one_mul          := one_mul,
     mul_one          := mul_one,
     left_distrib     := mul.left_distrib,

library/data/real/basic.lean

@@ -1029,10 +1029,17 @@ definition neg (x : ℝ) : ℝ :=
                                    quot.sound (rneg_well_defined Hab)))
 prefix [priority real.prio] `-` := neg
 
-definition zero : ℝ := quot.mk r_zero
+open rat -- no coercions before
+
+definition of_rat [coercion] (a : ℚ) : ℝ := quot.mk (s.r_const a)
+definition of_num [coercion] [reducible] (n : num) : ℝ := of_rat (rat.of_num n)
+
+--definition zero : ℝ := 0
+--definition one : ℝ := 1
+--definition zero : ℝ := quot.mk r_zero
 --notation 0 := zero
 
-definition one : ℝ := quot.mk r_one
+--definition one : ℝ := quot.mk r_one
 
 theorem add_comm (x y : ℝ) : x + y = y + x :=
   quot.induction_on₂ x y (λ s t, quot.sound (r_add_comm s t))
@@ -1040,13 +1047,13 @@ theorem add_comm (x y : ℝ) : x + y = y + x :=
 theorem add_assoc (x y z : ℝ) : x + y + z = x + (y + z) :=
   quot.induction_on₃ x y z (λ s t u, quot.sound (r_add_assoc s t u))
 
-theorem zero_add (x : ℝ) : zero + x = x :=
+theorem zero_add (x : ℝ) : 0 + x = x :=
   quot.induction_on x (λ s, quot.sound (r_zero_add s))
 
-theorem add_zero (x : ℝ) : x + zero = x :=
+theorem add_zero (x : ℝ) : x + 0 = x :=
   quot.induction_on x (λ s, quot.sound (r_add_zero s))
 
-theorem neg_cancel (x : ℝ) : -x + x = zero :=
+theorem neg_cancel (x : ℝ) : -x + x = 0 :=
   quot.induction_on x (λ s, quot.sound (r_neg_cancel s))
 
 theorem mul_assoc (x y z : ℝ) : x * y * z = x * (y * z) :=
@@ -1055,10 +1062,10 @@ theorem mul_assoc (x y z : ℝ) : x * y * z = x * (y * z) :=
 theorem mul_comm (x y : ℝ) : x * y = y * x :=
   quot.induction_on₂ x y (λ s t, quot.sound (r_mul_comm s t))
 
-theorem one_mul (x : ℝ) : one * x = x :=
+theorem one_mul (x : ℝ) : 1 * x = x :=
   quot.induction_on x (λ s, quot.sound (r_one_mul s))
 
-theorem mul_one (x : ℝ) : x * one = x :=
+theorem mul_one (x : ℝ) : x * 1 = x :=
   quot.induction_on x (λ s, quot.sound (r_mul_one s))
 
 theorem distrib (x y z : ℝ) : x * (y + z) = x * y + x * z :=
@@ -1067,8 +1074,8 @@ theorem distrib (x y z : ℝ) : x * (y + z) = x * y + x * z :=
 theorem distrib_l (x y z : ℝ) : (x + y) * z = x * z + y * z :=
   by rewrite [mul_comm, distrib, {x * _}mul_comm, {y * _}mul_comm] -- this shouldn't be necessary
 
-theorem zero_ne_one : ¬ zero = one :=
-  take H : zero = one,
+theorem zero_ne_one : ¬ (0 : ℝ) = 1 :=
+  take H : 0 = 1,
   absurd (quot.exact H) (r_zero_nequiv_one)
 
 protected definition comm_ring [reducible] : algebra.comm_ring ℝ :=
@@ -1076,7 +1083,7 @@ protected definition comm_ring [reducible] : algebra.comm_ring ℝ :=
     fapply algebra.comm_ring.mk,
     exact add,
     exact add_assoc,
-    exact zero,
+    exact of_num 0,
     exact zero_add,
     exact add_zero,
     exact neg,
@@ -1084,7 +1091,7 @@ protected definition comm_ring [reducible] : algebra.comm_ring ℝ :=
     exact add_comm,
     exact mul,
     exact mul_assoc,
-    apply one,
+    apply of_num 1,
     apply one_mul,
     apply mul_one,
     apply distrib,
@@ -1092,10 +1099,6 @@ protected definition comm_ring [reducible] : algebra.comm_ring ℝ :=
     apply mul_comm
   end
 
-open rat -- no coercions before
-
-definition of_rat [coercion] (a : ℚ) : ℝ := quot.mk (s.r_const a)
-
 theorem of_rat_add (a b : ℚ) : of_rat a + of_rat b = of_rat (a + b) :=
    quot.sound (s.r_add_consts a b)
 

library/data/real/division.lean

@@ -584,10 +584,10 @@ postfix [priority real.prio] `⁻¹` := inv
 theorem le_total (x y : ℝ) : x ≤ y ∨ y ≤ x :=
   quot.induction_on₂ x y (λ s t, s.r_le_total s t)
 
-theorem mul_inv' (x : ℝ) : x ≢ zero → x * x⁻¹ = one :=
+theorem mul_inv' (x : ℝ) : x ≢ 0 → x * x⁻¹ = 1 :=
   quot.induction_on x (λ s H, quot.sound (s.r_mul_inv s H))
 
-theorem inv_mul' (x : ℝ) : x ≢ zero → x⁻¹ * x = one :=
+theorem inv_mul' (x : ℝ) : x ≢ 0 → x⁻¹ * x = 1 :=
   by rewrite real.mul_comm; apply mul_inv'
 
 theorem neq_of_sep {x y : ℝ} (H : x ≢ y) : ¬ x = y :=
@@ -599,11 +599,11 @@ theorem sep_of_neq {x y : ℝ} : ¬ x = y → x ≢ y :=
 theorem sep_is_neq (x y : ℝ) : (x ≢ y) = (¬ x = y) :=
   propext (iff.intro neq_of_sep sep_of_neq)
 
-theorem mul_inv (x : ℝ) : x ≠ zero → x * x⁻¹ = one := !sep_is_neq ▸ !mul_inv'
+theorem mul_inv (x : ℝ) : x ≠ 0 → x * x⁻¹ = 1 := !sep_is_neq ▸ !mul_inv'
 
-theorem inv_mul (x : ℝ) : x ≠ zero → x⁻¹ * x = one := !sep_is_neq ▸ !inv_mul'
+theorem inv_mul (x : ℝ) : x ≠ 0 → x⁻¹ * x = 1 := !sep_is_neq ▸ !inv_mul'
 
-theorem inv_zero : zero⁻¹ = zero := quot.sound (s.r_inv_zero)
+theorem inv_zero : (0 : ℝ)⁻¹ = 0 := quot.sound (s.r_inv_zero)
 
 theorem lt_or_eq_of_le (x y : ℝ) : x ≤ y → x < y ∨ x = y :=
   quot.induction_on₂ x y (λ s t H, or.elim (s.r_lt_or_equiv_of_le s t H)

library/data/real/order.lean

@@ -17,8 +17,6 @@ local notation 0 := rat.of_num 0
 local notation 1 := rat.of_num 1
 local notation 2 := subtype.tag (of_num 2) dec_trivial
 
-----------------------------------------------------------------------------------------------------
-
 namespace s
 definition pos (s : seq) := ∃ n : ℕ+, n⁻¹ < (s n)
 
@@ -1026,8 +1024,8 @@ definition lt (x y : ℝ) := quot.lift_on₂ x y (λ a b, s.r_lt a b) s.r_lt_wel
 infix [priority real.prio] `<` := lt
 
 definition le (x y : ℝ) := quot.lift_on₂ x y (λ a b, s.r_le a b) s.r_le_well_defined
-infix [priority real.prio] `≤` := le
 infix [priority real.prio] `<=` := le
+infix [priority real.prio] `≤` := le
 
 definition gt [reducible] (a b : ℝ) := lt b a
 definition ge [reducible] (a b : ℝ) := le b a
@@ -1084,7 +1082,7 @@ theorem add_lt_add_left_var (x y z : ℝ) : x < y → z + x < z + y :=
 theorem add_lt_add_left (x y : ℝ) : x < y → ∀ z : ℝ, z + x < z + y :=
   take H z, add_lt_add_left_var x y z H
 
-theorem zero_lt_one : zero < one := s.r_zero_lt_one
+theorem zero_lt_one : (0 : ℝ) < (1 : ℝ) := s.r_zero_lt_one
 
 theorem le_of_lt_or_eq (x y : ℝ) : x < y ∨ x = y → x ≤ y :=
     (quot.induction_on₂ x y (λ s t H, or.elim H (take H', begin