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)
 

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,