feat(library/algebra): finish/move more general theorems from reals to algebra)

library/algebra/field.lean

@@ -448,6 +448,9 @@ section discrete_field
   theorem div_div_div_div' : (a / b) / (c / d) = (a * d) / (b * c) :=
     by rewrite [div_div_eq_mul_div', div_mul_eq_mul_div, div_div_eq_div_mul']
 
+  theorem div_helper (H : a ≠ 0) : (1 / (a * b)) * a = 1 / b :=
+    by rewrite [div_mul_eq_mul_div, one_mul, (div_mul_right' H)]
+
 end discrete_field
 
 end algebra

library/algebra/ordered_field.lean

@@ -288,6 +288,17 @@ theorem nonneg_le_nonneg_of_squares_le  (Ha : a ≥ 0) (Hb : b ≥ 0) (H : a * a
     apply (not_le_of_gt H') H
   end
 
+  theorem div_two : (a + a) / (1 + 1) = a :=
+    symm (iff.mp' (eq_div_iff_mul_eq (ne_of_gt (add_pos zero_lt_one zero_lt_one)))
+           (by rewrite [left_distrib, *mul_one]))
+
+
+  theorem mul_le_mul_of_mul_div_le (H : a * (b / c) ≤ d) (Hc : c > 0) : b * a ≤ d * c :=
+    begin
+      rewrite [-mul_div_assoc at H, mul.comm b],
+      apply le_mul_of_div_le Hc H
+    end
+
 end linear_ordered_field
 
 structure discrete_linear_ordered_field [class] (A : Type) extends linear_ordered_field A,

library/algebra/ordered_group_2.lean

@@ -0,0 +1,561 @@
+/-
+Copyright (c) 2014 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jeremy Avigad
+
+Partially ordered additive groups, modeled on Isabelle's library. We could refine the structures,
+but we would have to declare more inheritance paths.
+-/
+
+import logic.eq data.unit data.sigma data.prod
+import algebra.function algebra.binary
+import algebra.group algebra.order
+open eq eq.ops   -- note: ⁻¹ will be overloaded
+
+namespace algebra
+
+variable {A : Type}
+
+/- partially ordered monoids, such as the natural numbers -/
+
+structure ordered_cancel_comm_monoid [class] (A : Type) extends add_comm_monoid A,
+  add_left_cancel_semigroup A, add_right_cancel_semigroup A, order_pair A :=
+(add_le_add_left : ∀a b, le a b → ∀c, le (add c a) (add c b))
+(le_of_add_le_add_left : ∀a b c, le (add a b) (add a c) → le b c)
+
+section
+  variables [s : ordered_cancel_comm_monoid A]
+  variables {a b c d e : A}
+  include s
+
+  theorem add_le_add_left (H : a ≤ b) (c : A) : c + a ≤ c + b :=
+  !ordered_cancel_comm_monoid.add_le_add_left H c
+
+  theorem add_le_add_right (H : a ≤ b) (c : A) : a + c ≤ b + c :=
+  (add.comm c a) ▸ (add.comm c b) ▸ (add_le_add_left H c)
+
+  theorem add_le_add (Hab : a ≤ b) (Hcd : c ≤ d) : a + c ≤ b + d :=
+  le.trans (add_le_add_right Hab c) (add_le_add_left Hcd b)
+
+  theorem add_lt_add_left (H : a < b) (c : A) : c + a < c + b :=
+  have H1 : c + a ≤ c + b, from add_le_add_left (le_of_lt H) c,
+  have H2 : c + a ≠ c + b, from
+    take H3 : c + a = c + b,
+    have H4 : a = b, from add.left_cancel H3,
+    ne_of_lt H H4,
+  sorry--lt_of_le_of_ne H1 H2
+
+  theorem add_lt_add_right (H : a < b) (c : A) : a + c < b + c :=
+  begin
+    rewrite [add.comm, {b + _}add.comm],
+    exact (add_lt_add_left H c)
+  end
+
+  theorem le_add_of_nonneg_right (H : b ≥ 0) : a ≤ a + b :=
+  begin
+    have H1 : a + b ≥ a + 0, from add_le_add_left H a,
+    rewrite add_zero at H1,
+    exact H1
+  end
+
+  theorem le_add_of_nonneg_left (H : b ≥ 0) : a ≤ b + a :=
+  begin
+    have H1 : 0 + a ≤ b + a, from add_le_add_right H a,
+    rewrite zero_add at H1,
+    exact H1
+  end
+
+  theorem add_lt_add (Hab : a < b) (Hcd : c < d) : a + c < b + d :=
+  lt.trans (add_lt_add_right Hab c) (add_lt_add_left Hcd b)
+
+  theorem add_lt_add_of_le_of_lt (Hab : a ≤ b) (Hcd : c < d) : a + c < b + d :=
+  lt_of_le_of_lt (add_le_add_right Hab c) (add_lt_add_left Hcd b)
+
+  theorem add_lt_add_of_lt_of_le (Hab : a < b) (Hcd : c ≤ d) : a + c < b + d :=
+  lt_of_lt_of_le (add_lt_add_right Hab c) (add_le_add_left Hcd b)
+
+  theorem lt_add_of_pos_right (H : b > 0) : a < a + b := !add_zero ▸ add_lt_add_left H a
+
+  theorem lt_add_of_pos_left (H : b > 0) : a < b + a := !zero_add ▸ add_lt_add_right H a
+
+  -- here we start using le_of_add_le_add_left.
+  theorem le_of_add_le_add_left (H : a + b ≤ a + c) : b ≤ c :=
+  !ordered_cancel_comm_monoid.le_of_add_le_add_left H
+
+  theorem le_of_add_le_add_right (H : a + b ≤ c + b) : a ≤ c :=
+  le_of_add_le_add_left (show b + a ≤ b + c, begin rewrite [add.comm, {b + _}add.comm], exact H end)
+
+  theorem lt_of_add_lt_add_left (H : a + b < a + c) : b < c :=
+  have H1 : b ≤ c, from le_of_add_le_add_left (le_of_lt H),
+  have H2 : b ≠ c, from
+    assume H3 : b = c, lt.irrefl _ (H3 ▸ H),
+  sorry --lt_of_le_of_ne H1 H2
+
+  theorem lt_of_add_lt_add_right (H : a + b < c + b) : a < c :=
+  lt_of_add_lt_add_left ((add.comm a b) ▸ (add.comm c b) ▸ H)
+
+  theorem add_le_add_left_iff (a b c : A) : a + b ≤ a + c ↔ b ≤ c :=
+  iff.intro le_of_add_le_add_left (assume H, add_le_add_left H _)
+
+  theorem add_le_add_right_iff (a b c : A) : a + b ≤ c + b ↔ a ≤ c :=
+  iff.intro le_of_add_le_add_right (assume H, add_le_add_right H _)
+
+  theorem add_lt_add_left_iff (a b c : A) : a + b < a + c ↔ b < c :=
+  iff.intro lt_of_add_lt_add_left (assume H, add_lt_add_left H _)
+
+  theorem add_lt_add_right_iff (a b c : A) : a + b < c + b ↔ a < c :=
+  iff.intro lt_of_add_lt_add_right (assume H, add_lt_add_right H _)
+
+  -- here we start using properties of zero.
+  theorem add_nonneg (Ha : 0 ≤ a) (Hb : 0 ≤ b) : 0 ≤ a + b :=
+  !zero_add ▸ (add_le_add Ha Hb)
+
+  theorem add_pos (Ha : 0 < a) (Hb : 0 < b) : 0 < a + b :=
+  !zero_add ▸ (add_lt_add Ha Hb)
+
+  theorem add_pos_of_pos_of_nonneg (Ha : 0 < a) (Hb : 0 ≤ b) : 0 < a + b :=
+  !zero_add ▸ (add_lt_add_of_lt_of_le Ha Hb)
+
+  theorem add_pos_of_nonneg_of_pos (Ha : 0 ≤ a) (Hb : 0 < b) : 0 < a + b :=
+  !zero_add ▸ (add_lt_add_of_le_of_lt Ha Hb)
+
+  theorem add_nonpos (Ha : a ≤ 0) (Hb : b ≤ 0) : a + b ≤ 0 :=
+  !zero_add ▸ (add_le_add Ha Hb)
+
+  theorem add_neg (Ha : a < 0) (Hb : b < 0) : a + b < 0 :=
+  !zero_add ▸ (add_lt_add Ha Hb)
+
+  theorem add_neg_of_neg_of_nonpos (Ha : a < 0) (Hb : b ≤ 0) : a + b < 0 :=
+  !zero_add ▸ (add_lt_add_of_lt_of_le Ha Hb)
+
+  theorem add_neg_of_nonpos_of_neg (Ha : a ≤ 0) (Hb : b < 0) : a + b < 0 :=
+  !zero_add ▸ (add_lt_add_of_le_of_lt Ha Hb)
+
+  -- TODO: add nonpos version (will be easier with simplifier)
+  theorem add_eq_zero_iff_eq_zero_and_eq_zero_of_nonneg_of_nonneg
+    (Ha : 0 ≤ a) (Hb : 0 ≤ b) : a + b = 0 ↔ a = 0 ∧ b = 0 :=
+  iff.intro
+    (assume Hab : a + b = 0,
+      have Ha' : a ≤ 0, from
+        calc
+          a     = a + 0 : by rewrite add_zero
+            ... ≤ a + b : add_le_add_left Hb
+            ... = 0     : Hab,
+      have Haz : a = 0, from le.antisymm Ha' Ha,
+      have Hb' : b ≤ 0, from
+        calc
+          b     = 0 + b : by rewrite zero_add
+            ... ≤ a + b : add_le_add_right Ha
+            ... = 0     : Hab,
+      have Hbz : b = 0, from le.antisymm Hb' Hb,
+      and.intro Haz Hbz)
+    (assume Hab : a = 0 ∧ b = 0,
+     obtain Ha' Hb', from Hab,
+     by rewrite [Ha', Hb', add_zero])
+
+  theorem le_add_of_nonneg_of_le (Ha : 0 ≤ a) (Hbc : b ≤ c) : b ≤ a + c :=
+  !zero_add ▸ add_le_add Ha Hbc
+
+  theorem le_add_of_le_of_nonneg (Hbc : b ≤ c) (Ha : 0 ≤ a) : b ≤ c + a :=
+  !add_zero ▸ add_le_add Hbc Ha
+
+  theorem lt_add_of_pos_of_le (Ha : 0 < a) (Hbc : b ≤ c) : b < a + c :=
+  !zero_add ▸ add_lt_add_of_lt_of_le Ha Hbc
+
+  theorem lt_add_of_le_of_pos (Hbc : b ≤ c) (Ha : 0 < a) : b < c + a :=
+  !add_zero ▸ add_lt_add_of_le_of_lt Hbc Ha
+
+  theorem add_le_of_nonpos_of_le (Ha : a ≤ 0) (Hbc : b ≤ c) : a + b ≤ c :=
+  !zero_add ▸ add_le_add Ha Hbc
+
+  theorem add_le_of_le_of_nonpos (Hbc : b ≤ c) (Ha : a ≤ 0) : b + a ≤ c :=
+  !add_zero ▸ add_le_add Hbc Ha
+
+  theorem add_lt_of_neg_of_le (Ha : a < 0) (Hbc : b ≤ c) : a + b < c :=
+  !zero_add ▸ add_lt_add_of_lt_of_le Ha Hbc
+
+  theorem add_lt_of_le_of_neg (Hbc : b ≤ c) (Ha : a < 0) : b + a < c :=
+  !add_zero ▸ add_lt_add_of_le_of_lt Hbc Ha
+
+  theorem lt_add_of_nonneg_of_lt (Ha : 0 ≤ a) (Hbc : b < c) : b < a + c :=
+  !zero_add ▸ add_lt_add_of_le_of_lt Ha Hbc
+
+  theorem lt_add_of_lt_of_nonneg (Hbc : b < c) (Ha : 0 ≤ a) : b < c + a :=
+  !add_zero ▸ add_lt_add_of_lt_of_le Hbc Ha
+
+  theorem lt_add_of_pos_of_lt (Ha : 0 < a) (Hbc : b < c) : b < a + c :=
+  !zero_add ▸ add_lt_add Ha Hbc
+
+  theorem lt_add_of_lt_of_pos (Hbc : b < c) (Ha : 0 < a) : b < c + a :=
+  !add_zero ▸ add_lt_add Hbc Ha
+
+  theorem add_lt_of_nonpos_of_lt (Ha : a ≤ 0) (Hbc : b < c) : a + b < c :=
+  !zero_add ▸ add_lt_add_of_le_of_lt Ha Hbc
+
+  theorem add_lt_of_lt_of_nonpos (Hbc : b < c) (Ha : a ≤ 0)  : b + a < c :=
+  !add_zero ▸ add_lt_add_of_lt_of_le Hbc Ha
+
+  theorem add_lt_of_neg_of_lt (Ha : a < 0) (Hbc : b < c) : a + b < c :=
+  !zero_add ▸ add_lt_add Ha Hbc
+
+  theorem add_lt_of_lt_of_neg (Hbc : b < c) (Ha : a < 0) : b + a < c :=
+  !add_zero ▸ add_lt_add Hbc Ha
+end
+
+-- TODO: add properties of max and min
+
+/- partially ordered groups -/
+
+structure ordered_comm_group [class] (A : Type) extends add_comm_group A, order_pair A :=
+(add_le_add_left : ∀a b, le a b → ∀c, le (add c a) (add c b))
+
+theorem ordered_comm_group.le_of_add_le_add_left [s : ordered_comm_group A] {a b c : A} (H : a + b ≤ a + c) : b ≤ c :=
+assert H' : -a + (a + b) ≤ -a + (a + c), from ordered_comm_group.add_le_add_left _ _ H _,
+by rewrite *neg_add_cancel_left at H'; exact H'
+
+definition ordered_comm_group.to_ordered_cancel_comm_monoid [instance] [coercion] [reducible]
+    [s : ordered_comm_group A] : ordered_cancel_comm_monoid A :=
+⦃ ordered_cancel_comm_monoid, s,
+  add_left_cancel       := @add.left_cancel A s,
+  add_right_cancel      := @add.right_cancel A s,
+  le_of_add_le_add_left := @ordered_comm_group.le_of_add_le_add_left A s ⦄
+
+section
+  variables [s : ordered_comm_group A] (a b c d e : A)
+  include s
+
+  theorem neg_le_neg {a b : A} (H : a ≤ b) : -b ≤ -a :=
+  have H1 : 0 ≤ -a + b, from !add.left_inv ▸ !(add_le_add_left H),
+  !add_neg_cancel_right ▸ !zero_add ▸ add_le_add_right H1 (-b)
+
+  theorem le_of_neg_le_neg {a b : A} (H : -b ≤ -a) : a ≤ b :=
+  neg_neg a ▸ neg_neg b ▸ neg_le_neg H
+
+  theorem neg_le_neg_iff_le : -a ≤ -b ↔ b ≤ a :=
+  iff.intro le_of_neg_le_neg neg_le_neg
+
+  theorem nonneg_of_neg_nonpos {a : A} (H : -a ≤ 0) : 0 ≤ a :=
+  le_of_neg_le_neg (neg_zero⁻¹ ▸ H)
+
+  theorem neg_nonpos_of_nonneg {a : A} (H : 0 ≤ a) : -a ≤ 0 :=
+  neg_zero ▸ neg_le_neg H
+
+  theorem neg_nonpos_iff_nonneg : -a ≤ 0 ↔ 0 ≤ a :=
+  iff.intro nonneg_of_neg_nonpos neg_nonpos_of_nonneg
+
+  theorem nonpos_of_neg_nonneg {a : A} (H : 0 ≤ -a) : a ≤ 0 :=
+  le_of_neg_le_neg (neg_zero⁻¹ ▸ H)
+
+  theorem neg_nonneg_of_nonpos {a : A} (H : a ≤ 0) : 0 ≤ -a :=
+  neg_zero ▸ neg_le_neg H
+
+  theorem neg_nonneg_iff_nonpos : 0 ≤ -a ↔ a ≤ 0 :=
+  iff.intro nonpos_of_neg_nonneg neg_nonneg_of_nonpos
+
+  theorem neg_lt_neg {a b : A} (H : a < b) : -b < -a :=
+  have H1 : 0 < -a + b, from !add.left_inv ▸ !(add_lt_add_left H),
+  !add_neg_cancel_right ▸ !zero_add ▸ add_lt_add_right H1 (-b)
+
+  theorem lt_of_neg_lt_neg {a b : A} (H : -b < -a) : a < b :=
+  neg_neg a ▸ neg_neg b ▸ neg_lt_neg H
+
+  theorem neg_lt_neg_iff_lt : -a < -b ↔ b < a :=
+  iff.intro lt_of_neg_lt_neg neg_lt_neg
+
+  theorem pos_of_neg_neg {a : A} (H : -a < 0) : 0 < a :=
+  lt_of_neg_lt_neg (neg_zero⁻¹ ▸ H)
+
+  theorem neg_neg_of_pos {a : A} (H : 0 < a) : -a < 0 :=
+  neg_zero ▸ neg_lt_neg H
+
+  theorem neg_neg_iff_pos : -a < 0 ↔ 0 < a :=
+  iff.intro pos_of_neg_neg neg_neg_of_pos
+
+  theorem neg_of_neg_pos {a : A} (H : 0 < -a) : a < 0 :=
+  lt_of_neg_lt_neg (neg_zero⁻¹ ▸ H)
+
+  theorem neg_pos_of_neg {a : A} (H : a < 0) : 0 < -a :=
+  neg_zero ▸ neg_lt_neg H
+
+  theorem neg_pos_iff_neg : 0 < -a ↔ a < 0 :=
+  iff.intro neg_of_neg_pos neg_pos_of_neg
+
+  theorem le_neg_iff_le_neg : a ≤ -b ↔ b ≤ -a := !neg_neg ▸ !neg_le_neg_iff_le
+
+  theorem neg_le_iff_neg_le : -a ≤ b ↔ -b ≤ a := !neg_neg ▸ !neg_le_neg_iff_le
+
+  theorem lt_neg_iff_lt_neg : a < -b ↔ b < -a := !neg_neg ▸ !neg_lt_neg_iff_lt
+
+  theorem neg_lt_iff_neg_lt : -a < b ↔ -b < a := !neg_neg ▸ !neg_lt_neg_iff_lt
+
+  theorem sub_nonneg_iff_le : 0 ≤ a - b ↔ b ≤ a := !sub_self ▸ !add_le_add_right_iff
+
+  theorem sub_nonpos_iff_le : a - b ≤ 0 ↔ a ≤ b := !sub_self ▸ !add_le_add_right_iff
+
+  theorem sub_pos_iff_lt : 0 < a - b ↔ b < a := !sub_self ▸ !add_lt_add_right_iff
+
+  theorem sub_neg_iff_lt : a - b < 0 ↔ a < b := !sub_self ▸ !add_lt_add_right_iff
+
+  theorem add_le_iff_le_neg_add : a + b ≤ c ↔ b ≤ -a + c :=
+  have H: a + b ≤ c ↔ -a + (a + b) ≤ -a + c, from iff.symm (!add_le_add_left_iff),
+  !neg_add_cancel_left ▸ H
+
+  theorem add_le_iff_le_sub_left : a + b ≤ c ↔ b ≤ c - a :=
+  by rewrite [sub_eq_add_neg, {c+_}add.comm]; apply add_le_iff_le_neg_add
+
+  theorem add_le_iff_le_sub_right : a + b ≤ c ↔ a ≤ c - b :=
+  have H: a + b ≤ c ↔ a + b - b ≤ c - b, from iff.symm (!add_le_add_right_iff),
+  !add_neg_cancel_right ▸ H
+
+  theorem le_add_iff_neg_add_le : a ≤ b + c ↔ -b + a ≤ c :=
+  assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff),
+  by rewrite neg_add_cancel_left at H; exact H
+
+  theorem le_add_iff_sub_left_le : a ≤ b + c ↔ a - b ≤ c :=
+  by rewrite [sub_eq_add_neg, {a+_}add.comm]; apply le_add_iff_neg_add_le
+
+  theorem le_add_iff_sub_right_le : a ≤ b + c ↔ a - c ≤ b :=
+  assert H: a ≤ b + c ↔ a - c ≤ b + c - c, from iff.symm (!add_le_add_right_iff),
+  by rewrite add_neg_cancel_right at H; exact H
+
+  theorem add_lt_iff_lt_neg_add_left : a + b < c ↔ b < -a + c :=
+  assert H: a + b < c ↔ -a + (a + b) < -a + c, from iff.symm (!add_lt_add_left_iff),
+  begin rewrite neg_add_cancel_left at H, exact H end
+
+  theorem add_lt_iff_lt_neg_add_right : a + b < c ↔ a < -b + c :=
+  by rewrite add.comm; apply add_lt_iff_lt_neg_add_left
+
+  theorem add_lt_iff_lt_sub_left : a + b < c ↔ b < c - a :=
+  begin
+    rewrite [sub_eq_add_neg, {c+_}add.comm],
+    apply add_lt_iff_lt_neg_add_left
+  end
+
+  theorem add_lt_add_iff_lt_sub_right : a + b < c ↔ a < c - b :=
+  assert H: a + b < c ↔ a + b - b < c - b, from iff.symm (!add_lt_add_right_iff),
+  by rewrite add_neg_cancel_right at H; exact H
+
+  theorem lt_add_iff_neg_add_lt_left : a < b + c ↔ -b + a < c :=
+  assert H: a < b + c ↔ -b + a < -b + (b + c), from iff.symm (!add_lt_add_left_iff),
+  by rewrite neg_add_cancel_left at H; exact H
+
+  theorem lt_add_iff_neg_add_lt_right : a < b + c ↔ -c + a < b :=
+  by rewrite add.comm; apply lt_add_iff_neg_add_lt_left
+
+  theorem lt_add_iff_sub_lt_left : a < b + c ↔ a - b < c :=
+  by rewrite [sub_eq_add_neg, {a + _}add.comm]; apply lt_add_iff_neg_add_lt_left
+
+  theorem lt_add_iff_sub_lt_right : a < b + c ↔ a - c < b :=
+  by rewrite add.comm; apply lt_add_iff_sub_lt_left
+
+  -- TODO: the Isabelle library has varations on a + b ≤ b ↔ a ≤ 0
+  theorem le_iff_le_of_sub_eq_sub {a b c d : A} (H : a - b = c - d) : a ≤ b ↔ c ≤ d :=
+  calc
+    a ≤ b ↔ a - b ≤ 0   : iff.symm (sub_nonpos_iff_le a b)
+      ... = (c - d ≤ 0) : by rewrite H
+      ... ↔ c ≤ d       : sub_nonpos_iff_le c d
+
+  theorem lt_iff_lt_of_sub_eq_sub {a b c d : A} (H : a - b = c - d) : a < b ↔ c < d :=
+  calc
+    a < b ↔ a - b < 0   : iff.symm (sub_neg_iff_lt a b)
+      ... = (c - d < 0) : by rewrite H
+      ... ↔ c < d       : sub_neg_iff_lt c d
+
+  theorem sub_le_sub_left {a b : A} (H : a ≤ b) (c : A) : c - b ≤ c - a :=
+  add_le_add_left (neg_le_neg H) c
+
+  theorem sub_le_sub_right {a b : A} (H : a ≤ b) (c : A) : a - c ≤ b - c := add_le_add_right H (-c)
+
+  theorem sub_le_sub {a b c d : A} (Hab : a ≤ b) (Hcd : c ≤ d) : a - d ≤ b - c :=
+  add_le_add Hab (neg_le_neg Hcd)
+
+  theorem sub_lt_sub_left {a b : A} (H : a < b) (c : A) : c - b < c - a :=
+  add_lt_add_left (neg_lt_neg H) c
+
+  theorem sub_lt_sub_right {a b : A} (H : a < b) (c : A) : a - c < b - c := add_lt_add_right H (-c)
+
+  theorem sub_lt_sub {a b c d : A} (Hab : a < b) (Hcd : c < d) : a - d < b - c :=
+  add_lt_add Hab (neg_lt_neg Hcd)
+
+  theorem sub_lt_sub_of_le_of_lt {a b c d : A} (Hab : a ≤ b) (Hcd : c < d) : a - d < b - c :=
+  add_lt_add_of_le_of_lt Hab (neg_lt_neg Hcd)
+
+  theorem sub_lt_sub_of_lt_of_le {a b c d : A} (Hab : a < b) (Hcd : c ≤ d) : a - d < b - c :=
+  add_lt_add_of_lt_of_le Hab (neg_le_neg Hcd)
+
+  theorem sub_le_self (a : A) {b : A} (H : b ≥ 0) : a - b ≤ a :=
+  calc
+    a - b = a + -b : rfl
+      ... ≤ a + 0  : add_le_add_left (neg_nonpos_of_nonneg H)
+      ... = a      : by rewrite add_zero
+
+  theorem sub_lt_self (a : A) {b : A} (H : b > 0) : a - b < a :=
+  calc
+    a - b = a + -b : rfl
+      ... < a + 0  : add_lt_add_left (neg_neg_of_pos H)
+      ... = a      : by rewrite add_zero
+end
+
+structure decidable_linear_ordered_comm_group [class] (A : Type)
+    extends add_comm_group A, decidable_linear_order A :=
+    (add_le_add_left : ∀ a b, le a b → ∀ c, le (add c a) (add c b))
+
+private theorem add_le_add_left' (A : Type) (s : decidable_linear_ordered_comm_group A) (a b : A) :
+                a ≤ b → (∀ c : A, c + a ≤ c + b) :=
+  decidable_linear_ordered_comm_group.add_le_add_left a b
+
+definition decidable_linear_ordered_comm_group.to_ordered_comm_group [instance] [reducible] [coercion]
+           (A : Type) [s : decidable_linear_ordered_comm_group A] : ordered_comm_group A :=
+   ⦃ordered_comm_group, s,
+   le_of_lt := @le_of_lt A s,
+   add_le_add_left := add_le_add_left' A s,
+   lt_of_le_of_lt := @lt_of_le_of_lt A s,
+   lt_of_lt_of_le := @lt_of_lt_of_le A s⦄
+
+section
+  variables [s : decidable_linear_ordered_comm_group A]
+  variables {a b c d e : A}
+  include s
+
+  theorem eq_zero_of_neg_eq (H : -a = a) : a = 0 :=
+  lt.by_cases
+    (assume H1 : a < 0,
+      have H2: a > 0, from H ▸ neg_pos_of_neg H1,
+      absurd H1 (lt.asymm H2))
+    (assume H1 : a = 0, H1)
+    (assume H1 : a > 0,
+      have H2: a < 0, from H ▸ neg_neg_of_pos H1,
+      absurd H1 (lt.asymm H2))
+
+  definition abs (a : A) : A := if 0 ≤ a then a else -a
+
+  theorem abs_of_nonneg (H : a ≥ 0) : abs a = a := if_pos H
+
+  theorem abs_of_pos (H : a > 0) : abs a = a := if_pos (le_of_lt H)
+
+  theorem abs_of_neg (H : a < 0) : abs a = -a := if_neg (not_le_of_gt H)
+
+  theorem abs_zero : abs 0 = (0:A) := abs_of_nonneg (le.refl _)
+
+  theorem abs_of_nonpos (H : a ≤ 0) : abs a = -a :=
+  decidable.by_cases
+    (assume H1 : a = 0, by rewrite [H1, abs_zero, neg_zero])
+    (assume H1 : a ≠ 0,
+      have H2 : a < 0, from lt_of_le_of_ne H H1,
+      abs_of_neg H2)
+
+  theorem abs_neg (a : A) : abs (-a) = abs a :=
+  or.elim (le.total 0 a)
+    (assume H1 : 0 ≤ a, by rewrite [abs_of_nonpos (neg_nonpos_of_nonneg H1), neg_neg, abs_of_nonneg H1])
+    (assume H1 : a ≤ 0, by rewrite [abs_of_nonneg (neg_nonneg_of_nonpos H1), abs_of_nonpos H1])
+
+  theorem abs_nonneg (a : A) : abs a ≥ 0 :=
+  or.elim (le.total 0 a)
+    (assume H : 0 ≤ a, by rewrite (abs_of_nonneg H); exact H)
+    (assume H : a ≤ 0,
+      calc
+          0 ≤ -a    : neg_nonneg_of_nonpos H
+        ... = abs a : abs_of_nonpos H)
+
+  theorem abs_abs (a : A) : abs (abs a) = abs a := abs_of_nonneg !abs_nonneg
+
+  theorem le_abs_self (a : A) : a ≤ abs a :=
+  or.elim (le.total 0 a)
+    (assume H : 0 ≤ a, abs_of_nonneg H ▸ !le.refl)
+    (assume H : a ≤ 0, le.trans H !abs_nonneg)
+
+  theorem neg_le_abs_self (a : A) : -a ≤ abs a :=
+  !abs_neg ▸ !le_abs_self
+
+  theorem eq_zero_of_abs_eq_zero (H : abs a = 0) : a = 0 :=
+  have H1 : a ≤ 0, from H ▸ le_abs_self a,
+  have H2 : -a ≤ 0, from H ▸ abs_neg a ▸ le_abs_self (-a),
+  le.antisymm H1 (nonneg_of_neg_nonpos H2)
+
+  theorem abs_eq_zero_iff_eq_zero (a : A) : abs a = 0 ↔ a = 0 :=
+  iff.intro eq_zero_of_abs_eq_zero (assume H, congr_arg abs H ⬝ !abs_zero)
+
+  theorem abs_pos_of_pos (H : a > 0) : abs a > 0 :=
+  by rewrite (abs_of_pos H); exact H
+
+  theorem abs_pos_of_neg (H : a < 0) : abs a > 0 :=
+  !abs_neg ▸ abs_pos_of_pos (neg_pos_of_neg H)
+
+  theorem abs_pos_of_ne_zero (H : a ≠ 0) : abs a > 0 :=
+  or.elim (lt_or_gt_of_ne H) abs_pos_of_neg abs_pos_of_pos
+
+  theorem abs_sub (a b : A) : abs (a - b) = abs (b - a) :=
+  by rewrite [-neg_sub, abs_neg]
+
+  theorem abs.by_cases {P : A → Prop} {a : A} (H1 : P a) (H2 : P (-a)) : P (abs a) :=
+  or.elim (le.total 0 a)
+    (assume H : 0 ≤ a, (abs_of_nonneg H)⁻¹ ▸ H1)
+    (assume H : a ≤ 0, (abs_of_nonpos H)⁻¹ ▸ H2)
+
+  theorem abs_le_of_le_of_neg_le (H1 : a ≤ b) (H2 : -a ≤ b) : abs a ≤ b :=
+  abs.by_cases H1 H2
+
+  theorem abs_lt_of_lt_of_neg_lt (H1 : a < b) (H2 : -a < b) : abs a < b :=
+  abs.by_cases H1 H2
+
+  -- the triangle inequality
+  section
+    private lemma aux1 {a b : A} (H1 : a + b ≥ 0) (H2 : a ≥ 0) : abs (a + b) ≤ abs a + abs b :=
+    decidable.by_cases
+      (assume H3 : b ≥ 0,
+          calc
+            abs (a + b) ≤ abs (a + b)   : le.refl
+                ... = a + b             : by rewrite (abs_of_nonneg H1)
+                ... = abs a + b         : by rewrite (abs_of_nonneg H2)
+                ... = abs a + abs b     : by rewrite (abs_of_nonneg H3))
+      (assume H3 : ¬ b ≥ 0,
+        assert H4 : b ≤ 0, from le_of_lt (lt_of_not_ge H3),
+        calc
+          abs (a + b) = a + b     : by rewrite (abs_of_nonneg H1)
+              ... = abs a + b     : by rewrite (abs_of_nonneg H2)
+              ... ≤ abs a + 0     : add_le_add_left H4
+              ... ≤ abs a + -b    : add_le_add_left (neg_nonneg_of_nonpos H4)
+              ... = abs a + abs b : by rewrite (abs_of_nonpos H4))
+
+    private lemma aux2 {a b : A} (H1 : a + b ≥ 0) : abs (a + b) ≤ abs a + abs b :=
+    or.elim (le.total b 0)
+      (assume H2 : b ≤ 0,
+        have H3 : ¬ a < 0, from
+          assume H4 : a < 0,
+          have H5 : a + b < 0, from !add_zero ▸ add_lt_add_of_lt_of_le H4 H2,
+          not_lt_of_ge H1 H5,
+        aux1 H1 (le_of_not_gt H3))
+      (assume H2 : 0 ≤ b,
+        begin
+          have H3 : abs (b + a) ≤ abs b + abs a,
+          begin
+            rewrite add.comm at H1,
+            exact aux1 H1 H2
+          end,
+          rewrite [add.comm, {abs a + _}add.comm],
+          exact H3
+        end)
+
+    theorem abs_add_le_abs_add_abs (a b : A) : abs (a + b) ≤ abs a + abs b :=
+    or.elim (le.total 0 (a + b))
+      (assume H2 : 0 ≤ a + b, aux2 H2)
+      (assume H2 : a + b ≤ 0,
+        assert H3 : -a + -b = -(a + b), by rewrite neg_add,
+        assert H4 : -(a + b) ≥ 0, from iff.mp' (neg_nonneg_iff_nonpos (a+b)) H2,
+        have H5   : -a + -b ≥ 0, begin rewrite -H3 at H4, exact H4 end,
+        calc
+          abs (a + b) = abs (-a + -b)   : by rewrite [-abs_neg, neg_add]
+              ... ≤ abs (-a) + abs (-b) : aux2 H5
+              ... = abs a + abs b       : by rewrite *abs_neg)
+  end
+
+  theorem abs_sub_abs_le_abs_sub (a b : A) : abs a - abs b ≤ abs (a - b) :=
+  have H1 : abs a - abs b + abs b ≤ abs (a - b) + abs b, from
+    calc
+      abs a - abs b + abs b = abs a : by rewrite sub_add_cancel
+        ... = abs (a - b + b)       : by rewrite sub_add_cancel
+        ... ≤ abs (a - b) + abs b   : abs_add_le_abs_add_abs,
+  algebra.le_of_add_le_add_right H1
+end
+
+end algebra

library/algebra/ring.lean

@@ -57,6 +57,9 @@ section semiring
   assume H1 : b = 0,
   have H2 : a * b = 0, from H1⁻¹ ▸ mul_zero a,
   H H2
+
+  theorem distrib_three_right (a b c d : A) : (a + b + c) * d = a * d + b * d + c * d :=
+    by rewrite *right_distrib
 end semiring
 
 /- comm semiring -/