diff --git a/library/algebra/order.lean b/library/algebra/order.lean
index 86b3c1ed2..b9eebfe1e 100644
--- a/library/algebra/order.lean
+++ b/library/algebra/order.lean
@@ -3,16 +3,8 @@ Copyright (c) 2014 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Author: Jeremy Avigad
 
-Various types of orders. We develop weak orders "≤" and strict orders "<" separately. We also
-consider structures with both, where the two are related by
-
-  x < y ↔ (x ≤ y ∧ x ≠ y)   (order_pair)
-  x ≤ y ↔ (x < y ∨ x = y)   (strong_order_pair)
-
-These might not hold constructively in some applications, but we can define additional structures
-with both < and ≤ as needed.
+Weak orders "≤", strict orders "<", and structures that include both.
 -/
-
 import logic.eq logic.connectives
 open eq eq.ops
 
@@ -28,7 +20,7 @@ structure has_le [class] (A : Type) :=
 structure has_lt [class] (A : Type) :=
 (lt : A → A → Prop)
 
-infixl `<=`   := has_le.le
+infixl `<=`  := has_le.le
 infixl `≤`   := has_le.le
 infixl `<`   := has_lt.lt
 
@@ -58,7 +50,7 @@ section
 
   theorem le.antisymm {a b : A} : a ≤ b → b ≤ a → a = b := !weak_order.le_antisymm
 
-  -- Alternate syntax. A definition does not migrate well.
+  -- Alternate syntax. (Abbreviations do not migrate well.)
   theorem eq_of_le_of_ge {a b : A} : a ≤ b → b ≤ a → a = b := !le.antisymm
 end
 
@@ -100,8 +92,6 @@ end
 
 /- well-founded orders -/
 
--- TODO: do these duplicate what Leo has done? if so, eliminate
-
 structure wf_strict_order [class] (A : Type) extends strict_order A :=
 (wf_rec : ∀P : A → Type, (∀x, (∀y, lt y x → P y) → P x) → ∀x, P x)
 
@@ -120,7 +110,6 @@ structure order_pair [class] (A : Type) extends weak_order A, has_lt A :=
 (lt_of_lt_of_le : ∀ a b c, lt a b → le b c → lt a c)
 (lt_of_le_of_lt : ∀ a b c, le a b → lt b c → lt a c)
 (lt_irrefl : ∀ a, ¬ lt a a)
---lt_iff_le_and_ne : a < b ↔ (a ≤ b ∧ a ≠ b)
 
 section
   variable [s : order_pair A]
@@ -141,7 +130,6 @@ section
   definition order_pair.to_strict_order [trans-instance] [coercion] [reducible] : strict_order A :=
   ⦃ strict_order, s, lt_irrefl := lt_irrefl s, lt_trans := lt_trans s ⦄
 
-
   theorem gt_of_gt_of_ge [trans] (H1 : a > b) (H2 : b ≥ c) : a > c := lt_of_le_of_lt H2 H1
 
   theorem gt_of_ge_of_gt [trans] (H1 : a ≥ b) (H2 : b > c) : a > c := lt_of_lt_of_le H2 H1
@@ -155,7 +143,7 @@ section
   lt.irrefl _ (lt_of_le_of_lt H H1)
 end
 
-structure strong_order_pair [class] (A : Type) extends weak_order A, has_lt A := --order_pair A :=
+structure strong_order_pair [class] (A : Type) extends weak_order A, has_lt A :=
 (le_iff_lt_or_eq : ∀a b, le a b ↔ lt a b ∨ a = b)
 (lt_irrefl : ∀ a, ¬ lt a a)
 
@@ -166,147 +154,57 @@ theorem lt_or_eq_of_le [s : strong_order_pair A] {a b : A} (le_ab : a ≤ b) : a
 iff.mp le_iff_lt_or_eq le_ab
 
 theorem le_of_lt_or_eq [s : strong_order_pair A] {a b : A} (lt_or_eq : a < b ∨ a = b) : a ≤ b :=
-  iff.mp' le_iff_lt_or_eq lt_or_eq
+iff.mp' le_iff_lt_or_eq lt_or_eq
 
 private theorem lt_irrefl' [s : strong_order_pair A] (a : A) : ¬ a < a :=
-  !strong_order_pair.lt_irrefl
+!strong_order_pair.lt_irrefl
 
 private theorem le_of_lt' [s : strong_order_pair A] (a b : A) : a < b → a ≤ b :=
-  take Hlt, le_of_lt_or_eq (or.inl Hlt)
+take Hlt, le_of_lt_or_eq (or.inl Hlt)
 
 private theorem lt_iff_le_and_ne [s : strong_order_pair A] {a b : A} : a < b ↔ (a ≤ b ∧ a ≠ b) :=
-  iff.intro
-    (take Hlt, and.intro (le_of_lt_or_eq (or.inl Hlt)) (take Hab, absurd (Hab ▸ Hlt) !lt_irrefl'))
-    (take Hand,
-     have Hor : a < b ∨ a = b, from lt_or_eq_of_le (and.left Hand),
-     or_resolve_left Hor (and.right Hand))
+iff.intro
+  (take Hlt, and.intro (le_of_lt_or_eq (or.inl Hlt)) (take Hab, absurd (Hab ▸ Hlt) !lt_irrefl'))
+  (take Hand,
+   have Hor : a < b ∨ a = b, from lt_or_eq_of_le (and.left Hand),
+   or_resolve_left Hor (and.right Hand))
 
 theorem lt_of_le_of_ne [s : strong_order_pair A] {a b : A} : a ≤ b → a ≠ b → a < b :=
-  take H1 H2, iff.mp' lt_iff_le_and_ne (and.intro H1 H2)
+take H1 H2, iff.mp' lt_iff_le_and_ne (and.intro H1 H2)
 
 private theorem ne_of_lt' [s : strong_order_pair A] {a b : A} (H : a < b) : a ≠ b :=
-  and.right ((iff.mp lt_iff_le_and_ne) H)
+and.right ((iff.mp lt_iff_le_and_ne) H)
 
 private theorem lt_of_lt_of_le' [s : strong_order_pair A] (a b c : A) : a < b → b ≤ c → a < c :=
-  assume lt_ab : a < b,
-  assume le_bc : b ≤ c,
-  have le_ac : a ≤ c, from le.trans (le_of_lt' _ _ lt_ab) le_bc,
-  have ne_ac : a ≠ c, from
-    assume eq_ac : a = c,
-    have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
-    have eq_ab : a = b, from le.antisymm  (le_of_lt' _ _ lt_ab) le_ba,
-    show false, from ne_of_lt' lt_ab eq_ab,
-  show a < c, from iff.mp' (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
+assume lt_ab : a < b,
+assume le_bc : b ≤ c,
+have le_ac : a ≤ c, from le.trans (le_of_lt' _ _ lt_ab) le_bc,
+have ne_ac : a ≠ c, from
+  assume eq_ac : a = c,
+  have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
+  have eq_ab : a = b, from le.antisymm  (le_of_lt' _ _ lt_ab) le_ba,
+  show false, from ne_of_lt' lt_ab eq_ab,
+show a < c, from iff.mp' (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
 
-  theorem lt_of_le_of_lt' [s : strong_order_pair A] (a b c : A) : a ≤ b → b < c → a < c :=
-  assume le_ab : a ≤ b,
-  assume lt_bc : b < c,
-  have le_ac : a ≤ c, from le.trans le_ab (le_of_lt' _ _ lt_bc),
-  have ne_ac : a ≠ c, from
-    assume eq_ac : a = c,
-    have le_cb : c ≤ b, from eq_ac ▸ le_ab,
-    have eq_bc : b = c, from le.antisymm  (le_of_lt' _ _ lt_bc) le_cb,
-    show false, from ne_of_lt' lt_bc eq_bc,
-  show a < c, from iff.mp' (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
-
-
-definition strong_order_pair.to_order_pair [trans-instance] [coercion] [reducible] [s : strong_order_pair A]
-        : order_pair A :=
-  ⦃ order_pair, s,
-    lt_irrefl := lt_irrefl',
-    le_of_lt := le_of_lt',
-    lt_of_le_of_lt := lt_of_le_of_lt',
-    lt_of_lt_of_le := lt_of_lt_of_le'
-  ⦄
-
--- We can also construct a strong order pair by defining a strict order, and then defining
--- x ≤ y ↔ x < y ∨ x = y
-
-structure strict_order_with_le [class] (A : Type) extends strict_order A, has_le A :=
-(le_iff_lt_or_eq : ∀a b, le a b ↔ lt a b ∨ a = b)
-
-private theorem le_refl (s : strict_order_with_le A) (a : A) : a ≤ a :=
-iff.mp (iff.symm !strict_order_with_le.le_iff_lt_or_eq) (or.intro_right _ rfl)
-
-private theorem le_trans (s : strict_order_with_le A) (a b c : A) (le_ab : a ≤ b) (le_bc : b ≤ c) : a ≤ c :=
-or.elim (iff.mp !strict_order_with_le.le_iff_lt_or_eq le_ab)
-  (assume lt_ab : a < b,
-    or.elim (iff.mp !strict_order_with_le.le_iff_lt_or_eq le_bc)
-      (assume lt_bc : b < c,
-        iff.elim_right
-          !strict_order_with_le.le_iff_lt_or_eq (or.intro_left _ (lt.trans lt_ab lt_bc)))
-      (assume eq_bc : b = c, eq_bc ▸ le_ab))
-  (assume eq_ab : a = b,
-    eq_ab⁻¹ ▸ le_bc)
-
-private theorem le_antisymm (s : strict_order_with_le A) (a b : A) (le_ab : a ≤ b) (le_ba : b ≤ a) : a = b :=
-or.elim (iff.mp !strict_order_with_le.le_iff_lt_or_eq le_ab)
-  (assume lt_ab : a < b,
-    or.elim (iff.mp !strict_order_with_le.le_iff_lt_or_eq le_ba)
-      (assume lt_ba : b < a, absurd (lt.trans lt_ab lt_ba) (lt.irrefl a))
-      (assume eq_ba : b = a, eq_ba⁻¹))
-  (assume eq_ab : a = b, eq_ab)
-
-private theorem lt_iff_le_ne (s : strict_order_with_le A) (a b : A) : a < b ↔ a ≤ b ∧ a ≠ b :=
-iff.intro
-  (assume lt_ab : a < b,
-    have le_ab : a ≤ b, from
-      iff.elim_right !strict_order_with_le.le_iff_lt_or_eq (or.intro_left _ lt_ab),
-    show a ≤ b ∧ a ≠ b, from and.intro le_ab (ne_of_lt lt_ab))
-  (assume H : a ≤ b ∧ a ≠ b,
-    have H1 : a < b ∨ a = b, from
-      iff.mp !strict_order_with_le.le_iff_lt_or_eq (and.elim_left H),
-    show a < b, from or_resolve_left H1 (and.elim_right H))
-
-private theorem le_of_lt' (s : strict_order_with_le A) (a b : A) : a < b → a ≤ b :=
-  take Hlt, and.left (iff.mp (lt_iff_le_ne s _ _) Hlt)
-
-private theorem lt_trans (s : strict_order_with_le A) (a b c: A) (lt_ab : a < b) (lt_bc : b < c) : a < c :=
-  have le_ab : a ≤ b, from le_of_lt' s _ _ lt_ab,
-  have le_bc : b ≤ c, from le_of_lt' s _ _ lt_bc,
-  have le_ac : a ≤ c, from le_trans s _ _ _ le_ab le_bc,
-  have ne_ac : a ≠ c, from
-    assume eq_ac : a = c,
-      have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
-      have eq_ab : a = b, from le_antisymm s a b le_ab le_ba,
-      have ne_ab : a ≠ b, from and.elim_right ((iff.mp (lt_iff_le_ne s a b)) lt_ab),
-      ne_ab eq_ab,
-  show a < c, from (iff.mp' !lt_iff_le_ne) (and.intro le_ac ne_ac)
-
-  theorem lt_of_lt_of_le' (s : strict_order_with_le A) (a b c : A) : a < b → b ≤ c → a < c :=
-  assume lt_ab : a < b,
-  assume le_bc : b ≤ c,
-  have le_ac : a ≤ c, from le_trans s _ _ _ (le_of_lt' s _ _ lt_ab) le_bc,
-  have ne_ac : a ≠ c, from
-    assume eq_ac : a = c,
-    have le_ba : b ≤ a, from eq_ac⁻¹ ▸ le_bc,
-    have eq_ab : a = b, from le_antisymm s _ _ (le_of_lt' s _ _ lt_ab) le_ba,
-    show false, from ne_of_lt lt_ab eq_ab,
-  show a < c, from iff.mp' (lt_iff_le_ne s _ _) (and.intro le_ac ne_ac)
-
-  theorem lt_of_le_of_lt'' (s : strict_order_with_le A) (a b c : A) : a ≤ b → b < c → a < c :=
-  assume le_ab : a ≤ b,
-  assume lt_bc : b < c,
-  have le_ac : a ≤ c, from le_trans s _ _ _ le_ab (le_of_lt' s _ _ lt_bc),
-  have ne_ac : a ≠ c, from
-    assume eq_ac : a = c,
-    have le_cb : c ≤ b, from eq_ac ▸ le_ab,
-    have eq_bc : b = c, from le_antisymm s _ _ (le_of_lt' s _ _ lt_bc) le_cb,
-    show false, from ne_of_lt lt_bc eq_bc,
-  show a < c, from iff.mp' (lt_iff_le_ne s _ _) (and.intro le_ac ne_ac)
-
-
-definition strict_order_with_le.to_order_pair [trans-instance] [coercion] [reducible] [s : strict_order_with_le A] :
-  strong_order_pair A :=
-⦃ strong_order_pair, s,
-  le_refl          := le_refl s,
-  le_trans         := le_trans s,
-  le_antisymm      := le_antisymm s ⦄
-  --le_of_lt         := le_of_lt' s,
-  --lt_of_le_of_lt   := lt_of_le_of_lt' s,
-  --lt_of_lt_of_le   := lt_of_lt_of_le' s ⦄
-  --lt_iff_le_and_ne := lt_iff_le_ne s ⦄
+theorem lt_of_le_of_lt' [s : strong_order_pair A] (a b c : A) : a ≤ b → b < c → a < c :=
+assume le_ab : a ≤ b,
+assume lt_bc : b < c,
+have le_ac : a ≤ c, from le.trans le_ab (le_of_lt' _ _ lt_bc),
+have ne_ac : a ≠ c, from
+  assume eq_ac : a = c,
+  have le_cb : c ≤ b, from eq_ac ▸ le_ab,
+  have eq_bc : b = c, from le.antisymm  (le_of_lt' _ _ lt_bc) le_cb,
+  show false, from ne_of_lt' lt_bc eq_bc,
+show a < c, from iff.mp' (lt_iff_le_and_ne) (and.intro le_ac ne_ac)
 
+definition strong_order_pair.to_order_pair [trans-instance] [coercion] [reducible]
+    [s : strong_order_pair A] : order_pair A :=
+⦃ order_pair, s,
+  lt_irrefl := lt_irrefl',
+  le_of_lt := le_of_lt',
+  lt_of_le_of_lt := lt_of_le_of_lt',
+  lt_of_lt_of_le := lt_of_lt_of_le'
+⦄
 
 /- linear orders -/
 
@@ -317,7 +215,7 @@ structure linear_strong_order_pair [class] (A : Type) extends strong_order_pair
 
 definition linear_strong_order_pair.to_linear_order_pair [trans-instance] [coercion] [reducible]
     [s : linear_strong_order_pair A] : linear_order_pair A :=
-  ⦃ linear_order_pair, s, strong_order_pair.to_order_pair⦄
+⦃ linear_order_pair, s, strong_order_pair.to_order_pair⦄
 
 section
   variable [s : linear_strong_order_pair A]