fix(library/algebra/ring.lean): allow degenerate semirings and rings, but not degenerate ordered_semirings and ordered_rings. Closes #478.

library/algebra/field.lean

@@ -17,7 +17,7 @@ namespace algebra
 
 variable {A : Type}
 
-structure division_ring [class] (A : Type) extends ring A, has_inv A :=
+structure division_ring [class] (A : Type) extends ring A, has_inv A, zero_ne_one_class A :=
   (mul_inv_cancel : ∀{a}, a ≠ zero → mul a (inv a) = one)
   (inv_mul_cancel : ∀{a}, a ≠ zero → mul (inv a) a = one)
 
@@ -40,21 +40,21 @@ section division_ring
   theorem inv_eq_one_div : a⁻¹ = 1 / a := !one_mul⁻¹
 
   theorem div_eq_mul_one_div : a / b = a * (1 / b) :=
-    by rewrite [↑divide, one_mul] 
+    by rewrite [↑divide, one_mul]
 
   theorem mul_one_div_cancel (H : a ≠ 0) : a * (1 / a) = 1 :=
     by rewrite [-inv_eq_one_div, (mul_inv_cancel H)]
 
   theorem one_div_mul_cancel (H : a ≠ 0) : (1 / a) * a = 1 :=
     by rewrite [-inv_eq_one_div, (inv_mul_cancel H)]
- 
+
   theorem div_self (H : a ≠ 0) : a / a = 1 := mul_inv_cancel H
 
   theorem mul_div_assoc : (a * b) / c = a * (b / c) := !mul.assoc
 
   theorem one_div_ne_zero (H : a ≠ 0) : 1 / a ≠ 0 :=
     assume H2 : 1 / a = 0,
-    have C1 : 0 = 1, from symm (by rewrite [-(mul_one_div_cancel H), H2, mul_zero]), 
+    have C1 : 0 = 1, from symm (by rewrite [-(mul_one_div_cancel H), H2, mul_zero]),
     absurd C1 zero_ne_one
 
   -- the analogue in group is called inv_one
@@ -93,7 +93,7 @@ section division_ring
 
   -- make "left" and "right" versions?
   theorem eq_one_div_of_mul_eq_one (H : a * b = 1) : b = 1 / a :=
-    have H2 : a ≠ 0, from 
+    have H2 : a ≠ 0, from
       (assume A : a = 0,
       have B : 0 = 1, by rewrite [-(zero_mul b), -A, H],
       absurd B zero_ne_one),
@@ -106,7 +106,7 @@ section division_ring
 
   -- which one is left and which is right?
   theorem eq_one_div_of_mul_eq_one_left (H : b * a = 1) : b = 1 / a :=
-    have H2 : a ≠ 0, from 
+    have H2 : a ≠ 0, from
       (assume A : a = 0,
       have B : 0 = 1, from symm (calc
         1 = b * a : symm H
@@ -121,9 +121,9 @@ section division_ring
         ... = b                 : mul_one)
 
   theorem one_div_mul_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : (1 / a) * (1 / b) = 1 / (b * a) :=
-    have H : (b * a) * ((1 / a) * (1 / b)) = 1, by 
+    have H : (b * a) * ((1 / a) * (1 / b)) = 1, by
       rewrite [mul.assoc, -(mul.assoc a), (mul_one_div_cancel Ha), one_mul, (mul_one_div_cancel Hb)],
-    eq_one_div_of_mul_eq_one H 
+    eq_one_div_of_mul_eq_one H
 
   theorem one_div_neg_one_eq_neg_one : 1 / (-1) = -1 :=
     have H : (-1) * (-1) = 1, by rewrite [-neg_eq_neg_one_mul, neg_neg],
@@ -135,10 +135,10 @@ section division_ring
       a + a * -1 = a * 1 + a * -1 : mul_one
              ... = a * (1 + -1)   : left_distrib
              ... = a * 0          : add.right_inv
-             ... = 0              : mul_zero, 
+             ... = 0              : mul_zero,
     symm (neg_eq_of_add_eq_zero H)
 
-  theorem one_div_neg_eq_neg_one_div (H : a ≠ 0) : 1 / (- a) = - (1 / a) := 
+  theorem one_div_neg_eq_neg_one_div (H : a ≠ 0) : 1 / (- a) = - (1 / a) :=
     have H1 : -1 ≠ 0, from
       (assume H2 : -1 = 0, absurd (symm (calc
           1 = -(-1) : neg_neg
@@ -146,7 +146,7 @@ section division_ring
         ... = 0     : neg_zero)) zero_ne_one),
     calc
       1 / (- a) = 1 / ((-1) * a)        : neg_eq_neg_one_mul
-            ... = (1 / a) * (1 / (- 1)) : one_div_mul_one_div H H1 
+            ... = (1 / a) * (1 / (- 1)) : one_div_mul_one_div H H1
             ... = (1 / a) * (-1)        : one_div_neg_one_eq_neg_one
             ... = - (1 / a)             : mul_neg_one_eq_neg
 
@@ -184,7 +184,7 @@ section division_ring
 
   theorem div_mul_cancel (Hb : b ≠ 0) : a / b * b = a :=
     by rewrite [↑divide, mul.assoc, (inv_mul_cancel Hb), mul_one]
- 
+
   theorem div_add_div_same : a / c + b / c = (a + b) / c := !right_distrib⁻¹
 
   theorem inv_mul_add_mul_inv_eq_inv_add_inv (Ha : a ≠ 0) (Hb : b ≠ 0) :
@@ -197,7 +197,7 @@ section division_ring
     by rewrite [(mul_sub_left_distrib (1 / a)), (one_div_mul_cancel Ha), mul_sub_right_distrib,
       one_mul, mul.assoc, (mul_one_div_cancel Hb), mul_one, one_mul]
 
-  theorem div_eq_one_iff_eq (Hb : b ≠ 0) : a / b = 1 ↔ a = b := 
+  theorem div_eq_one_iff_eq (Hb : b ≠ 0) : a / b = 1 ↔ a = b :=
     iff.intro
     (assume H1 : a / b = 1, symm (calc
       b   = 1 * b     : one_mul
@@ -268,10 +268,10 @@ section field
 
   theorem div_mul_eq_mul_div (Hc : c ≠ 0) : (b / c) * a = (b * a) / c :=
     by rewrite [↑divide, mul.assoc, (mul.comm c⁻¹), -mul.assoc]
-  
+
   -- this one is odd -- I am not sure what to call it, but again, the prefix is right
   theorem div_mul_eq_mul_div_comm (Hc : c ≠ 0) : (b / c) * a = b * (a / c) :=
-    by rewrite [(div_mul_eq_mul_div Hc), -(one_mul c), -(div_mul_div (ne.symm zero_ne_one) Hc), div_one, one_mul] 
+    by rewrite [(div_mul_eq_mul_div Hc), -(one_mul c), -(div_mul_div (ne.symm zero_ne_one) Hc), div_one, one_mul]
 
   theorem div_add_div (Hb : b ≠ 0) (Hd : d ≠ 0) :
       (a / b) + (c / d) = ((a * d) + (b * c)) / (b * d) :=
@@ -282,7 +282,7 @@ section field
       (a / b) - (c / d) = ((a * d) - (b * c)) / (b * d) :=
       by rewrite [↑sub, neg_eq_neg_one_mul, -mul_div_assoc, (div_add_div Hb Hd),
          -mul.assoc, (mul.comm b), mul.assoc, -neg_eq_neg_one_mul]
-   
+
   theorem mul_eq_mul_of_div_eq_div (Hb : b ≠ 0) (Hd : d ≠ 0) (H : a / b = c / d) : a * d = c * b :=
     by rewrite [-mul_one, mul.assoc, (mul.comm d), -mul.assoc, -(div_self Hb),
          -(div_mul_eq_mul_div_comm Hb), H, (div_mul_eq_mul_div Hd), (div_mul_cancel Hd)]
@@ -323,9 +323,9 @@ section discrete_field
   decidable.by_cases
     (assume H : x = 0, or.inl H)
     (assume H1 : x ≠ 0,
-      or.inr (by rewrite [-one_mul, -(inv_mul_cancel H1), mul.assoc, H, mul_zero])) 
+      or.inr (by rewrite [-one_mul, -(inv_mul_cancel H1), mul.assoc, H, mul_zero]))
 
-  definition discrete_field.to_integral_domain [instance] [reducible] [coercion] : 
+  definition discrete_field.to_integral_domain [instance] [reducible] [coercion] :
     integral_domain A :=
   ⦃ integral_domain, s,
     eq_zero_or_eq_zero_of_mul_eq_zero := discrete_field.eq_zero_or_eq_zero_of_mul_eq_zero⦄

library/algebra/ordered_ring.lean

@@ -22,7 +22,7 @@ absurd H (lt.irrefl a)
 
 structure ordered_semiring [class] (A : Type)
   extends has_mul A, has_zero A, has_lt A, -- TODO: remove hack for improving performance
-    semiring A, ordered_cancel_comm_monoid A :=
+    semiring A, ordered_cancel_comm_monoid A, zero_ne_one_class A :=
 (mul_le_mul_of_nonneg_left: ∀a b c, le a b → le zero c → le (mul c a) (mul c b))
 (mul_le_mul_of_nonneg_right: ∀a b c, le a b → le zero c → le (mul a c) (mul b c))
 (mul_lt_mul_of_pos_left: ∀a b c, lt a b → lt zero c → lt (mul c a) (mul c b))
@@ -147,7 +147,7 @@ section
       not_lt_of_le H3 H)
 end
 
-structure ordered_ring [class] (A : Type) extends ring A, ordered_comm_group A :=
+structure ordered_ring [class] (A : Type) extends ring A, ordered_comm_group A, zero_ne_one_class A :=
 (mul_nonneg : ∀a b, le zero a → le zero b → le zero (mul a b))
 (mul_pos : ∀a b, lt zero a → lt zero b → lt zero (mul a b))
 

library/algebra/ring.lean

@@ -44,7 +44,7 @@ theorem zero_ne_one [s: zero_ne_one_class A] : 0 ≠ 1 := @zero_ne_one_class.zer
 /- semiring -/
 
 structure semiring [class] (A : Type) extends add_comm_monoid A, monoid A, distrib A,
-    mul_zero_class A, zero_ne_one_class A
+    mul_zero_class A
 
 section semiring
   variables [s : semiring A] (a b c : A)
@@ -148,7 +148,7 @@ end comm_semiring
 
 /- ring -/
 
-structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A, zero_ne_one_class A
+structure ring [class] (A : Type) extends add_comm_group A, monoid A, distrib A
 
 theorem ring.mul_zero [s : ring A] (a : A) : a * 0 = 0 :=
 have H : a * 0 + 0 = a * 0 + a * 0, from calc

library/data/int/basic.lean

@@ -629,7 +629,6 @@ section
     mul_one        := mul_one,
     left_distrib   := mul.left_distrib,
     right_distrib  := mul.right_distrib,
-    zero_ne_one    := zero_ne_one,
     mul_comm       := mul.comm,
     eq_zero_or_eq_zero_of_mul_eq_zero := @eq_zero_or_eq_zero_of_mul_eq_zero⦄
 end

library/data/int/order.lean

@@ -233,7 +233,8 @@ section
     mul_nonneg       := @mul_nonneg,
     mul_pos          := @mul_pos,
     le_iff_lt_or_eq  := le_iff_lt_or_eq,
-    le_total         := le.total⦄
+    le_total         := le.total,
+    zero_ne_one      := zero_ne_one⦄
 
   protected definition decidable_linear_ordered_comm_ring [instance] [reducible] :
     algebra.decidable_linear_ordered_comm_ring int :=

library/data/nat/basic.lean

@@ -307,7 +307,6 @@ section
     right_distrib  := mul.right_distrib,
     zero_mul       := zero_mul,
     mul_zero       := mul_zero,
-    zero_ne_one    := ne.symm (succ_ne_zero zero),
     mul_comm       := mul.comm⦄
 end
 

library/data/nat/order.lean

@@ -161,6 +161,7 @@ section
     lt_iff_le_ne               := lt_iff_le_and_ne,
     add_le_add_left            := @add_le_add_left,
     le_of_add_le_add_left      := @le_of_add_le_add_left,
+    zero_ne_one                := ne.symm (succ_ne_zero zero),
     mul_le_mul_of_nonneg_left  := (take a b c H1 H2, mul_le_mul_left H1 c),
     mul_le_mul_of_nonneg_right := (take a b c H1 H2, mul_le_mul_right H1 c),
     mul_lt_mul_of_pos_left     := @mul_lt_mul_of_pos_left,