diff --git a/hott/init/nat.hlean b/hott/init/nat.hlean
index 497a17979..2587f6abf 100644
--- a/hott/init/nat.hlean
+++ b/hott/init/nat.hlean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn, Leonardo de Moura
 -/
 prelude
-import init.wf init.tactic init.num init.types init.path
+import init.tactic init.num init.types init.path
 open eq eq.ops decidable
 open algebra sum
 set_option class.force_new true
@@ -185,25 +185,6 @@ namespace nat
   protected theorem le_of_eq_sum_lt {a b : ℕ} (H : a = b ⊎ a < b) : a ≤ b :=
   sum.rec_on H !nat.le_of_eq !nat.le_of_lt
 
-  -- less-than is well-founded
-  definition lt.wf [instance] : well_founded (lt : ℕ → ℕ → Type₀) :=
-  begin
-    constructor, intro n, induction n with n IH,
-    { constructor, intros n H, exfalso, exact !not_lt_zero H},
-    { constructor, intros m H,
-      assert aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m,
-        { intros n₁ hlt, induction hlt,
-          { intro p, injection p with q, exact q ▸ IH},
-          { intro p, injection p with q, exact (acc.inv (q ▸ IH) a)}},
-      apply aux H rfl},
-  end
-
-  definition measure {A : Type} : (A → ℕ) → A → A → Type₀ :=
-  inv_image lt
-
-  definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
-  inv_image.wf f lt.wf
-
   theorem succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
   succ_le_succ
 
diff --git a/hott/init/types.hlean b/hott/init/types.hlean
index f01da8acf..bac36fc19 100644
--- a/hott/init/types.hlean
+++ b/hott/init/types.hlean
@@ -5,7 +5,7 @@ Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn, Jakob von Raumer
 -/
 
 prelude
-import init.num init.wf
+import init.num init.relation
 open iff
 
 -- Empty type
@@ -92,61 +92,4 @@ namespace prod
 
   definition flip [unfold 3] {A B : Type} (a : A × B) : B × A := pair (pr2 a) (pr1 a)
 
-  open well_founded
-
-  section
-  variables {A B : Type}
-  variable  (Ra  : A → A → Type)
-  variable  (Rb  : B → B → Type)
-
-  -- Lexicographical order based on Ra and Rb
-  inductive lex : A × B → A × B → Type :=
-  | left  : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
-  | right : ∀a {b₁ b₂},     Rb b₁ b₂ → lex (a, b₁)  (a, b₂)
-
-  -- Relational product based on Ra and Rb
-  inductive rprod : A × B → A × B → Type :=
-  intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
-  end
-
-  section
-  parameters {A B : Type}
-  parameters {Ra  : A → A → Type} {Rb  : B → B → Type}
-  local infix `≺`:50 := lex Ra Rb
-
-  definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
-  acc.rec_on aca
-    (λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
-      λb, acc.rec_on (acb b)
-        (λxb acb
-          (iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
-          acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
-            have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
-              @prod.lex.rec_on A B Ra Rb (λp₁ p₂ h, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
-                               p (xa, xb) lt
-                (λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
-                  show acc (lex Ra Rb) (a₁, b₁), from
-                  have Ra₁  : Ra a₁ xa, from eq.rec_on eq₂ H,
-                  iHa a₁ Ra₁ b₁)
-                (λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
-                  show acc (lex Ra Rb) (a, b₁), from
-                  have Rb₁  : Rb b₁ xb, from eq.rec_on eq₃ H,
-                  have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
-                  eq.rec_on eq₂' (iHb b₁ Rb₁)),
-            aux rfl rfl)))
-
-  -- The lexicographical order of well founded relations is well-founded
-  definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
-  well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
-
-  -- Relational product is a subrelation of the lex
-  definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
-  λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
-
-  -- The relational product of well founded relations is well-founded
-  definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
-  subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
-
-  end
-
 end prod
diff --git a/hott/init/wf.hlean b/hott/init/wf.hlean
index b5fc9cc71..00346cd30 100644
--- a/hott/init/wf.hlean
+++ b/hott/init/wf.hlean
@@ -3,9 +3,10 @@ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 -/
-
 prelude
-import init.relation init.tactic
+import init.relation init.tactic init.funext
+
+open eq
 
 inductive acc.{l₁ l₂} {A : Type.{l₁}} (R : A → A → Type.{l₂}) : A → Type.{max l₁ l₂} :=
 intro : ∀x, (∀ y, R y x → acc R y) → acc R x
@@ -13,15 +14,38 @@ intro : ∀x, (∀ y, R y x → acc R y) → acc R x
 namespace acc
   variables {A : Type} {R : A → A → Type}
 
+  definition acc_eq {a : A} (H₁ H₂ : acc R a) : H₁ = H₂ :=
+  begin
+    induction H₁ with a K₁ IH₁,
+    induction H₂ with a K₂ IH₂,
+    apply eq.ap (intro a),
+    apply eq_of_homotopy, intro a,
+    apply eq_of_homotopy, intro r,
+    apply IH₁
+  end
+
   definition inv {x y : A} (H₁ : acc R x) (H₂ : R y x) : acc R y :=
   acc.rec_on H₁ (λ x₁ ac₁ iH H₂, ac₁ y H₂) H₂
+
+  -- dependent elimination for acc
+  protected definition drec [recursor]
+      {C : Π (a : A), acc R a → Type}
+      (h₁ : Π (x : A) (acx : Π (y : A), R y x → acc R y),
+              (Π (y : A) (ryx : R y x), C y (acx y ryx)) → C x (acc.intro x acx))
+      {a : A} (h₂ : acc R a) : C a h₂ :=
+  begin
+    refine acc.rec _ h₂ h₂,
+    intro x acx ih h₂,
+    exact transport (C x) !acc_eq (h₁ x acx (λ y ryx, ih y ryx (acx y ryx)))
+  end
+
 end acc
 
 inductive well_founded [class] {A : Type} (R : A → A → Type) : Type :=
-intro : (∀ a, acc R a) → well_founded R
+intro : (Π a, acc R a) → well_founded R
 
 namespace well_founded
-  definition apply [coercion] {A : Type} {R : A → A → Type} (wf : well_founded R) : ∀a, acc R a :=
+  definition apply [coercion] {A : Type} {R : A → A → Type} (wf : well_founded R) : Πa, acc R a :=
   take a, well_founded.rec_on wf (λp, p) a
 
   section
@@ -30,60 +54,42 @@ namespace well_founded
 
   hypothesis [Hwf : well_founded R]
 
-  definition recursion {C : A → Type} (a : A) (H : Πx, (Πy, y ≺ x → C y) → C x) : C a :=
+  theorem recursion {C : A → Type} (a : A) (H : Πx, (Πy, y ≺ x → C y) → C x) : C a :=
   acc.rec_on (Hwf a) (λ x₁ ac₁ iH, H x₁ iH)
 
-  definition induction {C : A → Type} (a : A) (H : ∀x, (∀y, y ≺ x → C y) → C x) : C a :=
+  theorem induction {C : A → Type} (a : A) (H : Πx, (Πy, y ≺ x → C y) → C x) : C a :=
   recursion a H
 
-  parameter {C : A → Type}
-  parameter F : Πx, (Πy, y ≺ x → C y) → C x
+  variable {C : A → Type}
+  variable F : Πx, (Πy, y ≺ x → C y) → C x
 
   definition fix_F (x : A) (a : acc R x) : C x :=
   acc.rec_on a (λ x₁ ac₁ iH, F x₁ iH)
 
-  definition fix_F_eq (x : A) (r : acc R x) :
-    fix_F x r = F x (λ (y : A) (p : y ≺ x), fix_F y (acc.inv r p)) :=
-  acc.rec_on r (λ x H ih, rfl)
+  theorem fix_F_eq (x : A) (r : acc R x) :
+    fix_F F x r = F x (λ (y : A) (p : y ≺ x), fix_F F y (acc.inv r p)) :=
+  begin
+    induction r using acc.drec,
+    reflexivity -- proof is star due to proof irrelevance
+  end
+  end
 
-  -- Remark: after we prove function extensionality from univalence, we can drop this hypothesis
-  hypothesis F_ext : Π (x : A) (f g : Π y, y ≺ x → C y),
-                       (Π (y : A) (p : y ≺ x), f y p = g y p) → F x f = F x g
-
-  lemma fix_F_inv (x : A) (r : acc R x) : Π (s : acc R x), fix_F x r = fix_F x s :=
-  acc.rec_on r (λ
-    (x₁  : A)
-    (h₁  : Π y, y ≺ x₁ → acc R y)
-    (ih₁ : Π y (hlt : y ≺ x₁) (s : acc R y), fix_F y (h₁ y hlt) = fix_F y s)
-    (s : acc R x₁),
-    have aux₁ : Π (s : acc R x₁) (h₁ : Π y, y ≺ x₁ → acc R y) (ih₁ : Π y (hlt : y ≺ x₁) (s : acc R y),
-                    fix_F y (h₁ y hlt) = fix_F y s), fix_F x₁ (acc.intro x₁ h₁) = fix_F x₁ s, from
-      λ s, acc.rec_on s (λ
-        (x₂ : A)
-        (h₂ : Π y, y ≺ x₂ → acc R y)
-        (ih₂ : _)
-        (h₁ : Π y, y ≺ x₂ → acc R y)
-        (ih₁ : Π y (hlt : y ≺ x₂) (s : acc R y), fix_F y (h₁ y hlt) = fix_F y s),
-        calc fix_F x₂ (acc.intro x₂ h₁)
-                   = F x₂ (λ (y : A) (p : y ≺ x₂), fix_F y (h₁ y p))  : rfl
-               ... = F x₂ (λ (y : A) (p : y ≺ x₂), fix_F y (h₂ y p))  : F_ext x₂ _ _ (λ (y : A) (p : y ≺ x₂), ih₁ y p (h₂ y p))
-               ... = fix_F x₂ (acc.intro x₂ h₂)                       : rfl),
-    show fix_F x₁ (acc.intro x₁ h₁) = fix_F x₁ s, from
-    aux₁ s h₁ ih₁)
+  variables {A : Type} {C : A → Type} {R : A → A → Type}
 
   -- Well-founded fixpoint
-  definition fix (x : A) : C x :=
-  fix_F x (Hwf x)
+  definition fix [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) : C x :=
+  fix_F F x (Hwf x)
 
   -- Well-founded fixpoint satisfies fixpoint equation
-  definition fix_eq (x : A) : fix x = F x (λy h, fix y) :=
-  calc
-    fix x
-        = fix_F x (Hwf x)                         : rfl
-    ... = F x (λy h, fix_F y (acc.inv (Hwf x) h)) : fix_F_eq x (Hwf x)
-    ... = F x (λy h, fix_F y (Hwf y))             : F_ext x _ _ (λ y h, fix_F_inv y _ _)
-    ... = F x (λy h, fix y)                       : rfl
-
+  theorem fix_eq [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) :
+    fix F x = F x (λy h, fix F y) :=
+  begin
+    refine fix_F_eq F x (Hwf x) ⬝ _,
+    apply ap (F x),
+    apply eq_of_homotopy, intro a,
+    apply eq_of_homotopy, intro r,
+    apply ap (fix_F F a),
+    apply acc.acc_eq
   end
 end well_founded
 
@@ -97,17 +103,20 @@ well_founded.intro (λ (a : A),
 -- Subrelation of a well-founded relation is well-founded
 namespace subrelation
 section
+  universe variable u
   parameters {A : Type} {R Q : A → A → Type}
   parameters (H₁ : subrelation Q R)
   parameters (H₂ : well_founded R)
 
   definition accessible {a : A} (ac : acc R a) : acc Q a :=
-  acc.rec_on ac
-    (λ (x : A) (ax : _) (iH : ∀ (y : A), R y x → acc Q y),
-      acc.intro x (λ (y : A) (lt : Q y x), iH y (H₁ lt)))
+  using H₁,
+  begin
+    induction ac with x ax ih, constructor,
+    exact λ (y : A) (lt : Q y x), ih y (H₁ lt)
+  end
 
   definition wf : well_founded Q :=
-  well_founded.intro (λ a, accessible (H₂ a))
+  well_founded.intro (λ a, accessible proof (@apply A R H₂ a) qed)
 end
 end subrelation
 
@@ -118,14 +127,16 @@ section
   parameters (f : A → B)
   parameters (H : well_founded R)
 
+  private definition acc_aux {b : B} (ac : acc R b) : Π x, f x = b → acc (inv_image R f) x :=
+  begin
+    induction ac with x acx ih,
+    intro z e, constructor,
+    intro y lt, subst x,
+    exact ih (f y) lt y rfl
+  end
+
   definition accessible {a : A} (ac : acc R (f a)) : acc (inv_image R f) a :=
-  have gen : ∀x, f x = f a → acc (inv_image R f) x, from
-    acc.rec_on ac
-      (λx acx (iH : ∀y, R y x → (∀z, f z = y → acc (inv_image R f) z))
-          (z : A) (eq₁ : f z = x),
-        acc.intro z (λ (y : A) (lt : R (f y) (f z)),
-          iH (f y) (eq.rec_on eq₁ lt) y rfl)),
-  gen a rfl
+  acc_aux ac a rfl
 
   definition wf : well_founded (inv_image R f) :=
   well_founded.intro (λ a, accessible (H (f a)))
@@ -139,22 +150,99 @@ section
   local notation `R⁺` := tc R
 
   definition accessible {z} (ac: acc R z) : acc R⁺ z :=
-  acc.rec_on ac
-    (λ x acx (iH : ∀y, R y x → acc R⁺ y),
-      acc.intro x (λ (y : A) (lt : R⁺ y x),
-        have gen : x = x → acc R⁺ y, from
-          tc.rec_on lt
-            (λa b (H : R a b) (Heq : b = x),
-               iH a (eq.rec_on Heq H))
-            (λa b c (H₁ : R⁺ a b) (H₂ : R⁺ b c)
-                (iH₁ : b = x → acc R⁺ a)
-                (iH₂ : c = x → acc R⁺ b)
-                (Heq : c = x),
-              acc.inv (iH₂ Heq) H₁),
-        gen rfl))
+  begin
+    induction ac with x acx ih,
+    constructor, intro y rel,
+    induction rel with a b rab a b c rab rbc ih₁ ih₂,
+      {exact ih a rab},
+      {exact acc.inv (ih₂ acx ih) rab}
+  end
 
   definition wf (H : well_founded R) : well_founded R⁺ :=
   well_founded.intro (λ a, accessible (H a))
-
 end
 end tc
+
+namespace nat
+
+  -- less-than is well-founded
+  definition lt.wf [instance] : well_founded (lt : ℕ → ℕ → Type₀) :=
+  begin
+    constructor, intro n, induction n with n IH,
+    { constructor, intros n H, exfalso, exact !not_lt_zero H},
+    { constructor, intros m H,
+      assert aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m,
+        { intros n₁ hlt, induction hlt,
+          { intro p, injection p with q, exact q ▸ IH},
+          { intro p, injection p with q, exact (acc.inv (q ▸ IH) a)}},
+      apply aux H rfl},
+  end
+
+  definition measure {A : Type} : (A → ℕ) → A → A → Type₀ :=
+  inv_image lt
+
+  definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
+  inv_image.wf f lt.wf
+
+end nat
+
+namespace prod
+
+  open well_founded prod.ops
+
+  section
+  variables {A B : Type}
+  variable  (Ra  : A → A → Type)
+  variable  (Rb  : B → B → Type)
+
+  -- Lexicographical order based on Ra and Rb
+  inductive lex : A × B → A × B → Type :=
+  | left  : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
+  | right : ∀a {b₁ b₂},     Rb b₁ b₂ → lex (a, b₁)  (a, b₂)
+
+  -- Relational product based on Ra and Rb
+  inductive rprod : A × B → A × B → Type :=
+  intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
+  end
+
+  section
+  parameters {A B : Type}
+  parameters {Ra  : A → A → Type} {Rb  : B → B → Type}
+  local infix `≺`:50 := lex Ra Rb
+
+  definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
+  acc.rec_on aca
+    (λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
+      λb, acc.rec_on (acb b)
+        (λxb acb
+          (iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
+          acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
+            have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
+              @prod.lex.rec_on A B Ra Rb (λp₁ p₂ h, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
+                               p (xa, xb) lt
+                (λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
+                  show acc (lex Ra Rb) (a₁, b₁), from
+                  have Ra₁  : Ra a₁ xa, from eq.rec_on eq₂ H,
+                  iHa a₁ Ra₁ b₁)
+                (λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
+                  show acc (lex Ra Rb) (a, b₁), from
+                  have Rb₁  : Rb b₁ xb, from eq.rec_on eq₃ H,
+                  have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
+                  eq.rec_on eq₂' (iHb b₁ Rb₁)),
+            aux rfl rfl)))
+
+  -- The lexicographical order of well founded relations is well-founded
+  definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
+  well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
+
+  -- Relational product is a subrelation of the lex
+  definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
+  λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
+
+  -- The relational product of well founded relations is well-founded
+  definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
+  subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
+
+  end
+
+end prod
diff --git a/library/init/nat.lean b/library/init/nat.lean
index 3488756eb..f11181388 100644
--- a/library/init/nat.lean
+++ b/library/init/nat.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn, Leonardo de Moura
 -/
 prelude
-import init.wf init.tactic init.num
+import init.relation init.tactic init.num
 open eq.ops decidable or
 
 notation `ℕ` := nat
@@ -189,20 +189,6 @@ namespace nat
   protected theorem le_of_eq_or_lt {a b : ℕ} (H : a = b ∨ a < b) : a ≤ b :=
   or.elim H !nat.le_of_eq !nat.le_of_lt
 
-  -- less-than is well-founded
-  definition lt.wf [instance] : well_founded lt :=
-  well_founded.intro (nat.rec
-    (!acc.intro (λn H, absurd H (not_lt_zero n)))
-    (λn IH, !acc.intro (λm H,
-      or.elim (nat.eq_or_lt_of_le (le_of_succ_le_succ H))
-        (λe, eq.substr e IH) (acc.inv IH))))
-
-  definition measure {A : Type} : (A → ℕ) → A → A → Prop :=
-  inv_image lt
-
-  definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
-  inv_image.wf f lt.wf
-
   theorem succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
   succ_le_succ
 
diff --git a/library/init/prod.lean b/library/init/prod.lean
index 6085c43c7..f4a8027cf 100644
--- a/library/init/prod.lean
+++ b/library/init/prod.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura, Jeremy Avigad
 -/
 prelude
-import init.num init.wf
+import init.num init.relation
 
 definition pair [constructor] := @prod.mk
 notation A × B := prod A B
@@ -30,60 +30,4 @@ namespace prod
   | (a, b) := rfl
   end
 
-  open well_founded
-
-  section
-  variables {A B : Type}
-  variable  (Ra  : A → A → Prop)
-  variable  (Rb  : B → B → Prop)
-
-  -- Lexicographical order based on Ra and Rb
-  inductive lex : A × B → A × B → Prop :=
-  | left  : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
-  | right : ∀a {b₁ b₂},     Rb b₁ b₂ → lex (a, b₁)  (a, b₂)
-
-  -- Relational product based on Ra and Rb
-  inductive rprod : A × B → A × B → Prop :=
-  intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
-  end
-
-  section
-  parameters {A B : Type}
-  parameters {Ra  : A → A → Prop} {Rb  : B → B → Prop}
-  local infix `≺`:50 := lex Ra Rb
-
-  definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
-  acc.rec_on aca
-    (λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
-      λb, acc.rec_on (acb b)
-        (λxb acb
-          (iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
-          acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
-            have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
-              @prod.lex.rec_on A B Ra Rb (λp₁ p₂, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
-                               p (xa, xb) lt
-                (λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
-                  show acc (lex Ra Rb) (a₁, b₁), from
-                  have Ra₁  : Ra a₁ xa, from eq.rec_on eq₂ H,
-                  iHa a₁ Ra₁ b₁)
-                (λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
-                  show acc (lex Ra Rb) (a, b₁), from
-                  have Rb₁  : Rb b₁ xb, from eq.rec_on eq₃ H,
-                  have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
-                  eq.rec_on eq₂' (iHb b₁ Rb₁)),
-            aux rfl rfl)))
-
-  -- The lexicographical order of well founded relations is well-founded
-  definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
-  well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
-
-  -- Relational product is a subrelation of the lex
-  definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
-  λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
-
-  -- The relational product of well founded relations is well-founded
-  definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
-  subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
-
-  end
 end prod
diff --git a/library/init/wf.lean b/library/init/wf.lean
index 979e586e8..a8fd38fba 100644
--- a/library/init/wf.lean
+++ b/library/init/wf.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 -/
 prelude
-import init.relation init.tactic
+import init.relation init.tactic init.nat init.prod
 
 inductive acc {A : Type} (R : A → A → Prop) : A → Prop :=
 intro : ∀x, (∀ y, R y x → acc R y) → acc R x
@@ -141,3 +141,81 @@ section
   well_founded.intro (λ a, accessible (H a))
 end
 end tc
+
+namespace nat
+
+  -- less-than is well-founded
+  definition lt.wf [instance] : well_founded lt :=
+  well_founded.intro (nat.rec
+    (!acc.intro (λn H, absurd H (not_lt_zero n)))
+    (λn IH, !acc.intro (λm H,
+      or.elim (nat.eq_or_lt_of_le (le_of_succ_le_succ H))
+        (λe, eq.substr e IH) (acc.inv IH))))
+
+  definition measure {A : Type} : (A → ℕ) → A → A → Prop :=
+  inv_image lt
+
+  definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
+  inv_image.wf f lt.wf
+
+end nat
+
+namespace prod
+  open well_founded
+
+  section
+  variables {A B : Type}
+  variable  (Ra  : A → A → Prop)
+  variable  (Rb  : B → B → Prop)
+
+  -- Lexicographical order based on Ra and Rb
+  inductive lex : A × B → A × B → Prop :=
+  | left  : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
+  | right : ∀a {b₁ b₂},     Rb b₁ b₂ → lex (a, b₁)  (a, b₂)
+
+  -- Relational product based on Ra and Rb
+  inductive rprod : A × B → A × B → Prop :=
+  intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
+  end
+
+  section
+  parameters {A B : Type}
+  parameters {Ra  : A → A → Prop} {Rb  : B → B → Prop}
+  local infix `≺`:50 := lex Ra Rb
+
+  definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
+  acc.rec_on aca
+    (λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
+      λb, acc.rec_on (acb b)
+        (λxb acb
+          (iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
+          acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
+            have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
+              @prod.lex.rec_on A B Ra Rb (λp₁ p₂, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
+                               p (xa, xb) lt
+                (λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
+                  show acc (lex Ra Rb) (a₁, b₁), from
+                  have Ra₁  : Ra a₁ xa, from eq.rec_on eq₂ H,
+                  iHa a₁ Ra₁ b₁)
+                (λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
+                  show acc (lex Ra Rb) (a, b₁), from
+                  have Rb₁  : Rb b₁ xb, from eq.rec_on eq₃ H,
+                  have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
+                  eq.rec_on eq₂' (iHb b₁ Rb₁)),
+            aux rfl rfl)))
+
+  -- The lexicographical order of well founded relations is well-founded
+  definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
+  well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
+
+  -- Relational product is a subrelation of the lex
+  definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
+  λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
+
+  -- The relational product of well founded relations is well-founded
+  definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
+  subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
+
+  end
+
+end prod
diff --git a/library/init/wf_k.lean b/library/init/wf_k.lean
index 03dde4d75..159e596ca 100644
--- a/library/init/wf_k.lean
+++ b/library/init/wf_k.lean
@@ -2,7 +2,7 @@
 -- Released under Apache 2.0 license as described in the file LICENSE.
 -- Author: Leonardo de Moura
 prelude
-import init.nat
+import init.wf
 
 namespace well_founded
   -- This is an auxiliary definition that useful for generating a new "proof" for (well_founded R)