diff --git a/tests/lean/run/coe6.lean b/tests/lean/run/coe6.lean
new file mode 100644
index 000000000..b97f520a7
--- /dev/null
+++ b/tests/lean/run/coe6.lean
@@ -0,0 +1,15 @@
+import data.unit
+open unit
+
+variable int : Type.{1}
+variable nat : Type.{1}
+variable izero : int
+variable nzero : nat
+variable isucc : int → int
+variable nsucc : nat → nat
+definition f [coercion] (a : unit) : int := izero
+definition g [coercion] (a : unit) : nat := nzero
+
+set_option pp.coercion true
+check isucc star
+check nsucc star
diff --git a/tests/lean/run/coe8.lean b/tests/lean/run/coe8.lean
new file mode 100644
index 000000000..93b8c2da0
--- /dev/null
+++ b/tests/lean/run/coe8.lean
@@ -0,0 +1,16 @@
+import logic
+
+variable nat : Type.{1}
+variable int : Type.{1}
+variable of_nat : nat → int
+coercion of_nat
+variable nat_add : nat → nat → nat
+variable int_add : int → int → int
+infixl `+`:65 := int_add
+infixl `+`:65 := nat_add
+
+print "================"
+variable tst (n m : nat) : @eq int (of_nat n + of_nat m) (n + m)
+
+check tst
+exit
diff --git a/tests/lean/run/ex.lean b/tests/lean/run/ex.lean
new file mode 100644
index 000000000..6bad54d9b
--- /dev/null
+++ b/tests/lean/run/ex.lean
@@ -0,0 +1,3 @@
+import standard
+set_option pp.implicit true
+check ∃x, x = 0
\ No newline at end of file
diff --git a/tests/lean/run/protected.lean b/tests/lean/run/protected.lean
new file mode 100644
index 000000000..72457ab52
--- /dev/null
+++ b/tests/lean/run/protected.lean
@@ -0,0 +1,10 @@
+import logic
+
+namespace foo
+  definition C [protected] := true
+  definition D := true
+end foo
+
+open foo
+check foo.C
+check D
diff --git a/tests/lean/run/t3.lean b/tests/lean/run/t3.lean
new file mode 100644
index 000000000..4cbddd8f4
--- /dev/null
+++ b/tests/lean/run/t3.lean
@@ -0,0 +1,22 @@
+variable int : Type.{1}
+variable nat : Type.{1}
+namespace int
+variable plus : int → int → int
+end int
+
+namespace nat
+variable plus : nat → nat → nat
+end nat
+
+open int nat
+
+variables a b : int
+
+
+check plus a b
+
+variable f : int → int → int
+variable g : nat → nat → int
+notation A `+`:65 B:65 := f A (g B B)
+variable n : nat
+check a + n
diff --git a/tests/lean/run/tac1.lean b/tests/lean/run/tac1.lean
new file mode 100644
index 000000000..47e648b3c
--- /dev/null
+++ b/tests/lean/run/tac1.lean
@@ -0,0 +1,6 @@
+import tools.tactic
+open tactic
+
+definition mytac := apply @and_intro; apply @refl
+
+check @mytac
diff --git a/tests/lean/run/tt1.lean b/tests/lean/run/tt1.lean
new file mode 100644
index 000000000..4753423a5
--- /dev/null
+++ b/tests/lean/run/tt1.lean
@@ -0,0 +1,8 @@
+import data.prod data.num logic.core.quantifiers
+open prod
+
+check (true, false, 10)
+
+-- definition a f := f
+
+check fun x, x ∧ x
diff --git a/tests/lean/slow/nat_bug1.lean b/tests/lean/slow/nat_bug1.lean
new file mode 100644
index 000000000..a571458fe
--- /dev/null
+++ b/tests/lean/slow/nat_bug1.lean
@@ -0,0 +1,23 @@
+----------------------------------------------------------------------------------------------------
+-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
+-- Released under Apache 2.0 license as described in the file LICENSE.
+-- Author: Floris van Doorn
+----------------------------------------------------------------------------------------------------
+import logic
+open tactic num
+
+inductive nat : Type :=
+zero : nat,
+succ : nat → nat
+
+notation `ℕ`:max := nat
+
+abbreviation plus (x y : ℕ) : ℕ
+:= nat_rec x (λ n r, succ r) y
+
+definition to_nat [coercion] [inline] (n : num) : ℕ
+:= num_rec zero (λ n, pos_num_rec (succ zero) (λ n r, plus r (plus r (succ zero))) (λ n r, plus r r) n) n
+
+print "=================="
+theorem nat_rec_zero {P : ℕ → Type} (x : P 0) (f : ∀m, P m → P (succ m)) : nat_rec x f 0 = x :=
+refl _
diff --git a/tests/lean/slow/nat_bug2.lean b/tests/lean/slow/nat_bug2.lean
new file mode 100644
index 000000000..925eeb164
--- /dev/null
+++ b/tests/lean/slow/nat_bug2.lean
@@ -0,0 +1,1412 @@
+----------------------------------------------------------------------------------------------------
+-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
+-- Released under Apache 2.0 license as described in the file LICENSE.
+-- Author: Floris van Doorn
+----------------------------------------------------------------------------------------------------
+import logic struc.binary
+open tactic num binary eq_ops
+open decidable
+
+namespace nat
+inductive nat : Type :=
+zero : nat,
+succ : nat → nat
+
+notation `ℕ`:max := nat
+
+abbreviation plus (x y : ℕ) : ℕ
+:= nat_rec x (λ n r, succ r) y
+
+definition to_nat [coercion] [inline] (n : num) : ℕ
+:= num_rec zero (λ n, pos_num_rec (succ zero) (λ n r, plus r (plus r (succ zero))) (λ n r, plus r r) n) n
+
+namespace helper_tactics
+  definition apply_refl := apply @refl
+  tactic_hint apply_refl
+end helper_tactics
+open helper_tactics
+
+theorem nat_rec_zero {P : ℕ → Type} (x : P 0) (f : ∀m, P m → P (succ m)) : nat_rec x f 0 = x
+
+theorem nat_rec_succ {P : ℕ → Type} (x : P 0) (f : ∀m, P m → P (succ m)) (n : ℕ) : nat_rec x f (succ n) = f n (nat_rec x f n)
+
+theorem induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : ∀ (n : ℕ) (IH : P n), P (succ n)) : P a
+:= nat_rec H1 H2 a
+
+definition rec_on {P : ℕ → Type} (n : ℕ) (H1 : P 0) (H2 : ∀m, P m → P (succ m)) : P n
+:= nat_rec H1 H2 n
+
+-------------------------------------------------- succ pred
+
+theorem succ_ne_zero (n : ℕ) : succ n ≠ 0
+:= assume H : succ n = 0,
+     have H2 : true = false, from
+     let f := (nat_rec false (fun a b, true)) in
+       calc true = f (succ n) : _
+             ... = f 0        : {H}
+             ... = false      : _,
+     absurd H2 true_ne_false
+
+definition pred (n : ℕ) := nat_rec 0 (fun m x, m) n
+
+theorem pred_zero : pred 0 = 0
+
+theorem pred_succ (n : ℕ) : pred (succ n) = n
+
+theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n)
+:= induction_on n
+    (or_intro_left _ (refl 0))
+    (take m IH, or_intro_right _
+      (show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
+
+theorem zero_or_succ2 (n : ℕ) : n = 0 ∨ ∃k, n = succ k
+:= or_imp_or (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists_intro (pred n) H)
+
+theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n
+:= induction_on n H1 (take m IH, H2 m)
+
+theorem discriminate {B : Prop} {n : ℕ} (H1: n = 0 → B) (H2 : ∀m, n = succ m → B) : B
+:= or_elim (zero_or_succ n)
+    (take H3 : n = 0, H1 H3)
+    (take H3 : n = succ (pred n), H2 (pred n) H3)
+
+theorem succ_inj {n m : ℕ} (H : succ n = succ m) : n = m
+:= calc
+    n = pred (succ n) : pred_succ n⁻¹
+  ... = pred (succ m) : {H}
+  ... = m             : pred_succ m
+
+theorem succ_ne_self (n : ℕ) : succ n ≠ n
+:= induction_on n
+    (take H : 1 = 0,
+      have ne : 1 ≠ 0, from succ_ne_zero 0,
+      absurd H ne)
+    (take k IH H, IH (succ_inj H))
+
+theorem decidable_eq [instance] (n m : ℕ) : decidable (n = m)
+:= have general : ∀n, decidable (n = m), from
+     rec_on m
+       (take n,
+         rec_on n
+           (inl (refl 0))
+           (λ m iH, inr (succ_ne_zero m)))
+       (λ (m' : ℕ) (iH1 : ∀n, decidable (n = m')),
+         take n, rec_on n
+           (inr (ne_symm (succ_ne_zero m')))
+           (λ (n' : ℕ) (iH2 : decidable (n' = succ m')),
+             have d1 : decidable (n' = m'), from iH1 n',
+             decidable.rec_on d1
+               (assume Heq : n' = m', inl (congr_arg succ Heq))
+               (assume Hne : n' ≠ m',
+                 have H1 : succ n' ≠ succ m', from
+                   assume Heq, absurd (succ_inj Heq) Hne,
+                 inr H1))),
+  general n
+
+theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
+    (H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a
+:= have stronger : P a ∧ P (succ a), from
+    induction_on a
+      (and_intro H1 H2)
+      (take k IH,
+        have IH1 : P k, from and_elim_left IH,
+        have IH2 : P (succ k), from and_elim_right IH,
+          and_intro IH2 (H3 k IH1 IH2)),
+    and_elim_left stronger
+
+theorem sub_induction {P : ℕ → ℕ → Prop} (n m : ℕ) (H1 : ∀m, P 0 m)
+   (H2 : ∀n, P (succ n) 0) (H3 : ∀n m, P n m → P (succ n) (succ m)) : P n m
+:= have general : ∀m, P n m, from induction_on n
+    (take m : ℕ, H1 m)
+    (take k : ℕ,
+      assume IH : ∀m, P k m,
+      take m : ℕ,
+      discriminate
+        (assume Hm : m = 0,
+          Hm⁻¹ ▸ (H2 k))
+        (take l : ℕ,
+          assume Hm : m = succ l,
+            Hm⁻¹ ▸ (H3 k l (IH l)))),
+  general m
+
+-------------------------------------------------- add
+definition add (x y : ℕ) : ℕ := plus x y
+infixl `+`:65 := add
+theorem add_zero_right (n : ℕ) : n + 0 = n
+theorem add_succ_right (n m : ℕ) : n + succ m = succ (n + m)
+---------- comm, assoc
+
+theorem add_zero_left (n : ℕ) : 0 + n = n
+:= induction_on n
+    (add_zero_right 0)
+    (take m IH, show 0 + succ m = succ m, from
+      calc
+        0 + succ m = succ (0 + m) : add_succ_right _ _
+            ... = succ m : {IH})
+
+theorem add_succ_left (n m : ℕ) : (succ n) + m = succ (n + m)
+:= induction_on m
+    (calc
+      succ n + 0 = succ n : add_zero_right (succ n)
+        ... = succ (n + 0) : {symm (add_zero_right n)})
+    (take k IH,
+      calc
+        succ n + succ k = succ (succ n + k) : add_succ_right _ _
+          ... = succ (succ (n + k)) : {IH}
+          ... = succ (n + succ k) : {symm (add_succ_right _ _)})
+
+theorem add_comm (n m : ℕ) : n + m = m + n
+:= induction_on m
+    (trans (add_zero_right _) (symm (add_zero_left _)))
+    (take k IH,
+      calc
+        n + succ k = succ (n+k) : add_succ_right _ _
+          ... = succ (k + n) : {IH}
+          ... = succ k + n : symm (add_succ_left _ _))
+
+theorem add_move_succ (n m : ℕ) : succ n + m = n + succ m
+:= calc
+    succ n + m = succ (n + m) : add_succ_left n m
+      ... = n +succ m : symm (add_succ_right n m)
+
+theorem add_comm_succ (n m : ℕ) : n + succ m = m + succ n
+:= calc
+    n + succ m = succ n + m : symm (add_move_succ n m)
+      ... = m + succ n : add_comm (succ n) m
+
+theorem add_assoc (n m k : ℕ) : (n + m) + k = n + (m + k)
+:= induction_on k
+    (calc
+      (n + m) + 0 = n + m : add_zero_right _
+        ... = n + (m + 0) : {symm (add_zero_right m)})
+    (take l IH,
+      calc
+        (n + m) + succ l = succ ((n + m) + l) : add_succ_right _ _
+          ... = succ (n + (m + l)) : {IH}
+          ... = n + succ (m + l) : symm (add_succ_right _ _)
+          ... = n + (m + succ l) : {symm (add_succ_right _ _)})
+
+theorem add_left_comm (n m k : ℕ) : n + (m + k) = m + (n + k)
+:= left_comm add_comm add_assoc n m k
+
+theorem add_right_comm (n m k : ℕ) : n + m + k = n + k + m
+:= right_comm add_comm add_assoc n m k
+
+
+---------- inversion
+
+theorem add_cancel_left {n m k : ℕ} : n + m = n + k → m = k
+:=
+  induction_on n
+    (take H : 0 + m = 0 + k,
+      calc
+        m = 0 + m : symm (add_zero_left m)
+          ... = 0 + k : H
+          ... = k : add_zero_left k)
+    (take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
+      have H2 : succ (n + m) = succ (n + k),
+      from calc
+        succ (n + m) = succ n + m : symm (add_succ_left n m)
+          ... = succ n + k : H
+          ... = succ (n + k) : add_succ_left n k,
+      have H3 : n + m = n + k, from succ_inj H2,
+      IH H3)
+
+--rename to and_cancel_right
+theorem add_cancel_right {n m k : ℕ} (H : n + m = k + m) : n = k
+:=
+  have H2 : m + n = m + k,
+    from calc
+      m + n = n + m : add_comm m n
+      ... = k + m : H
+      ... = m + k : add_comm k m,
+    add_cancel_left H2
+
+theorem add_eq_zero_left {n m : ℕ} : n + m = 0 → n = 0
+:=
+  induction_on n
+    (take (H : 0 + m = 0), refl 0)
+    (take k IH,
+      assume (H : succ k + m = 0),
+      absurd
+        (show succ (k + m) = 0, from
+          calc
+            succ (k + m) = succ k + m : symm (add_succ_left k m)
+              ... = 0 : H)
+        (succ_ne_zero (k + m)))
+
+theorem add_eq_zero_right {n m : ℕ} (H : n + m = 0) : m = 0
+:= add_eq_zero_left (trans (add_comm m n) H)
+
+theorem add_eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0
+:= and_intro (add_eq_zero_left H) (add_eq_zero_right H)
+
+-- add_eq_self below
+
+---------- misc
+
+theorem add_one (n:ℕ) : n + 1 = succ n
+:=
+  calc
+    n + 1 = succ (n + 0) : add_succ_right _ _
+      ... = succ n : {add_zero_right _}
+
+theorem add_one_left (n:ℕ) : 1 + n = succ n
+:=
+  calc
+    1 + n = succ (0 + n) : add_succ_left _ _
+      ... = succ n : {add_zero_left _}
+
+--the following theorem has a terrible name, but since the name is not a substring or superstring of another name, it is at least easy to globally replace it
+theorem induction_plus_one {P : ℕ → Prop} (a : ℕ) (H1 : P 0)
+    (H2 : ∀ (n : ℕ) (IH : P n), P (n + 1)) : P a
+:= nat_rec H1 (take n IH, (add_one n) ▸ (H2 n IH)) a
+
+-------------------------------------------------- mul
+
+definition mul (n m : ℕ) := nat_rec 0 (fun m x, x + n) m
+infixl `*`:75 := mul
+
+theorem mul_zero_right (n:ℕ) : n * 0 = 0
+theorem mul_succ_right (n m:ℕ) : n * succ m = n * m + n
+set_option unifier.max_steps 100000
+---------- comm, distr, assoc, identity
+
+theorem mul_zero_left (n:ℕ) : 0 * n = 0
+:= induction_on n
+    (mul_zero_right 0)
+    (take m IH,
+      calc
+        0 * succ m = 0 * m + 0 : mul_succ_right _ _
+          ... = 0 * m : add_zero_right _
+          ... = 0 : IH)
+
+theorem mul_succ_left (n m:ℕ) : (succ n) * m = (n * m) + m
+:= induction_on m
+    (calc
+      succ n * 0 = 0     : mul_zero_right _
+        ... = n * 0      : symm (mul_zero_right _)
+        ... = n * 0 + 0  : symm (add_zero_right _))
+    (take k IH,
+      calc
+        succ n * succ k = (succ n * k) + succ n : mul_succ_right _ _
+            ... = (n * k) + k + succ n : { IH }
+            ... = (n * k) + (k + succ n) : add_assoc _ _ _
+            ... = (n * k) + (n + succ k) : {add_comm_succ _ _}
+            ... = (n * k) + n + succ k : symm (add_assoc _ _ _)
+            ... = (n * succ k) + succ k : {symm (mul_succ_right n k)})
+
+theorem mul_comm (n m:ℕ) : n * m = m * n
+:= induction_on m
+    (trans (mul_zero_right _) (symm (mul_zero_left _)))
+    (take k IH,
+      calc
+        n * succ k = n * k + n : mul_succ_right _ _
+          ... = k * n + n : {IH}
+          ... = (succ k) * n : symm (mul_succ_left _ _))
+
+theorem mul_add_distr_left (n m k : ℕ) : (n + m) * k = n * k + m * k
+:= induction_on k
+    (calc
+      (n + m) * 0 = 0 : mul_zero_right _
+        ... = 0 + 0 : symm (add_zero_right _)
+        ... = n * 0 + 0 : refl _
+        ... = n * 0 + m * 0 : refl _)
+    (take l IH, calc
+        (n + m) * succ l = (n + m) * l + (n + m) : mul_succ_right _ _
+          ... = n * l + m * l + (n + m) : {IH}
+          ... = n * l + m * l + n + m : symm (add_assoc _ _ _)
+          ... = n * l + n + m * l + m : {add_right_comm _ _ _}
+          ... = n * l + n + (m * l + m) : add_assoc _ _ _
+          ... = n * succ l + (m * l + m) : {symm (mul_succ_right _ _)}
+          ... = n * succ l + m * succ l : {symm (mul_succ_right _ _)})
+
+theorem mul_add_distr_right (n m k : ℕ) : n * (m + k) = n * m + n * k
+:= calc
+    n * (m + k) = (m + k) * n : mul_comm _ _
+      ... = m * n + k * n : mul_add_distr_left _ _ _
+      ... = n * m + k * n : {mul_comm _ _}
+      ... = n * m + n * k : {mul_comm _ _}
+
+theorem mul_assoc (n m k:ℕ) : (n * m) * k = n * (m * k)
+:= induction_on k
+    (calc
+      (n * m) * 0 = 0 : mul_zero_right _
+        ... = n * 0 : symm (mul_zero_right _)
+        ... = n * (m * 0) : {symm (mul_zero_right _)})
+    (take l IH,
+      calc
+        (n * m) * succ l = (n * m) * l + n * m : mul_succ_right _ _
+          ... = n * (m * l) + n * m : {IH}
+          ... = n * (m * l + m) : symm (mul_add_distr_right _ _ _)
+          ... = n * (m * succ l) : {symm (mul_succ_right _ _)})
+
+theorem mul_comm_left (n m k : ℕ) : n * (m * k) = m * (n * k)
+:= left_comm mul_comm mul_assoc n m k
+
+theorem mul_comm_right (n m k : ℕ) : n * m * k = n * k * m
+:= right_comm mul_comm mul_assoc n m k
+
+theorem mul_one_right (n : ℕ) : n * 1 = n
+:= calc
+    n * 1 = n * 0 + n : mul_succ_right n 0
+      ... = 0 + n : {mul_zero_right n}
+      ... = n : add_zero_left n
+
+theorem mul_one_left (n : ℕ) : 1 * n = n
+:= calc
+    1 * n = n * 1 : mul_comm _ _
+      ... = n : mul_one_right n
+
+---------- inversion
+
+theorem mul_eq_zero {n m : ℕ} (H : n * m = 0) : n = 0 ∨ m = 0
+:=
+  discriminate
+    (take Hn : n = 0, or_intro_left _ Hn)
+    (take (k : ℕ),
+      assume (Hk : n = succ k),
+      discriminate
+        (take (Hm : m = 0), or_intro_right _ Hm)
+        (take (l : ℕ),
+          assume (Hl : m = succ l),
+          have Heq : succ (k * succ l + l) = n * m, from
+            symm (calc
+              n * m = n * succ l : { Hl }
+                ... = succ k * succ l : { Hk }
+                ... = k * succ l + succ l : mul_succ_left _ _
+                ... = succ (k * succ l + l) : add_succ_right _ _),
+          absurd (trans Heq H) (succ_ne_zero _)))
+
+-- see more under "positivity" below
+-------------------------------------------------- le
+
+definition le (n m:ℕ) : Prop := ∃k, n + k = m
+infix  `<=`:50 := le
+infix  `≤`:50  := le
+
+theorem le_intro {n m k : ℕ} (H : n + k = m) : n ≤ m
+:= exists_intro k H
+
+theorem le_elim {n m : ℕ} (H : n ≤ m) : ∃ k, n + k = m
+:= H
+
+---------- partial order (totality is part of lt)
+
+theorem le_intro2 (n m : ℕ) : n ≤ n + m
+:= le_intro (refl (n + m))
+
+theorem le_refl (n : ℕ) : n ≤ n
+:= le_intro (add_zero_right n)
+
+theorem zero_le (n : ℕ) : 0 ≤ n
+:= le_intro (add_zero_left n)
+
+theorem le_zero {n : ℕ} (H : n ≤ 0) : n = 0
+:=
+  obtain (k : ℕ) (Hk : n + k = 0), from le_elim H,
+  add_eq_zero_left Hk
+
+theorem not_succ_zero_le (n : ℕ) : ¬ succ n ≤ 0
+:= assume H : succ n ≤ 0,
+     have H2 : succ n = 0, from le_zero H,
+     absurd H2 (succ_ne_zero n)
+
+theorem le_zero_inv {n : ℕ} (H : n ≤ 0) : n = 0
+:= obtain (k : ℕ) (Hk : n + k = 0), from le_elim H,
+   add_eq_zero_left Hk
+
+theorem le_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k
+:= obtain (l1 : ℕ) (Hl1 : n + l1 = m), from le_elim H1,
+   obtain (l2 : ℕ) (Hl2 : m + l2 = k), from le_elim H2,
+   le_intro
+     (calc
+       n + (l1 + l2) =  n + l1 + l2 : symm (add_assoc n l1 l2)
+         ... = m + l2 : { Hl1 }
+         ... = k : Hl2)
+
+theorem le_antisym {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m
+:= obtain (k : ℕ) (Hk : n + k = m), from (le_elim H1),
+   obtain (l : ℕ) (Hl : m + l = n), from (le_elim H2),
+   have L1 : k + l = 0, from
+     add_cancel_left
+       (calc
+         n + (k + l) = n + k + l : { symm (add_assoc n k l) }
+           ... = m + l : { Hk }
+           ... = n : Hl
+           ... = n + 0 : symm (add_zero_right n)),
+   have L2 : k = 0, from add_eq_zero_left L1,
+   calc
+     n = n + 0 : symm (add_zero_right n)
+      ... = n  + k : { symm L2 }
+      ... = m : Hk
+
+---------- interaction with add
+
+theorem add_le_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k + n ≤ k + m
+:= obtain (l : ℕ) (Hl : n + l = m), from (le_elim H),
+   le_intro
+     (calc
+        k + n + l  = k + (n + l) : add_assoc k n l
+               ... = k + m : { Hl })
+
+theorem add_le_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n + k ≤ m + k
+:= (add_comm k m) ▸ (add_comm k n) ▸ (add_le_left H k)
+
+theorem add_le {n m k l : ℕ} (H1 : n ≤ k) (H2 : m ≤ l) : n + m ≤ k + l
+:= le_trans (add_le_right H1 m) (add_le_left H2 k)
+
+theorem add_le_left_inv {n m k : ℕ} (H : k + n ≤ k + m) : n ≤ m
+:=
+  obtain (l : ℕ) (Hl : k + n + l = k + m), from (le_elim H),
+  le_intro (add_cancel_left
+    (calc
+        k + (n + l)  = k + n + l : symm (add_assoc k n l)
+          ... = k + m : Hl))
+
+theorem add_le_right_inv {n m k : ℕ} (H : n + k ≤ m + k) : n ≤ m
+:= add_le_left_inv (add_comm m k ▸ add_comm n k ▸ H)
+
+---------- interaction with succ and pred
+
+theorem succ_le {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m
+:= add_one m ▸ add_one n ▸ add_le_right H 1
+
+theorem succ_le_cancel {n m : ℕ} (H : succ n ≤ succ m) :  n ≤ m
+:= add_le_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)
+
+theorem self_le_succ (n : ℕ) : n ≤ succ n
+:= le_intro (add_one n)
+
+theorem le_imp_le_succ {n m : ℕ} (H : n ≤ m) : n ≤ succ m
+:= le_trans H (self_le_succ m)
+
+theorem succ_le_left_or {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
+:= obtain (k : ℕ) (Hk : n + k = m), from (le_elim H),
+   discriminate
+    (assume H3 : k = 0,
+      have Heq : n = m,
+        from calc
+          n = n + 0 : (add_zero_right n)⁻¹
+            ... = n + k : {H3⁻¹}
+            ... = m : Hk,
+      or_intro_right _ Heq)
+    (take l:ℕ,
+      assume H3 : k = succ l,
+      have Hlt : succ n ≤ m, from
+        (le_intro
+          (calc
+            succ n + l = n + succ l : add_move_succ n l
+              ... = n + k : {H3⁻¹}
+              ... = m : Hk)),
+      or_intro_left _ Hlt)
+
+theorem succ_le_left {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m
+:= resolve_left (succ_le_left_or H1) H2
+
+theorem succ_le_right_inv {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m
+:= or_imp_or (succ_le_left_or H)
+     (take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2)
+     (take H2 : n = succ m, H2)
+
+theorem succ_le_left_inv {n m : ℕ} (H : succ n ≤ m) : n ≤ m ∧ n ≠ m
+:= obtain (k : ℕ) (H2 : succ n + k = m), from (le_elim H),
+   and_intro
+    (have H3 : n + succ k = m,
+      from calc
+        n + succ k = succ n + k : symm (add_move_succ n k)
+          ... = m : H2,
+      show n ≤ m, from le_intro H3)
+    (assume H3 : n = m,
+        have H4 : succ n ≤ n, from subst (symm H3) H,
+        have H5 : succ n = n, from le_antisym H4 (self_le_succ n),
+        show false, from absurd H5 (succ_ne_self n))
+
+theorem le_pred_self (n : ℕ) : pred n ≤ n
+:= case n
+    (subst (symm pred_zero) (le_refl 0))
+    (take k : ℕ, subst (symm (pred_succ k)) (self_le_succ k))
+
+theorem pred_le {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m
+:= discriminate
+    (take Hn : n = 0,
+      have H2 : pred n = 0,
+        from calc
+          pred n = pred 0 : {Hn}
+             ... = 0 : pred_zero,
+      subst (symm H2) (zero_le (pred m)))
+    (take k : ℕ,
+      assume Hn : n = succ k,
+      obtain (l : ℕ) (Hl : n + l = m), from le_elim H,
+      have H2 : pred n + l = pred m,
+        from calc
+          pred n + l = pred (succ k) + l : {Hn}
+            ... = k + l : {pred_succ k}
+            ... = pred (succ (k + l)) : symm (pred_succ (k + l))
+            ... = pred (succ k + l) : {symm (add_succ_left k l)}
+            ... = pred (n + l) : {symm Hn}
+            ... = pred m : {Hl},
+      le_intro H2)
+
+theorem pred_le_left_inv {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = succ m
+:= discriminate
+    (take Hn : n = 0,
+      or_intro_left _ (subst (symm Hn) (zero_le m)))
+    (take k : ℕ,
+      assume Hn : n = succ k,
+      have H2 : pred n = k,
+        from calc
+          pred n = pred (succ k) : {Hn}
+             ... = k : pred_succ k,
+      have H3 : k ≤ m, from subst H2 H,
+      have H4 : succ k ≤ m ∨ k = m, from succ_le_left_or H3,
+      show n ≤ m ∨ n = succ m, from
+        or_imp_or H4
+          (take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5)
+          (take H5 : k = m, show n = succ m, from subst H5 Hn))
+
+-- ### interaction with successor and predecessor
+
+theorem le_imp_succ_le_or_eq {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
+:=
+  obtain (k : ℕ) (Hk : n + k = m), from (le_elim H),
+  discriminate
+    (assume H3 : k = 0,
+      have Heq : n = m,
+        from calc
+          n = n + 0 : symm (add_zero_right n)
+            ... = n + k : {symm H3}
+            ... = m : Hk,
+      or_intro_right _ Heq)
+    (take l : nat,
+      assume H3 : k = succ l,
+      have Hlt : succ n ≤ m, from
+        (le_intro
+          (calc
+            succ n + l = n + succ l : add_move_succ n l
+              ... = n + k : {symm H3}
+              ... = m : Hk)),
+      or_intro_left _ Hlt)
+
+theorem le_ne_imp_succ_le {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m
+:= resolve_left (le_imp_succ_le_or_eq H1) H2
+
+theorem succ_le_imp_le_and_ne {n m : ℕ} (H : succ n ≤ m) : n ≤ m ∧ n ≠ m
+:=
+  and_intro
+    (le_trans (self_le_succ n) H)
+    (assume H2 : n = m,
+        have H3 : succ n ≤ n, from subst (symm H2) H,
+        have H4 : succ n = n, from le_antisym H3 (self_le_succ n),
+        show false, from absurd H4 (succ_ne_self n))
+
+theorem pred_le_self (n : ℕ) : pred n ≤ n
+:=
+  case n
+    (subst (symm pred_zero) (le_refl 0))
+    (take k : nat, subst (symm (pred_succ k)) (self_le_succ k))
+
+theorem pred_le_imp_le_or_eq {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = succ m
+:=
+  discriminate
+    (take Hn : n = 0,
+      or_intro_left _ (subst (symm Hn) (zero_le m)))
+    (take k : nat,
+      assume Hn : n = succ k,
+      have H2 : pred n = k,
+        from calc
+          pred n = pred (succ k) : {Hn}
+             ... = k : pred_succ k,
+      have H3 : k ≤ m, from subst H2 H,
+      have H4 : succ k ≤ m ∨ k = m, from le_imp_succ_le_or_eq H3,
+      show n ≤ m ∨ n = succ m, from
+        or_imp_or H4
+          (take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5)
+          (take H5 : k = m, show n = succ m, from subst H5 Hn))
+
+---------- interaction with mul
+
+theorem mul_le_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k * n ≤ k * m
+:=
+  obtain (l : ℕ) (Hl : n + l = m), from (le_elim H),
+  induction_on k
+    (have H2 : 0 * n = 0 * m,
+      from calc
+        0 * n = 0 : mul_zero_left n
+          ... = 0 * m : symm (mul_zero_left m),
+      show 0 * n ≤ 0 * m, from subst H2 (le_refl (0 * n)))
+    (take (l : ℕ),
+      assume IH : l * n ≤ l * m,
+      have H2 : l * n + n ≤ l * m + m, from add_le IH H,
+      have H3 : succ l * n ≤ l * m + m, from subst (symm (mul_succ_left l n)) H2,
+      show succ l * n ≤ succ l * m, from subst (symm (mul_succ_left l m)) H3)
+
+theorem mul_le_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n * k ≤ m * k
+:= mul_comm k m ▸ mul_comm k n ▸ (mul_le_left H k)
+
+theorem mul_le {n m k l : ℕ} (H1 : n ≤ k) (H2 : m ≤ l) : n * m ≤ k * l
+:= le_trans (mul_le_right H1 m) (mul_le_left H2 k)
+
+-- mul_le_[left|right]_inv below
+
+-------------------------------------------------- lt
+
+definition lt (n m : ℕ) := succ n ≤ m
+infix `<`:50 := lt
+
+theorem lt_intro {n m k : ℕ} (H : succ n + k = m) : n < m
+:= le_intro H
+
+theorem lt_elim {n m : ℕ} (H : n < m) : ∃ k, succ n + k = m
+:= le_elim H
+
+theorem lt_intro2 (n m : ℕ) : n < n + succ m
+:= lt_intro (add_move_succ n m)
+
+-------------------------------------------------- ge, gt
+
+definition ge (n m : ℕ) := m ≤ n
+infix `>=`:50 := ge
+infix `≥`:50  := ge
+
+definition gt (n m : ℕ) := m < n
+infix `>`:50  := gt
+
+---------- basic facts
+
+theorem lt_ne {n m : ℕ} (H : n < m) : n ≠ m
+:= and_elim_right (succ_le_left_inv H)
+
+theorem lt_irrefl (n : ℕ) : ¬ n < n
+:= assume H : n < n, absurd (refl n) (lt_ne H)
+
+theorem lt_zero (n : ℕ) : 0 < succ n
+:= succ_le (zero_le n)
+
+theorem lt_zero_inv (n : ℕ) : ¬ n < 0
+:= assume H : n < 0,
+      have H2 : succ n = 0, from le_zero_inv H,
+      absurd H2 (succ_ne_zero n)
+
+theorem lt_positive {n m : ℕ} (H : n < m) : ∃k, m = succ k
+:= discriminate
+    (take (Hm : m = 0), absurd (subst Hm H) (lt_zero_inv n))
+    (take (l : ℕ) (Hm : m = succ l), exists_intro l Hm)
+
+---------- interaction with le
+
+theorem lt_imp_le_succ {n m : ℕ} (H : n < m) : succ n ≤ m
+:= H
+
+theorem le_succ_imp_lt {n m : ℕ} (H : succ n ≤ m) : n < m
+:= H
+
+theorem self_lt_succ (n : ℕ) : n < succ n
+:= le_refl (succ n)
+
+theorem lt_imp_le {n m : ℕ} (H : n < m) : n ≤ m
+:= and_elim_left (succ_le_imp_le_and_ne H)
+
+theorem le_imp_lt_or_eq {n m : ℕ} (H : n ≤ m) : n < m ∨ n = m
+:= le_imp_succ_le_or_eq H
+
+theorem le_ne_imp_lt {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : n < m
+:= le_ne_imp_succ_le H1 H2
+
+theorem le_imp_lt_succ {n m : ℕ} (H : n ≤ m) : n < succ m
+:= succ_le H
+
+theorem lt_succ_imp_le {n m : ℕ} (H : n < succ m) : n ≤ m
+:= succ_le_cancel H
+
+---------- trans, antisym
+
+theorem lt_le_trans {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k
+:= le_trans H1 H2
+
+theorem le_lt_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k
+:= le_trans (succ_le H1) H2
+
+theorem lt_trans {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k
+:= lt_le_trans H1 (lt_imp_le H2)
+
+theorem le_imp_not_gt {n m : ℕ} (H : n ≤ m) : ¬ n > m
+:= assume H2 : m < n, absurd (le_lt_trans H H2) (lt_irrefl n)
+
+theorem lt_imp_not_ge {n m : ℕ} (H : n < m) : ¬ n ≥ m
+:= assume H2 : m ≤ n, absurd (lt_le_trans H H2) (lt_irrefl n)
+
+theorem lt_antisym {n m : ℕ} (H : n < m) : ¬ m < n
+:= le_imp_not_gt (lt_imp_le H)
+
+---------- interaction with add
+
+theorem add_lt_left {n m : ℕ} (H : n < m) (k : ℕ) : k + n < k + m
+:= add_succ_right k n ▸ add_le_left H k
+
+theorem add_lt_right {n m : ℕ} (H : n < m) (k : ℕ) : n + k < m + k
+:= add_comm k m ▸ add_comm k n ▸ add_lt_left H k
+
+theorem add_le_lt {n m k l : ℕ} (H1 : n ≤ k) (H2 : m < l) : n + m < k + l
+:= le_lt_trans (add_le_right H1 m) (add_lt_left H2 k)
+
+theorem add_lt_le {n m k l : ℕ} (H1 : n < k) (H2 : m ≤ l) : n + m < k + l
+:= lt_le_trans (add_lt_right H1 m) (add_le_left H2 k)
+
+theorem add_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n + m < k + l
+:= add_lt_le H1 (lt_imp_le H2)
+
+theorem add_lt_left_inv {n m k : ℕ} (H : k + n < k + m) : n < m
+:= add_le_left_inv (add_succ_right k n⁻¹ ▸ H)
+
+theorem add_lt_right_inv {n m k : ℕ} (H : n + k < m + k) : n < m
+:= add_lt_left_inv (add_comm m k ▸ add_comm n k ▸ H)
+
+---------- interaction with succ (see also the interaction with le)
+
+theorem succ_lt {n m : ℕ} (H : n < m) : succ n < succ m
+:= add_one m ▸ add_one n ▸ add_lt_right H 1
+
+theorem succ_lt_inv {n m : ℕ} (H : succ n < succ m) :  n < m
+:= add_lt_right_inv (add_one m⁻¹ ▸ add_one n⁻¹ ▸ H)
+
+theorem lt_self_succ (n : ℕ) : n < succ n
+:= le_refl (succ n)
+
+theorem succ_lt_right {n m : ℕ} (H : n < m) : n < succ m
+:= lt_trans H (lt_self_succ m)
+
+---------- totality of lt and le
+
+theorem le_or_lt (n m : ℕ) : n ≤ m ∨ m < n
+:= induction_on n
+    (or_intro_left _ (zero_le m))
+    (take (k : ℕ),
+      assume IH : k ≤ m ∨ m < k,
+      or_elim IH
+        (assume H : k ≤ m,
+          obtain (l : ℕ) (Hl : k + l = m), from le_elim H,
+          discriminate
+            (assume H2 : l = 0,
+              have H3 : m = k,
+                from calc
+                  m = k + l : symm Hl
+                    ... = k + 0 : {H2}
+                    ... = k : add_zero_right k,
+              have H4 : m < succ k, from subst  H3 (lt_self_succ m),
+              or_intro_right _ H4)
+            (take l2 : ℕ,
+              assume H2 : l = succ l2,
+              have H3 : succ k + l2 = m,
+                from calc
+                  succ k + l2 = k + succ l2 : add_move_succ k l2
+                    ... = k + l : {symm H2}
+                    ... = m : Hl,
+              or_intro_left _ (le_intro H3)))
+        (assume H : m < k, or_intro_right _ (succ_lt_right H)))
+
+theorem trichotomy_alt (n m : ℕ) : (n < m ∨ n = m) ∨ m < n
+:= or_imp_or (le_or_lt n m) (assume H : n ≤ m, le_imp_lt_or_eq H) (assume H : m < n, H)
+
+theorem trichotomy (n m : ℕ) : n < m ∨ n = m ∨ m < n
+:= iff_elim_left or_assoc (trichotomy_alt n m)
+
+theorem le_total (n m : ℕ) : n ≤ m ∨ m ≤ n
+:= or_imp_or (le_or_lt n m) (assume H : n ≤ m, H) (assume H : m < n, lt_imp_le H)
+
+-- interaction with mul under "positivity"
+
+theorem strong_induction_on {P : ℕ → Prop} (n : ℕ) (IH : ∀n, (∀m, m < n → P m) → P n) : P n
+:= have stronger : ∀k, k ≤ n → P k, from
+    induction_on n
+      (take (k : ℕ),
+        assume H : k ≤ 0,
+        have H2 : k = 0, from le_zero_inv H,
+        have H3 : ∀m, m < k → P m, from
+          (take m : ℕ,
+            assume H4 : m < k,
+            have H5 : m < 0, from subst H2 H4,
+            absurd H5 (lt_zero_inv m)),
+        show P k, from IH k H3)
+      (take l : ℕ,
+        assume IHl : ∀k, k ≤ l → P k,
+        take k : ℕ,
+        assume H : k ≤ succ l,
+        or_elim (succ_le_right_inv H)
+          (assume H2 : k ≤ l, show P k, from IHl k H2)
+          (assume H2 : k = succ l,
+            have H3 : ∀m, m < k → P m, from
+              (take m : ℕ,
+                assume H4 : m < k,
+                have H5 : m ≤ l, from lt_succ_imp_le (subst H2 H4),
+                show P m, from IHl m H5),
+            show P k, from IH k H3)),
+  stronger n (le_refl n)
+
+theorem case_strong_induction_on {P : ℕ → Prop} (a : ℕ) (H0 : P 0) (Hind : ∀(n : ℕ), (∀m, m ≤ n → P m) → P (succ n)) : P a
+:= strong_induction_on a
+     (take n, case n
+       (assume H : (∀m, m < 0 → P m), H0)
+       (take n, assume H : (∀m, m < succ n → P m),
+         Hind n (take m, assume H1 : m ≤ n, H m (le_imp_lt_succ H1))))
+
+theorem add_eq_self {n m : ℕ} (H : n + m = n) : m = 0
+:= discriminate
+    (take Hm : m = 0, Hm)
+    (take k : ℕ,
+      assume Hm : m = succ k,
+      have H2 : succ n + k = n,
+        from calc
+          succ n + k = n + succ k : add_move_succ n k
+            ... = n + m : {symm Hm}
+            ... = n : H,
+      have H3 : n < n, from lt_intro H2,
+      have H4 : n ≠ n, from lt_ne H3,
+      absurd (refl n) H4)
+
+-------------------------------------------------- positivity
+
+--  we use " _ > 0" as canonical way of denoting that a number is positive
+
+---------- basic
+
+theorem zero_or_positive (n : ℕ) : n = 0 ∨ n > 0
+:= or_imp_or (or_swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H) (take H : n > 0, H)
+
+theorem succ_positive {n m : ℕ} (H : n = succ m) : n > 0
+:= subst (symm H) (lt_zero m)
+
+theorem ne_zero_positive {n : ℕ} (H : n ≠ 0) : n > 0
+:= or_elim (zero_or_positive n) (take H2 : n = 0, absurd H2 H) (take H2 : n > 0, H2)
+
+theorem pos_imp_eq_succ {n : ℕ} (H : n > 0) : ∃l, n = succ l
+:= discriminate
+    (take H2, absurd (subst H2 H) (lt_irrefl 0))
+    (take l Hl, exists_intro l Hl)
+
+theorem add_positive_right (n : ℕ) {k : ℕ} (H : k > 0) : n + k > n
+:= obtain (l : ℕ) (Hl : k = succ l), from pos_imp_eq_succ H,
+   subst (symm Hl) (lt_intro2 n l)
+
+theorem add_positive_left (n : ℕ) {k : ℕ} (H : k > 0) : k + n > n
+:= subst (add_comm n k) (add_positive_right n H)
+
+
+-- Positivity
+-- ---------
+--
+-- Writing "t > 0" is the preferred way to assert that a natural number is positive.
+
+-- ### basic
+
+-- See also succ_pos.
+
+theorem succ_pos (n : ℕ) : 0 < succ n
+:= succ_le (zero_le n)
+
+theorem case_zero_pos {P : ℕ → Prop} (y : ℕ) (H0 : P 0) (H1 : ∀y, y > 0 → P y) : P y
+:= case y H0 (take y', H1 _ (succ_pos _))
+
+theorem succ_imp_pos {n m : ℕ} (H : n = succ m) : n > 0
+:= subst (symm H) (succ_pos m)
+
+theorem add_pos_right (n : ℕ) {k : ℕ} (H : k > 0) : n + k > n
+:= subst (add_zero_right n) (add_lt_left H n)
+
+theorem add_pos_left (n : ℕ) {k : ℕ} (H : k > 0) : k + n > n
+:= subst (add_comm n k) (add_pos_right n H)
+
+---------- mul
+
+theorem mul_positive {n m : ℕ} (Hn : n > 0) (Hm : m > 0) : n * m > 0
+:= obtain (k : ℕ) (Hk : n = succ k), from pos_imp_eq_succ Hn,
+   obtain (l : ℕ) (Hl : m = succ l), from pos_imp_eq_succ Hm,
+   succ_positive (calc
+     n * m = succ k * m : {Hk}
+       ... = succ k * succ l : {Hl}
+       ... = succ k * l + succ k : mul_succ_right (succ k) l
+       ... = succ (succ k * l + k) : add_succ_right _ _)
+
+theorem mul_positive_inv_left {n m : ℕ} (H : n * m > 0) : n > 0
+:= discriminate
+    (assume H2 : n = 0,
+      have H3 : n * m = 0,
+        from calc
+          n * m = 0 * m : {H2}
+            ... = 0 : mul_zero_left m,
+      have H4 : 0 > 0, from subst H3 H,
+      absurd H4 (lt_irrefl 0))
+    (take l : ℕ,
+      assume Hl : n = succ l,
+      subst (symm Hl) (lt_zero l))
+
+theorem mul_positive_inv_right {n m : ℕ} (H : n * m > 0) : m > 0
+:= mul_positive_inv_left (subst (mul_comm n m) H)
+
+theorem mul_left_inj {n m k : ℕ} (Hn : n > 0) (H : n * m = n * k) : m = k
+:=
+  have general : ∀m, n * m = n * k → m = k, from
+    induction_on k
+      (take m:ℕ,
+        assume H : n * m = n * 0,
+        have H2 : n * m = 0,
+          from calc
+            n * m = n * 0 : H
+              ... = 0 : mul_zero_right n,
+        have H3 : n = 0 ∨ m = 0, from mul_eq_zero H2,
+        resolve_right H3 (ne_symm (lt_ne Hn)))
+      (take (l : ℕ),
+        assume (IH : ∀ m, n * m = n * l → m = l),
+        take (m : ℕ),
+        assume (H : n * m = n * succ l),
+        have H2 :  n * succ l > 0, from mul_positive Hn (lt_zero l),
+        have H3 : m > 0, from mul_positive_inv_right (subst (symm H) H2),
+        obtain (l2:ℕ) (Hm : m = succ l2), from pos_imp_eq_succ H3,
+        have H4 : n * l2 + n = n * l + n,
+          from calc
+            n * l2 + n = n * succ l2 : symm (mul_succ_right n l2)
+              ... = n * m : {symm Hm}
+              ... = n * succ l : H
+              ... = n * l + n : mul_succ_right n l,
+        have H5 : n * l2 = n * l, from add_cancel_right H4,
+        calc
+          m = succ l2 : Hm
+        ... = succ l : {IH l2 H5}),
+  general m H
+
+theorem mul_right_inj {n m k : ℕ} (Hm : m > 0) (H : n * m = k * m) : n = k
+:= mul_left_inj Hm (subst (mul_comm k m) (subst (mul_comm n m) H))
+
+-- mul_eq_one below
+
+---------- interaction of mul with le and lt
+
+
+theorem mul_lt_left {n m k : ℕ} (Hk : k > 0) (H : n < m) : k * n < k * m
+:=
+  have H2 : k * n < k * n + k, from add_positive_right (k * n) Hk,
+  have H3 : k * n + k ≤ k * m, from subst (mul_succ_right k n) (mul_le_left H k),
+  lt_le_trans H2 H3
+
+theorem mul_lt_right {n m k : ℕ} (Hk : k > 0) (H : n < m)  : n * k < m * k
+:= subst (mul_comm k m) (subst (mul_comm k n) (mul_lt_left Hk H))
+
+theorem mul_le_lt {n m k l : ℕ} (Hk : k > 0) (H1 : n ≤ k) (H2 : m < l) : n * m < k * l
+:= le_lt_trans (mul_le_right H1 m) (mul_lt_left Hk H2)
+
+theorem mul_lt_le {n m k l : ℕ} (Hl : l > 0) (H1 : n < k) (H2 : m ≤ l) : n * m < k * l
+:= le_lt_trans (mul_le_left H2 n) (mul_lt_right Hl H1)
+
+theorem mul_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n * m < k * l
+:=
+  have H3 : n * m ≤ k * m, from mul_le_right (lt_imp_le H1) m,
+  have H4 : k * m < k * l, from mul_lt_left (le_lt_trans (zero_le n) H1) H2,
+  le_lt_trans H3 H4
+
+theorem mul_lt_left_inv {n m k : ℕ} (H : k * n < k * m) : n < m
+:=
+  have general : ∀ m, k * n < k * m → n < m, from
+    induction_on n
+      (take m : ℕ,
+        assume H2 : k * 0 < k * m,
+        have H3 : 0 < k * m, from mul_zero_right k ▸ H2,
+        show 0 < m, from mul_positive_inv_right H3)
+      (take l : ℕ,
+        assume IH : ∀ m, k * l < k * m → l < m,
+        take m : ℕ,
+        assume H2 : k * succ l < k * m,
+        have H3 : 0 < k * m, from le_lt_trans (zero_le _) H2,
+        have H4 : 0 < m, from mul_positive_inv_right H3,
+        obtain (l2 : ℕ) (Hl2 : m = succ l2), from pos_imp_eq_succ H4,
+        have H5 : k * l + k < k * m, from mul_succ_right k l ▸ H2,
+        have H6 : k * l + k < k * succ l2, from Hl2 ▸ H5,
+        have H7 : k * l + k < k * l2 + k, from mul_succ_right k l2 ▸ H6,
+        have H8 : k * l < k * l2, from add_lt_right_inv H7,
+        have H9 : l < l2, from IH l2 H8,
+        have H10 : succ l < succ l2, from succ_lt H9,
+        show succ l < m, from Hl2⁻¹ ▸ H10),
+  general m H
+
+theorem mul_lt_right_inv {n m k : ℕ} (H : n * k < m * k) : n < m
+:= mul_lt_left_inv (mul_comm m k ▸ mul_comm n k ▸ H)
+
+theorem mul_le_left_inv {n m k : ℕ} (H : succ k * n ≤ succ k * m) : n ≤ m
+:=
+  have H2 : succ k * n < succ k * m + succ k, from le_lt_trans H (lt_intro2 _ _),
+  have H3 : succ k * n < succ k * succ m, from subst (symm (mul_succ_right (succ k) m)) H2,
+  have H4 : n < succ m, from mul_lt_left_inv H3,
+  show n ≤ m, from lt_succ_imp_le H4
+
+theorem mul_le_right_inv {n m k : ℕ} (H : n * succ m ≤ k * succ m) : n ≤ k
+:= mul_le_left_inv (subst (mul_comm k (succ m)) (subst (mul_comm n (succ m)) H))
+
+theorem mul_eq_one_left {n m : ℕ} (H : n * m = 1) : n = 1
+:=
+  have H2 : n * m > 0, from subst (symm H) (lt_zero 0),
+  have H3 : n > 0, from mul_positive_inv_left H2,
+  have H4 : m > 0, from mul_positive_inv_right H2,
+  or_elim (le_or_lt n 1)
+    (assume H5 : n ≤ 1,
+      show n = 1, from le_antisym H5 H3)
+    (assume H5 : n > 1,
+      have H6 : n * m ≥ 2 * 1, from mul_le H5 H4,
+      have H7 : 1 ≥ 2, from subst (mul_one_right 2) (subst H H6),
+      absurd (self_lt_succ 1) (le_imp_not_gt H7))
+
+theorem mul_eq_one_right {n m : ℕ} (H : n * m = 1) : m = 1
+:= mul_eq_one_left (subst (mul_comm n m) H)
+
+theorem mul_eq_one {n m : ℕ} (H : n * m = 1) : n = 1 ∧ m = 1
+:= and_intro (mul_eq_one_left H) (mul_eq_one_right H)
+
+-------------------------------------------------- sub
+
+definition sub (n m : ℕ) : ℕ := nat_rec n (fun m x, pred x) m
+infixl `-`:65 := sub
+theorem sub_zero_right (n : ℕ) : n - 0 = n
+theorem sub_succ_right (n m : ℕ) : n - succ m = pred (n - m)
+
+theorem sub_zero_left (n : ℕ) : 0 - n = 0
+:= induction_on n (sub_zero_right 0)
+    (take k : ℕ,
+      assume IH : 0 - k = 0,
+      calc
+        0 - succ k = pred (0 - k) : sub_succ_right 0 k
+          ... = pred 0 : {IH}
+          ... = 0 : pred_zero)
+
+theorem sub_succ_succ (n m : ℕ) : succ n - succ m = n - m
+:= induction_on m
+    (calc
+      succ n - 1 = pred (succ n - 0) : sub_succ_right (succ n) 0
+        ... = pred (succ n) : {sub_zero_right (succ n)}
+        ... = n : pred_succ n
+        ... = n - 0 : symm (sub_zero_right n))
+    (take k : ℕ,
+      assume IH : succ n - succ k = n - k,
+      calc
+        succ n - succ (succ k) = pred (succ n - succ k) : sub_succ_right (succ n) (succ k)
+          ... = pred (n - k) : {IH}
+          ... = n - succ k : symm (sub_succ_right n k))
+
+theorem sub_one (n : ℕ) : n - 1 = pred n
+:= calc
+    n - 1 = pred (n - 0) : sub_succ_right n 0
+      ... = pred n : {sub_zero_right n}
+
+theorem sub_self (n : ℕ) : n - n = 0
+:= induction_on n (sub_zero_right 0) (take k IH, trans (sub_succ_succ k k) IH)
+
+theorem sub_add_add_right (n m k : ℕ) : (n + k) - (m + k) = n - m
+:= induction_on k
+    (calc
+      (n + 0) - (m + 0) = n - (m + 0) : {add_zero_right _}
+        ... = n - m : {add_zero_right _})
+    (take l : ℕ,
+      assume IH : (n + l) - (m + l) = n - m,
+      calc
+        (n + succ l) - (m + succ l) = succ (n + l) - (m + succ l) : {add_succ_right _ _}
+          ... = succ (n + l) - succ (m + l) : {add_succ_right _ _}
+          ... = (n + l) - (m + l) : sub_succ_succ _ _
+          ... =  n - m : IH)
+
+theorem sub_add_add_left (n m k : ℕ) : (k + n) - (k + m) = n - m
+:= subst (add_comm m k) (subst (add_comm n k) (sub_add_add_right n m k))
+
+theorem sub_add_left (n m : ℕ) : n + m - m = n
+:= induction_on m
+    (subst (symm (add_zero_right n)) (sub_zero_right n))
+    (take k : ℕ,
+      assume IH : n + k - k = n,
+      calc
+        n + succ k - succ k = succ (n + k) - succ k : {add_succ_right n k}
+          ... = n + k - k : sub_succ_succ _ _
+          ... = n : IH)
+
+theorem sub_sub (n m k : ℕ) : n - m - k = n - (m + k)
+:= induction_on k
+    (calc
+      n - m - 0 = n - m : sub_zero_right _
+        ... =  n - (m + 0) : {symm (add_zero_right m)})
+    (take l : ℕ,
+      assume IH : n - m - l = n - (m + l),
+      calc
+        n - m - succ l = pred (n - m - l) : sub_succ_right (n - m) l
+          ... = pred (n - (m + l)) : {IH}
+          ... = n - succ (m + l) : symm (sub_succ_right n (m + l))
+          ... = n - (m + succ l) : {symm (add_succ_right m l)})
+
+theorem succ_sub_sub (n m k : ℕ) : succ n - m - succ k = n - m - k
+:= calc
+    succ n - m - succ k = succ n - (m + succ k) : sub_sub _ _ _
+      ... = succ n - succ (m + k) : {add_succ_right m k}
+      ... = n - (m + k) : sub_succ_succ _ _
+      ... = n - m - k : symm (sub_sub n m k)
+
+theorem sub_add_right_eq_zero (n m : ℕ) : n - (n + m) = 0
+:= calc
+    n - (n + m) = n - n - m : symm (sub_sub n n m)
+      ... = 0 - m : {sub_self n}
+      ... = 0 : sub_zero_left m
+
+theorem sub_comm (m n k : ℕ) : m - n - k = m - k - n
+:= calc
+    m - n - k = m - (n + k) : sub_sub m n k
+      ... = m - (k + n) : {add_comm n k}
+      ... = m - k - n : symm (sub_sub m k n)
+
+theorem succ_sub_one (n : ℕ) : succ n - 1 = n
+:= sub_succ_succ n 0 ⬝ sub_zero_right n
+
+---------- mul
+
+theorem mul_pred_left (n m : ℕ) : pred n * m = n * m - m
+:= induction_on n
+    (calc
+      pred 0 * m = 0 * m : {pred_zero}
+        ... = 0 : mul_zero_left _
+        ... = 0 - m : symm (sub_zero_left m)
+        ... = 0 * m - m : {symm (mul_zero_left m)})
+    (take k : ℕ,
+      assume IH : pred k * m = k * m - m,
+      calc
+        pred (succ k) * m = k * m : {pred_succ k}
+          ... = k * m + m - m : symm (sub_add_left _ _)
+          ... = succ k * m - m : {symm (mul_succ_left k m)})
+
+theorem mul_pred_right (n m : ℕ) : n * pred m = n * m - n
+:= calc n * pred m = pred m * n : mul_comm _ _
+    ... = m * n - n : mul_pred_left m n
+    ... = n * m - n : {mul_comm m n}
+
+theorem mul_sub_distr_left (n m k : ℕ) : (n - m) * k = n * k - m * k
+:= induction_on m
+    (calc
+      (n - 0) * k = n * k : {sub_zero_right n}
+        ... = n * k - 0 : symm (sub_zero_right _)
+        ... = n * k - 0 * k : {symm (mul_zero_left _)})
+    (take l : ℕ,
+      assume IH : (n - l) * k = n * k - l * k,
+      calc
+        (n - succ l) * k = pred (n - l) * k : {sub_succ_right n l}
+          ... = (n - l) * k - k : mul_pred_left _ _
+          ... = n * k - l * k - k : {IH}
+          ... = n * k - (l * k + k) : sub_sub _ _ _
+          ... = n * k - (succ l * k) : {symm (mul_succ_left l k)})
+
+theorem mul_sub_distr_right (n m k : ℕ) : n * (m - k) = n * m - n * k
+:= calc
+    n * (m - k) = (m - k) * n : mul_comm _ _
+      ... = m * n - k * n : mul_sub_distr_left _ _ _
+      ... = n * m - k * n : {mul_comm _ _}
+      ... = n * m - n * k : {mul_comm _ _}
+
+-------------------------------------------------- max, min, iteration, maybe: sub, div
+
+theorem succ_sub {m n : ℕ} : m ≥ n → succ m - n  = succ (m - n)
+:= sub_induction n m
+    (take k,
+      assume H : 0 ≤ k,
+      calc
+        succ k - 0 = succ k : sub_zero_right (succ k)
+          ... = succ (k - 0) : {symm (sub_zero_right k)})
+    (take k,
+      assume H : succ k ≤ 0,
+      absurd H (not_succ_zero_le k))
+    (take k l,
+      assume IH : k ≤ l → succ l - k = succ (l - k),
+      take H : succ k ≤ succ l,
+      calc
+        succ (succ l) - succ k = succ l - k : sub_succ_succ (succ l) k
+          ... = succ (l - k) : IH (succ_le_cancel H)
+          ... = succ (succ l - succ k) : {symm (sub_succ_succ l k)})
+
+theorem le_imp_sub_eq_zero {n m : ℕ} (H : n ≤ m) : n - m = 0
+:= obtain (k : ℕ) (Hk : n + k = m), from le_elim H, subst Hk (sub_add_right_eq_zero n k)
+
+theorem add_sub_le {n m : ℕ} : n ≤ m → n + (m - n) = m
+:= sub_induction n m
+    (take k,
+      assume H : 0 ≤ k,
+      calc
+        0 + (k - 0) = k - 0 : add_zero_left (k - 0)
+          ... = k : sub_zero_right k)
+    (take k, assume H : succ k ≤ 0, absurd H (not_succ_zero_le k))
+    (take k l,
+      assume IH : k ≤ l → k + (l - k) = l,
+      take H : succ k ≤ succ l,
+      calc
+        succ k + (succ l - succ k) = succ k + (l - k) : {sub_succ_succ l k}
+          ... = succ (k + (l - k)) : add_succ_left k (l - k)
+          ... = succ l : {IH (succ_le_cancel H)})
+
+theorem add_sub_ge_left {n m : ℕ} : n ≥ m → n - m + m = n
+:= subst (add_comm m (n - m)) add_sub_le
+
+theorem add_sub_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n
+:= calc
+    n + (m - n) = n + 0 : {le_imp_sub_eq_zero H}
+      ... = n : add_zero_right n
+
+theorem add_sub_le_left {n m : ℕ} : n ≤ m → n - m + m = m
+:= subst (add_comm m (n - m)) add_sub_ge
+
+theorem le_add_sub_left (n m : ℕ) : n ≤ n + (m - n)
+:= or_elim (le_total n m)
+    (assume H : n ≤ m, subst (symm (add_sub_le H)) H)
+    (assume H : m ≤ n, subst (symm (add_sub_ge H)) (le_refl n))
+
+theorem le_add_sub_right (n m : ℕ) : m ≤ n + (m - n)
+:= or_elim (le_total n m)
+    (assume H : n ≤ m, subst (symm (add_sub_le H)) (le_refl m))
+    (assume H : m ≤ n, subst (symm (add_sub_ge H)) H)
+
+theorem sub_split {P : ℕ → Prop} {n m : ℕ} (H1 : n ≤ m → P 0) (H2 : ∀k, m + k = n -> P k)
+    : P (n - m)
+:= or_elim (le_total n m)
+    (assume H3 : n ≤ m, subst (symm (le_imp_sub_eq_zero H3)) (H1 H3))
+    (assume H3 : m ≤ n, H2 (n - m) (add_sub_le H3))
+
+theorem sub_le_self (n m : ℕ) : n - m ≤ n
+:=
+  sub_split
+    (assume H : n ≤ m, zero_le n)
+    (take k : ℕ, assume H : m + k = n, le_intro (subst (add_comm m k) H))
+
+theorem le_elim_sub (n m : ℕ) (H : n ≤ m) : ∃k, m - k = n
+:=
+  obtain (k : ℕ) (Hk : n + k = m), from le_elim H,
+  exists_intro k
+    (calc
+      m - k = n + k - k : {symm Hk}
+        ... = n : sub_add_left n k)
+
+theorem add_sub_assoc {m k : ℕ} (H : k ≤ m) (n : ℕ) : n + m - k = n + (m - k)
+:= have l1 : k ≤ m → n + m - k = n + (m - k), from
+    sub_induction k m
+      (take m : ℕ,
+        assume H : 0 ≤ m,
+        calc
+          n + m - 0 = n + m : sub_zero_right (n + m)
+            ... = n + (m - 0) : {symm (sub_zero_right m)})
+      (take k : ℕ, assume H : succ k ≤ 0, absurd H (not_succ_zero_le k))
+      (take k m,
+        assume IH : k ≤ m → n + m - k = n + (m - k),
+        take H : succ k ≤ succ m,
+        calc
+          n + succ m - succ k = succ (n + m) - succ k : {add_succ_right n m}
+            ... = n + m - k : sub_succ_succ (n + m) k
+            ... = n + (m - k) : IH (succ_le_cancel H)
+            ... = n + (succ m - succ k) : {symm (sub_succ_succ m k)}),
+  l1 H
+
+theorem sub_eq_zero_imp_le {n m : ℕ} : n - m = 0 → n ≤ m
+:= sub_split
+    (assume H1 : n ≤ m, assume H2 : 0 = 0, H1)
+    (take k : ℕ,
+      assume H1 : m + k = n,
+      assume H2 : k = 0,
+      have H3 : n = m, from subst (add_zero_right m) (subst H2 (symm H1)),
+      subst H3 (le_refl n))
+
+theorem sub_sub_split {P : ℕ → ℕ → Prop} {n m : ℕ} (H1 : ∀k, n = m + k -> P k 0)
+    (H2 : ∀k, m = n + k → P 0 k) : P (n - m) (m - n)
+:= or_elim (le_total n m)
+    (assume H3 : n ≤ m,
+      le_imp_sub_eq_zero H3⁻¹ ▸  (H2 (m - n) (add_sub_le H3⁻¹)))
+    (assume H3 : m ≤ n,
+      le_imp_sub_eq_zero H3⁻¹ ▸ (H1 (n - m) (add_sub_le H3⁻¹)))
+
+theorem sub_intro {n m k : ℕ} (H : n + m = k) : k - n = m
+:= have H2 : k - n + n = m + n, from
+    calc
+      k - n + n = k : add_sub_ge_left (le_intro H)
+        ... = n + m : symm H
+        ... = m + n : add_comm n m,
+   add_cancel_right H2
+
+theorem sub_lt {x y : ℕ} (xpos : x > 0) (ypos : y > 0) : x - y < x
+:= obtain (x' : ℕ) (xeq : x = succ x'), from pos_imp_eq_succ xpos,
+   obtain (y' : ℕ) (yeq : y = succ y'), from pos_imp_eq_succ ypos,
+   have xsuby_eq : x - y = x' - y', from
+     calc
+       x - y = succ x' - y : {xeq}
+         ... = succ x' - succ y' : {yeq}
+         ... = x' - y' : sub_succ_succ _ _,
+   have H1 : x' - y' ≤ x', from sub_le_self _ _,
+   have H2 : x' < succ x', from self_lt_succ _,
+   show x - y < x, from xeq⁻¹ ▸ xsuby_eq⁻¹ ▸ le_lt_trans H1 H2
+
+-- Max, min, iteration, and absolute difference
+-- --------------------------------------------
+
+definition max (n m : ℕ) : ℕ := n + (m - n)
+definition min (n m : ℕ) : ℕ := m - (m - n)
+
+theorem max_le {n m : ℕ} (H : n ≤ m) : n + (m - n) = m := add_sub_le H
+
+theorem max_ge {n m : ℕ} (H : n ≥ m) : n + (m - n) = n := add_sub_ge H
+
+theorem left_le_max (n m : ℕ) : n ≤ n + (m - n) := le_add_sub_left n m
+
+theorem right_le_max (n m : ℕ) : m ≤ max n m := le_add_sub_right n m
+
+-- ### absolute difference
+
+-- This section is still incomplete
+
+definition dist (n m : ℕ) := (n - m) + (m - n)
+
+theorem dist_comm (n m : ℕ) : dist n m = dist m n
+:= add_comm (n - m) (m - n)
+
+theorem dist_eq_zero {n m : ℕ} (H : dist n m = 0) : n = m
+:=
+  have H2 : n - m = 0, from add_eq_zero_left H,
+  have H3 : n ≤ m, from sub_eq_zero_imp_le H2,
+  have H4 : m - n = 0, from add_eq_zero_right H,
+  have H5 : m ≤ n, from sub_eq_zero_imp_le H4,
+  le_antisym H3 H5
+
+theorem dist_le {n m : ℕ} (H : n ≤ m) : dist n m = m - n
+:= calc
+    dist n m = (n - m) + (m - n) : refl _
+         ... = 0       + (m - n) : {le_imp_sub_eq_zero H}
+         ... = m - n             : add_zero_left (m - n)
+
+theorem dist_ge {n m : ℕ} (H : n ≥ m) : dist n m = n - m
+:= subst (dist_comm m n) (dist_le H)
+
+theorem dist_zero_right (n : ℕ) : dist n 0 = n
+:= trans (dist_ge (zero_le n)) (sub_zero_right n)
+
+theorem dist_zero_left (n : ℕ) : dist 0 n = n
+:= trans (dist_le (zero_le n)) (sub_zero_right n)
+
+theorem dist_intro {n m k : ℕ} (H : n + m = k) : dist k n = m
+:= calc
+    dist k n = k - n : dist_ge (le_intro H)
+      ... = m : sub_intro H
+
+theorem dist_add_right (n k m : ℕ) : dist (n + k) (m + k) = dist n m
+:=
+  calc
+    dist (n + k) (m + k) = ((n+k) - (m+k)) + ((m+k)-(n+k)) : refl _
+                     ... = (n - m) + ((m + k) - (n + k))   : {sub_add_add_right _ _ _}
+                     ... = (n - m) + (m - n)               : {sub_add_add_right _ _ _}
+
+theorem dist_add_left (k n m : ℕ) : dist (k + n) (k + m) = dist n m
+:= subst (add_comm m k) (subst (add_comm n k) (dist_add_right n k m))
+
+theorem dist_ge_add_right {n m : ℕ} (H : n ≥ m) : dist n m + m = n
+:= calc
+    dist n m + m = n - m + m : {dist_ge H}
+      ... = n : add_sub_ge_left H
+
+theorem dist_eq_intro {n m k l : ℕ} (H : n + m = k + l) : dist n k = dist l m
+:= calc
+    dist n k = dist (n + m) (k + m) : symm (dist_add_right n m k)
+      ... = dist (k + l) (k + m) : {H}
+      ... = dist l m : dist_add_left k l m
+
+end nat