refactor(library/data/nat/basic): mark some theorems as protected to avoid overloading

2015-10-14 12:27:09 -07:00 · 2015-10-14 12:27:09 -07:00 · 6df31d3406
commit 6df31d3406
parent 0aaa8a9770
68 changed files with 195 additions and 262 deletions
--- a/doc/lean/calc.org
+++ b/doc/lean/calc.org
@ -25,7 +25,7 @@ Here is an example

 #+BEGIN_SRC lean
  import data.nat
-  open nat
+  open nat algebra
  constants a b c d e : nat.
  axiom Ax1 : a = b.
  axiom Ax2 : b = c + 1.
@ -53,7 +53,7 @@ some form of transitivity. It can even combine different relations.

 #+BEGIN_SRC lean
  import data.nat
-  open nat
+  open nat algebra

  theorem T2 (a b c : nat) (H1 : a = b) (H2 : b = c + 1) : a ≠ 0
  := calc  a = b      : H1
--- a/doc/lean/library_style.org
+++ b/doc/lean/library_style.org
@ -242,7 +242,7 @@ an additional indent for every line after the first, as in the following
 example:
 #+BEGIN_SRC lean
 import data.nat
-open nat eq
+open nat eq algebra
 theorem add_right_inj {n m k : nat} : n + m = n + k → m = k :=
 nat.induction_on n
  (take H : 0 + m = 0 + k,
--- a/doc/lean/tutorial.org
+++ b/doc/lean/tutorial.org
@ -752,7 +752,7 @@ of two even numbers is an even number.

 #+BEGIN_SRC lean
  import data.nat
-  open nat
+  open nat algebra

  namespace hide
  definition even (a : nat) := ∃ b, a = 2*b
@ -762,7 +762,7 @@ of two even numbers is an even number.
    exists.intro (w1 + w2)
      (calc a + b  =  2*w1 + b      : {Hw1}
              ...  =  2*w1 + 2*w2   : {Hw2}
-              ...  =  2*(w1 + w2)   : eq.symm !mul.left_distrib)))
+              ...  =  2*(w1 + w2)   : eq.symm !left_distrib)))
  end hide
 #+END_SRC

@ -774,7 +774,7 @@ With this macro we can write the example above in a more natural way

 #+BEGIN_SRC lean
  import data.nat
-  open nat
+  open nat algebra

  namespace hide
  definition even (a : nat) := ∃ b, a = 2*b
@ -784,6 +784,6 @@ With this macro we can write the example above in a more natural way
    exists.intro (w1 + w2)
      (calc a + b  =  2*w1 + b      : {Hw1}
              ...  =  2*w1 + 2*w2   : {Hw2}
-              ...  =  2*(w1 + w2)   : eq.symm !mul.left_distrib)
+              ...  =  2*(w1 + w2)   : eq.symm !left_distrib)
  end hide
 #+END_SRC
--- a/library/algebra/group.lean
+++ b/library/algebra/group.lean
@ -131,10 +131,14 @@ definition add_comm_monoid.to_comm_monoid {A : Type} [s : add_comm_monoid A] : c
 ⦄

 section add_comm_monoid
+  variables [s : add_comm_monoid A]
+  include s

-  theorem add_comm_three {A : Type} [s : add_comm_monoid A] (a b c : A) : a + b + c = c + b + a :=
+  theorem add_comm_three  (a b c : A) : a + b + c = c + b + a :=
    by rewrite [{a + _}add.comm, {_ + c}add.comm, -*add.assoc]

+  theorem add.comm4 : ∀ (n m k l : A), n + m + (k + l) = n + k + (m + l) :=
+  comm4 add.comm add.assoc
 end add_comm_monoid

 /- group -/
--- a/library/algebra/group_power.lean
+++ b/library/algebra/group_power.lean
@ -81,7 +81,7 @@ theorem pow_mul (a : A) (m : ℕ) : ∀ n, a^(m * n) = (a^m)^n
 | (succ n) := by rewrite [nat.mul_succ, pow_add, pow_succ', pow_mul]

 theorem pow_comm (a : A) (m n : ℕ)  : a^m * a^n = a^n * a^m :=
-by rewrite [-*pow_add, nat.add.comm]
+by rewrite [-*pow_add, add.comm]

 end monoid

@ -219,7 +219,7 @@ theorem one_nmul (a : A) : 1 ⬝ a = a := pow_one a

 theorem add_nmul (m n : ℕ) (a : A) : (m + n) ⬝ a = (m ⬝ a) + (n ⬝ a) := pow_add a m n

-theorem mul_nmul (m n : ℕ) (a : A) : (m * n) ⬝ a = m ⬝ (n ⬝ a) := eq.subst (nat.mul.comm n m) (pow_mul a n m)
+theorem mul_nmul (m n : ℕ) (a : A) : (m * n) ⬝ a = m ⬝ (n ⬝ a) := eq.subst (mul.comm n m) (pow_mul a n m)

 theorem nmul_comm (m n : ℕ) (a : A) : (m ⬝ a) + (n ⬝ a) = (n ⬝ a) + (m ⬝ a) := pow_comm a m n

--- a/library/data/encodable.lean
+++ b/library/data/encodable.lean
@ -7,7 +7,7 @@ Type class for encodable types.
 Note that every encodable type is countable.
 -/
 import data.fintype data.list data.list.sort data.sum data.nat.div data.countable data.equiv data.finset
-open option list nat function
+open option list nat function algebra

 structure encodable [class] (A : Type) :=
 (encode : A → nat) (decode : nat → option A) (encodek : ∀ a, decode (encode a) = some a)
--- a/library/data/examples/vector.lean
+++ b/library/data/examples/vector.lean
@ -7,7 +7,7 @@ This file demonstrates how to encode vectors using indexed inductive families.
 In standard library we do not use this approach.
 -/
 import data.nat data.list data.fin
-open nat prod fin
+open nat prod fin algebra

 inductive vector (A : Type) : nat → Type :=
 | nil {} : vector A zero
--- a/library/data/fin.lean
+++ b/library/data/fin.lean
@ -245,7 +245,7 @@ lemma madd_comm (i j : fin (succ n)) : madd i j = madd j i :=
 by apply eq_of_veq; rewrite [*val_madd, add.comm (val i)]

 lemma zero_madd (i : fin (succ n)) : madd 0 i = i :=
-have H : madd (fin.zero n) i = i, 
+have H : madd (fin.zero n) i = i,
  by apply eq_of_veq; rewrite [val_madd, ↑fin.zero, nat.zero_add, mod_eq_of_lt (is_lt i)],
 H

@ -446,7 +446,7 @@ assert aux₁ : ∀ {v₁ v₂}, v₁ < n → v₂ < m → v₁ + v₂ * n < n*m
  take v₁ v₂, assume h₁ h₂,
    have   nat.succ v₂ ≤ m, from succ_le_of_lt h₂,
    assert nat.succ v₂ * n ≤ m * n,       from mul_le_mul_right _ this,
-    have   v₂ * n + n ≤ n * m,            by rewrite [-add_one at this, mul.right_distrib at this, one_mul at this, mul.comm m n at this]; exact this,
+    have   v₂ * n + n ≤ n * m,            by rewrite [-add_one at this, right_distrib at this, one_mul at this, mul.comm m n at this]; exact this,
    assert v₁ + (v₂ * n + n) < n + n * m, from add_lt_add_of_lt_of_le h₁ this,
    have   v₁ + v₂ * n + n < n * m + n,   by rewrite [add.assoc, add.comm (n*m) n]; exact this,
    lt_of_add_lt_add_right this,
--- a/library/data/finset/card.lean
+++ b/library/data/finset/card.lean
@ -191,7 +191,7 @@ begin
  induction s with a s' H IH,
    rewrite [Sum_empty, card_empty, zero_mul],
    rewrite [Sum_insert_of_not_mem _ H, IH, card_insert_of_not_mem H, add.comm,
-             mul.right_distrib, one_mul]
+             right_distrib, one_mul]
 end

 theorem Sum_one_eq_card (s : finset A) : (∑ x ∈ s, (1 : nat)) = card s :=
--- a/library/data/fintype/function.lean
+++ b/library/data/fintype/function.lean
@ -94,7 +94,7 @@ lemma length_all_lists : ∀ {n : nat}, length (@all_lists_of_len A _ n) = (card
 | 0         := calc length [[]] = 1 : length_cons
 | (succ n)  := calc length _ = card A * length (all_lists_of_len n) : length_cons_all
                         ... = card A * (card A ^ n) : length_all_lists
-                         ... = (card A ^ n) * card A : nat.mul.comm
+                         ... = (card A ^ n) * card A : mul.comm
                         ... = (card A) ^ (succ n) : pow_succ'


--- a/library/data/int/basic.lean
+++ b/library/data/int/basic.lean
@ -370,16 +370,16 @@ calc
    ... = nat_abs a : abstr_repr

 theorem padd_pneg (p : ℕ × ℕ) : padd p (pneg p) ≡ (0, 0) :=
-show pr1 p + pr2 p + 0 = pr2 p + pr1 p + 0, from !nat.add.comm ▸ rfl
+show pr1 p + pr2 p + 0 = pr2 p + pr1 p + 0, from !nat.add_comm ▸ rfl

 theorem padd_padd_pneg (p q : ℕ × ℕ) : padd (padd p q) (pneg q) ≡ p :=
 calc      pr1 p + pr1 q + pr2 q + pr2 p
-        = pr1 p + (pr1 q + pr2 q) + pr2 p : nat.add.assoc
-    ... = pr1 p + (pr1 q + pr2 q + pr2 p) : nat.add.assoc
-    ... = pr1 p + (pr2 q + pr1 q + pr2 p) : nat.add.comm
-    ... = pr1 p + (pr2 q + pr2 p + pr1 q) : by rewrite add.right_comm
-    ... = pr1 p + (pr2 p + pr2 q + pr1 q) : nat.add.comm
-    ... = pr2 p + pr2 q + pr1 q + pr1 p   : nat.add.comm
+        = pr1 p + (pr1 q + pr2 q) + pr2 p : algebra.add.assoc
+    ... = pr1 p + (pr1 q + pr2 q + pr2 p) : algebra.add.assoc
+    ... = pr1 p + (pr2 q + pr1 q + pr2 p) : algebra.add.comm
+    ... = pr1 p + (pr2 q + pr2 p + pr1 q) : algebra.add.right_comm
+    ... = pr1 p + (pr2 p + pr2 q + pr1 q) : algebra.add.comm
+    ... = pr2 p + pr2 q + pr1 q + pr1 p   : algebra.add.comm

 theorem add.left_inv (a : ℤ) : -a + a = 0 :=
 have H : repr (-a + a) ≡ repr 0, from
@ -450,21 +450,25 @@ theorem equiv_mul_prep {xa ya xb yb xn yn xm ym : ℕ}
 nat.add.cancel_right (calc
            xa*xn+ya*yn + (xb*ym+yb*xm) + (yb*xn+xb*yn + (xb*xn+yb*yn))
          = xa*xn+ya*yn + (yb*xn+xb*yn) + (xb*ym+yb*xm + (xb*xn+yb*yn)) : by rewrite add.comm4
-      ... = xa*xn+ya*yn + (yb*xn+xb*yn) + (xb*xn+yb*yn + (xb*ym+yb*xm)) : by rewrite {xb*ym+yb*xm +_}nat.add.comm
-      ... = xa*xn+yb*xn + (ya*yn+xb*yn) + (xb*xn+xb*ym + (yb*yn+yb*xm)) : by exact !congr_arg2 add.comm4 add.comm4
-      ... = ya*xn+xb*xn + (xa*yn+yb*yn) + (xb*yn+xb*xm + (yb*xn+yb*ym)) : by rewrite[-+mul.left_distrib,-+mul.right_distrib]; exact H1 ▸ H2 ▸ rfl
-      ... = ya*xn+xa*yn + (xb*xn+yb*yn) + (xb*yn+yb*xn + (xb*xm+yb*ym)) : by exact !congr_arg2 add.comm4 add.comm4
-      ... = xa*yn+ya*xn + (xb*xn+yb*yn) + (xb*yn+yb*xn + (xb*xm+yb*ym)) : by rewrite {xa*yn + _}nat.add.comm
-      ... = xa*yn+ya*xn + (xb*xn+yb*yn) + (yb*xn+xb*yn + (xb*xm+yb*ym)) : by rewrite {xb*yn + _}nat.add.comm
-      ... = xa*yn+ya*xn + (yb*xn+xb*yn) + (xb*xn+yb*yn + (xb*xm+yb*ym)) : by rewrite add.comm4
-      ... = xa*yn+ya*xn + (yb*xn+xb*yn) + (xb*xm+yb*ym + (xb*xn+yb*yn)) : by rewrite {xb*xn+yb*yn + _}nat.add.comm
+      ... = xa*xn+ya*yn + (yb*xn+xb*yn) + (xb*xn+yb*yn + (xb*ym+yb*xm)) : by rewrite {xb*ym+yb*xm +_}nat.add_comm
+      ... = xa*xn+yb*xn + (ya*yn+xb*yn) + (xb*xn+xb*ym + (yb*yn+yb*xm)) : by exact !congr_arg2 !add.comm4 !add.comm4
+      ... = ya*xn+xb*xn + (xa*yn+yb*yn) + (xb*yn+xb*xm + (yb*xn+yb*ym)) : by rewrite[-+left_distrib,-+right_distrib]; exact H1 ▸ H2 ▸ rfl
+      ... = ya*xn+xa*yn + (xb*xn+yb*yn) + (xb*yn+yb*xn + (xb*xm+yb*ym)) : by exact !congr_arg2 !add.comm4 !add.comm4
+      ... = xa*yn+ya*xn + (xb*xn+yb*yn) + (xb*yn+yb*xn + (xb*xm+yb*ym)) : by rewrite {xa*yn + _}nat.add_comm
+      ... = xa*yn+ya*xn + (xb*xn+yb*yn) + (yb*xn+xb*yn + (xb*xm+yb*ym)) : by rewrite {xb*yn + _}nat.add_comm
+      ... = xa*yn+ya*xn + (yb*xn+xb*yn) + (xb*xn+yb*yn + (xb*xm+yb*ym)) : by rewrite (!add.comm4)
+      ... = xa*yn+ya*xn + (yb*xn+xb*yn) + (xb*xm+yb*ym + (xb*xn+yb*yn)) : by rewrite {xb*xn+yb*yn + _}nat.add_comm
      ... = xa*yn+ya*xn + (xb*xm+yb*ym) + (yb*xn+xb*yn + (xb*xn+yb*yn)) : by rewrite add.comm4)

 theorem pmul_congr {p p' q q' : ℕ × ℕ} : p ≡ p' → q ≡ q' → pmul p q ≡ pmul p' q' := equiv_mul_prep

 theorem pmul_comm (p q : ℕ × ℕ) : pmul p q = pmul q p :=
-show (_,_) = (_,_), from !congr_arg2
-   (!congr_arg2 !mul.comm !mul.comm) (!nat.add.comm ⬝ (!congr_arg2 !mul.comm !mul.comm))
+show (_,_) = (_,_),
+begin
+  congruence,
+    { congruence, repeat rewrite mul.comm },
+    { rewrite algebra.add.comm, congruence, repeat rewrite mul.comm }
+end

 theorem mul.comm (a b : ℤ) : a * b = b * a :=
 eq_of_repr_equiv_repr
@ -476,9 +480,11 @@ eq_of_repr_equiv_repr
 private theorem pmul_assoc_prep {p1 p2 q1 q2 r1 r2 : ℕ} :
  ((p1*q1+p2*q2)*r1+(p1*q2+p2*q1)*r2, (p1*q1+p2*q2)*r2+(p1*q2+p2*q1)*r1) =
   (p1*(q1*r1+q2*r2)+p2*(q1*r2+q2*r1), p1*(q1*r2+q2*r1)+p2*(q1*r1+q2*r2)) :=
-by rewrite[+mul.left_distrib,+mul.right_distrib,*mul.assoc];
-   exact (congr_arg2 pair (!add.comm4 ⬝ (!congr_arg !nat.add.comm))
-                          (!add.comm4 ⬝ (!congr_arg !nat.add.comm)))
+begin
+  rewrite[+left_distrib,+right_distrib,*algebra.mul.assoc],
+  exact (congr_arg2 pair (!add.comm4 ⬝ (!congr_arg !nat.add_comm))
+                         (!add.comm4 ⬝ (!congr_arg !nat.add_comm)))
+end

 theorem pmul_assoc (p q r: ℕ × ℕ) : pmul (pmul p q) r = pmul p (pmul q r) := pmul_assoc_prep

@ -500,19 +506,23 @@ theorem one_mul (a : ℤ) : 1 * a = a :=
 mul.comm a 1 ▸ mul_one a

 private theorem mul_distrib_prep {a1 a2 b1 b2 c1 c2 : ℕ} :
- ((a1+b1)*c1+(a2+b2)*c2, (a1+b1)*c2+(a2+b2)*c1) =
+ ((a1+b1)*c1+(a2+b2)*c2,     (a1+b1)*c2+(a2+b2)*c1) =
 (a1*c1+a2*c2+(b1*c1+b2*c2), a1*c2+a2*c1+(b1*c2+b2*c1)) :=
-by rewrite[+mul.right_distrib] ⬝ (!congr_arg2 !add.comm4 !add.comm4)
+begin
+  rewrite +right_distrib, congruence,
+    {rewrite add.comm4},
+    {rewrite add.comm4}
+end

 theorem mul.right_distrib (a b c : ℤ) : (a + b) * c = a * c + b * c :=
 eq_of_repr_equiv_repr
  (calc
    repr ((a + b) * c) = pmul (repr (a + b)) (repr c) : repr_mul
-      ... ≡ pmul (padd (repr a) (repr b)) (repr c) : pmul_congr !repr_add equiv.refl
+      ... ≡ pmul (padd (repr a) (repr b)) (repr c)    : pmul_congr !repr_add equiv.refl
      ... = padd (pmul (repr a) (repr c)) (pmul (repr b) (repr c)) : mul_distrib_prep
-      ... = padd (repr (a * c)) (pmul (repr b) (repr c)) : repr_mul
-      ... = padd (repr (a * c)) (repr (b * c)) : repr_mul
-      ... ≡ repr (a * c + b * c) : repr_add)
+      ... = padd (repr (a * c)) (pmul (repr b) (repr c))           : repr_mul
+      ... = padd (repr (a * c)) (repr (b * c))                     : repr_mul
+      ... ≡ repr (a * c + b * c)                                   : repr_add)

 theorem mul.left_distrib (a b c : ℤ) : a * (b + c) = a * b + a * c :=
 calc
--- a/library/data/int/div.lean
+++ b/library/data/int/div.lean
@ -162,10 +162,10 @@ or.elim (nat.lt_or_ge m (n * k))
    have (-[1+m] + n * k) div k = -[1+m] div k + n, from calc
      (-[1+m] + n * k) div k
            = of_nat ((k * n - (m + 1)) div k) :
-                by rewrite [add.comm, neg_succ_of_nat_eq, of_nat_div, nat.mul.comm k n,
+                by rewrite [add.comm, neg_succ_of_nat_eq, of_nat_div, algebra.mul.comm k n,
                            of_nat_sub H3]
        ... = of_nat (n - m div k - 1)         :
-                nat.mul_sub_div_of_lt (!nat.mul.comm ▸ m_lt_nk)
+                nat.mul_sub_div_of_lt (!nat.mul_comm ▸ m_lt_nk)
        ... = -[1+m] div k + n :
                by rewrite [nat.sub_sub, of_nat_sub H4, add.comm, sub_eq_add_neg,
                            !neg_succ_of_nat_div (of_nat_lt_of_nat_of_lt H2),
--- a/library/data/list/comb.lean
+++ b/library/data/list/comb.lean
@ -6,7 +6,7 @@ Authors: Leonardo de Moura, Haitao Zhang
 List combinators.
 -/
 import data.list.basic data.equiv
-open nat prod decidable function helper_tactics
+open nat prod decidable function helper_tactics algebra

 namespace list
 variables {A B C : Type}
@ -368,7 +368,7 @@ theorem length_product : ∀ (l₁ : list A) (l₂ : list B), length (product l
 | (x::l₁) l₂ :=
  assert length (product l₁ l₂) = length l₁ * length l₂, from length_product l₁ l₂,
  by rewrite [product_cons, length_append, length_cons,
-              length_map, this, mul.right_distrib, one_mul, add.comm]
+              length_map, this, right_distrib, one_mul, add.comm]
 end product

 -- new for list/comb dependent map theory
--- a/library/data/nat/basic.lean
+++ b/library/data/nat/basic.lean
@ -102,15 +102,15 @@ general m

 /- addition -/

-theorem add_zero [simp] (n : ℕ) : n + 0 = n :=
+protected theorem add_zero [simp] (n : ℕ) : n + 0 = n :=
 rfl

 theorem add_succ [simp] (n m : ℕ) : n + succ m = succ (n + m) :=
 rfl

-theorem zero_add [simp] (n : ℕ) : 0 + n = n :=
+protected theorem zero_add [simp] (n : ℕ) : 0 + n = n :=
 nat.induction_on n
-    !add_zero
+    !nat.add_zero
    (take m IH, show 0 + succ m = succ m, from
      calc
        0 + succ m = succ (0 + m) : add_succ
@ -118,15 +118,15 @@ nat.induction_on n

 theorem succ_add [simp] (n m : ℕ) : (succ n) + m = succ (n + m) :=
 nat.induction_on m
-    (!add_zero ▸ !add_zero)
+    (!nat.add_zero ▸ !nat.add_zero)
    (take k IH, calc
      succ n + succ k = succ (succ n + k)    : add_succ
                  ... = succ (succ (n + k))  : IH
                  ... = succ (n + succ k)    : add_succ)

-theorem add.comm [simp] (n m : ℕ) : n + m = m + n :=
+protected theorem add_comm [simp] (n m : ℕ) : n + m = m + n :=
 nat.induction_on m
-    (by rewrite [add_zero, zero_add])
+    (by rewrite [nat.add_zero, nat.zero_add])
    (take k IH, calc
        n + succ k = succ (n+k)   : add_succ
               ... = succ (k + n) : IH
@ -135,9 +135,9 @@ nat.induction_on m
 theorem succ_add_eq_succ_add (n m : ℕ) : succ n + m = n + succ m :=
 !succ_add ⬝ !add_succ⁻¹

-theorem add.assoc [simp] (n m k : ℕ) : (n + m) + k = n + (m + k) :=
+protected theorem add_assoc [simp] (n m k : ℕ) : (n + m) + k = n + (m + k) :=
 nat.induction_on k
-    (by rewrite +add_zero)
+    (by rewrite +nat.add_zero)
    (take l IH,
      calc
        (n + m) + succ l = succ ((n + m) + l)  : add_succ
@ -145,19 +145,16 @@ nat.induction_on k
                     ... = n + succ (m + l)    : add_succ
                     ... = n + (m + succ l)    : add_succ)

-theorem add.left_comm [simp] : Π (n m k : ℕ), n + (m + k) = m + (n + k) :=
-left_comm add.comm add.assoc
+protected theorem add.left_comm : Π (n m k : ℕ), n + (m + k) = m + (n + k) :=
+left_comm nat.add_comm nat.add_assoc

-theorem add.right_comm : Π (n m k : ℕ), n + m + k = n + k + m :=
-right_comm add.comm add.assoc
-
-theorem add.comm4 : Π {n m k l : ℕ}, n + m + (k + l) = n + k + (m + l) :=
-comm4 add.comm add.assoc
+protected theorem add.right_comm : Π (n m k : ℕ), n + m + k = n + k + m :=
+right_comm nat.add_comm nat.add_assoc

 theorem add.cancel_left {n m k : ℕ} : n + m = n + k → m = k :=
 nat.induction_on n
  (take H : 0 + m = 0 + k,
-    !zero_add⁻¹ ⬝ H ⬝ !zero_add)
+    !nat.zero_add⁻¹ ⬝ H ⬝ !nat.zero_add)
  (take (n : ℕ) (IH : n + m = n + k → m = k) (H : succ n + m = succ n + k),
    have succ (n + m) = succ (n + k),
    from calc
@ -168,7 +165,7 @@ nat.induction_on n
    IH this)

 theorem add.cancel_right {n m k : ℕ} (H : n + m = k + m) : n = k :=
-have H2 : m + n = m + k, from !add.comm ⬝ H ⬝ !add.comm,
+have H2 : m + n = m + k, from !nat.add_comm ⬝ H ⬝ !nat.add_comm,
  add.cancel_left H2

 theorem eq_zero_of_add_eq_zero_right {n m : ℕ} : n + m = 0 → n = 0 :=
@ -183,20 +180,20 @@ nat.induction_on n
      !succ_ne_zero)

 theorem eq_zero_of_add_eq_zero_left {n m : ℕ} (H : n + m = 0) : m = 0 :=
-eq_zero_of_add_eq_zero_right (!add.comm ⬝ H)
+eq_zero_of_add_eq_zero_right (!nat.add_comm ⬝ H)

 theorem eq_zero_and_eq_zero_of_add_eq_zero {n m : ℕ} (H : n + m = 0) : n = 0 ∧ m = 0 :=
 and.intro (eq_zero_of_add_eq_zero_right H) (eq_zero_of_add_eq_zero_left H)

 theorem add_one [simp] (n : ℕ) : n + 1 = succ n :=
-!add_zero ▸ !add_succ
+!nat.add_zero ▸ !add_succ

 theorem one_add (n : ℕ) : 1 + n = succ n :=
-!zero_add ▸ !succ_add
+!nat.zero_add ▸ !succ_add

 /- multiplication -/

-theorem mul_zero [simp] (n : ℕ) : n * 0 = 0 :=
+protected theorem mul_zero [simp] (n : ℕ) : n * 0 = 0 :=
 rfl

 theorem mul_succ [simp] (n m : ℕ) : n * succ m = n * m + n :=
@ -204,75 +201,75 @@ rfl

 -- commutativity, distributivity, associativity, identity

-theorem zero_mul [simp] (n : ℕ) : 0 * n = 0 :=
+protected theorem zero_mul [simp] (n : ℕ) : 0 * n = 0 :=
 nat.induction_on n
-  !mul_zero
-  (take m IH, !mul_succ ⬝ !add_zero ⬝ IH)
+  !nat.mul_zero
+  (take m IH, !mul_succ ⬝ !nat.add_zero ⬝ IH)

 theorem succ_mul [simp] (n m : ℕ) : (succ n) * m = (n * m) + m :=
 nat.induction_on m
-  (by rewrite mul_zero)
+  (by rewrite nat.mul_zero)
  (take k IH, calc
    succ n * succ k = succ n * k + succ n   : mul_succ
                ... = n * k + k + succ n    : IH
-                ... = n * k + (k + succ n)  : add.assoc
-                ... = n * k + (succ n + k)  : add.comm
+                ... = n * k + (k + succ n)  : nat.add_assoc
+                ... = n * k + (succ n + k)  : nat.add_comm
                ... = n * k + (n + succ k)  : succ_add_eq_succ_add
-                ... = n * k + n + succ k    : add.assoc
+                ... = n * k + n + succ k    : nat.add_assoc
                ... = n * succ k + succ k   : mul_succ)

-theorem mul.comm [simp] (n m : ℕ) : n * m = m * n :=
+protected theorem mul_comm [simp] (n m : ℕ) : n * m = m * n :=
 nat.induction_on m
-  (!mul_zero ⬝ !zero_mul⁻¹)
+  (!nat.mul_zero ⬝ !nat.zero_mul⁻¹)
  (take k IH, calc
    n * succ k = n * k + n    : mul_succ
           ... = k * n + n    : IH
           ... = (succ k) * n : succ_mul)

-theorem mul.right_distrib (n m k : ℕ) : (n + m) * k = n * k + m * k :=
+protected theorem right_distrib (n m k : ℕ) : (n + m) * k = n * k + m * k :=
 nat.induction_on k
  (calc
-    (n + m) * 0 = 0             : mul_zero
-            ... = 0 + 0         : add_zero
-            ... = n * 0 + 0     : mul_zero
-            ... = n * 0 + m * 0 : mul_zero)
+    (n + m) * 0 = 0             : nat.mul_zero
+            ... = 0 + 0         : nat.add_zero
+            ... = n * 0 + 0     : nat.mul_zero
+            ... = n * 0 + m * 0 : nat.mul_zero)
  (take l IH, calc
    (n + m) * succ l = (n + m) * l + (n + m)    : mul_succ
                 ... = n * l + m * l + (n + m)  : IH
-                 ... = n * l + m * l + n + m    : add.assoc
-                 ... = n * l + n + m * l + m    : add.right_comm
-                 ... = n * l + n + (m * l + m)  : add.assoc
+                 ... = n * l + m * l + n + m    : nat.add_assoc
+                 ... = n * l + n + m * l + m    : nat.add.right_comm
+                 ... = n * l + n + (m * l + m)  : nat.add_assoc
                 ... = n * succ l + (m * l + m) : mul_succ
                 ... = n * succ l + m * succ l  : mul_succ)

-theorem mul.left_distrib (n m k : ℕ) : n * (m + k) = n * m + n * k :=
+protected theorem left_distrib (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.right_distrib
-          ... = n * m + k * n  : mul.comm
-          ... = n * m + n * k  : mul.comm
+  n * (m + k) = (m + k) * n    : nat.mul_comm
+          ... = m * n + k * n  : nat.right_distrib
+          ... = n * m + k * n  : nat.mul_comm
+          ... = n * m + n * k  : nat.mul_comm

-theorem mul.assoc [simp] (n m k : ℕ) : (n * m) * k = n * (m * k) :=
+protected theorem mul_assoc [simp] (n m k : ℕ) : (n * m) * k = n * (m * k) :=
 nat.induction_on k
  (calc
-    (n * m) * 0 = n * (m * 0) : mul_zero)
+    (n * m) * 0 = n * (m * 0) : nat.mul_zero)
  (take l IH,
    calc
      (n * m) * succ l = (n * m) * l + n * m : mul_succ
                   ... = n * (m * l) + n * m : IH
-                   ... = n * (m * l + m)     : mul.left_distrib
+                   ... = n * (m * l + m)     : nat.left_distrib
                   ... = n * (m * succ l)    : mul_succ)

-theorem mul_one [simp] (n : ℕ) : n * 1 = n :=
+protected theorem mul_one [simp] (n : ℕ) : n * 1 = n :=
 calc
  n * 1 = n * 0 + n : mul_succ
-    ... = 0 + n     : mul_zero
-    ... = n         : zero_add
+    ... = 0 + n     : nat.mul_zero
+    ... = n         : nat.zero_add

-theorem one_mul [simp] (n : ℕ) : 1 * n = n :=
+protected theorem one_mul [simp] (n : ℕ) : 1 * n = n :=
 calc
-  1 * n = n * 1 : mul.comm
-    ... = n     : mul_one
+  1 * n = n * 1 : nat.mul_comm
+    ... = n     : nat.mul_one

 theorem eq_zero_or_eq_zero_of_mul_eq_zero {n m : ℕ} : n * m = 0 → n = 0 ∨ m = 0 :=
 nat.cases_on n
@ -293,22 +290,21 @@ open algebra
 protected definition comm_semiring [reducible] [trans_instance] : algebra.comm_semiring nat :=
 ⦃algebra.comm_semiring,
 add            := nat.add,
-
- add_assoc      := add.assoc,
+ add_assoc      := nat.add_assoc,
 zero           := nat.zero,
- zero_add       := zero_add,
- add_zero       := add_zero,
- add_comm       := add.comm,
+ zero_add       := nat.zero_add,
+ add_zero       := nat.add_zero,
+ add_comm       := nat.add_comm,
 mul            := nat.mul,
- mul_assoc      := mul.assoc,
+ mul_assoc      := nat.mul_assoc,
 one            := nat.succ nat.zero,
- one_mul        := one_mul,
- mul_one        := mul_one,
- left_distrib   := mul.left_distrib,
- right_distrib  := mul.right_distrib,
- zero_mul       := zero_mul,
- mul_zero       := mul_zero,
- mul_comm       := mul.comm⦄
+ one_mul        := nat.one_mul,
+ mul_one        := nat.mul_one,
+ left_distrib   := nat.left_distrib,
+ right_distrib  := nat.right_distrib,
+ zero_mul       := nat.zero_mul,
+ mul_zero       := nat.mul_zero,
+ mul_comm       := nat.mul_comm⦄
 end nat

 section
--- a/library/data/nat/div.lean
+++ b/library/data/nat/div.lean
@ -263,7 +263,7 @@ else
      (calc
        ((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : eq_div_mul_add_mod
          ... = z * (x div y * y + x mod y)                           : eq_div_mul_add_mod
-          ... = z * (x div y * y) + z * (x mod y)                     : mul.left_distrib
+          ... = z * (x div y * y) + z * (x mod y)                     : left_distrib
          ... = (x div y) * (z * y) + z * (x mod y)                   : mul.left_comm)

 theorem mul_div_mul_right {x z y : ℕ} (zpos : z > 0) : (x * z) div (y * z) = x div y :=
@ -286,7 +286,7 @@ or.elim (eq_zero_or_pos z)
          (calc
            ((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : eq_div_mul_add_mod
              ... = z * (x div y * y + x mod y)                           : eq_div_mul_add_mod
-              ... = z * (x div y * y) + z * (x mod y)                     : mul.left_distrib
+              ... = z * (x div y * y) + z * (x mod y)                     : left_distrib
              ... = (x div y) * (z * y) + z * (x mod y)                   : mul.left_comm)))

 theorem mul_mod_mul_right (x z y : ℕ) : (x * z) mod (y * z) = (x mod y) * z :=
@ -300,7 +300,7 @@ theorem mul_mod_eq_mod_mul_mod (m n k : nat) : (m * n) mod k = ((m mod k) * n) m
 calc
  (m * n) mod k = (((m div k) * k + m mod k) * n) mod k : eq_div_mul_add_mod
            ... = ((m mod k) * n) mod k                 :
-                    by rewrite [mul.right_distrib, mul.right_comm, add.comm, add_mul_mod_self]
+                    by rewrite [right_distrib, mul.right_comm, add.comm, add_mul_mod_self]

 theorem mul_mod_eq_mul_mod_mod (m n k : nat) : (m * n) mod k = (m * (n mod k)) mod k :=
 !mul.comm ▸ !mul.comm ▸ !mul_mod_eq_mod_mul_mod
@ -558,7 +558,7 @@ calc
     ... = ((n - k div m) * m - (k mod m + 1)) div m                    :
               by rewrite [mul.comm m, mul_sub_right_distrib]
     ... = ((n - k div m - 1) * m + m - (k mod m + 1)) div m            :
-               by rewrite [H3 at {1}, mul.right_distrib, nat.one_mul]
+               by rewrite [H3 at {1}, right_distrib, nat.one_mul]
     ... = ((n - k div m - 1) * m + (m - (k mod m + 1))) div m          : {add_sub_assoc H5 _}
     ... = (m - (k mod m + 1)) div m + (n - k div m - 1)                :
               by rewrite [add.comm, (add_mul_div_self H4)]
@ -610,7 +610,7 @@ lemma div_lt_of_ne_zero : ∀ {n : nat}, n ≠ 0 → n div 2 < n
 | (succ n) h :=
  begin
    apply div_lt_of_lt_mul,
-    rewrite [-add_one, mul.right_distrib],
+    rewrite [-add_one, right_distrib],
    change n + 1 < (n * 1 + n) + (1 + 1),
    rewrite [mul_one, -add.assoc],
    apply add_lt_add_right,
--- a/library/data/nat/examples/fib2.lean
+++ b/library/data/nat/examples/fib2.lean
@ -6,7 +6,7 @@ Author: Leonardo de Moura
 Show that tail recursive fib is equal to standard one.
 -/
 import data.nat
-open nat
+open nat algebra

 definition fib : nat → nat
 | 0     := 1
--- a/library/data/nat/examples/partial_sum.lean
+++ b/library/data/nat/examples/partial_sum.lean
@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 -/
 import data.nat
-open nat
+open nat algebra

 definition partial_sum : nat → nat
 | 0        := 0
@ -19,10 +19,10 @@ rfl
 lemma two_mul_partial_sum_eq : ∀ n, 2 * partial_sum n = (succ n) * n
 | 0        := by reflexivity
 | (succ n) := calc
-   2 * (succ n + partial_sum n) = 2 * succ n + 2 * partial_sum n  : mul.left_distrib
+   2 * (succ n + partial_sum n) = 2 * succ n + 2 * partial_sum n  : left_distrib
                   ...   = 2 * succ n + succ n * n  : two_mul_partial_sum_eq
                   ...   = 2 * succ n + n * succ n  : mul.comm
-                   ...   = (2 + n) * succ n         : mul.right_distrib
+                   ...   = (2 + n) * succ n         : right_distrib
                   ...   = (n + 2) * succ n         : add.comm
                   ...   = (succ (succ n)) * succ n : rfl

--- a/library/data/nat/order.lean
+++ b/library/data/nat/order.lean
@ -6,7 +6,7 @@ Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad
 The order relation on the natural numbers.
 -/
 import data.nat.basic algebra.ordered_ring
-open eq.ops
+open eq.ops algebra

 namespace nat

@ -73,7 +73,7 @@ theorem lt_add_of_pos_right {n k : ℕ} (H : k > 0) : n < n + k :=

 theorem mul_le_mul_left {n m : ℕ} (k : ℕ) (H : n ≤ m) : k * n ≤ k * m :=
 obtain (l : ℕ) (Hl : n + l = m), from le.elim H,
-have k * n + k * l = k * m, by rewrite [-mul.left_distrib, Hl],
+have k * n + k * l = k * m, by rewrite [-left_distrib, Hl],
 le.intro this

 theorem mul_le_mul_right {n m : ℕ} (k : ℕ) (H : n ≤ m) : n * k ≤ m * k :=
--- a/library/data/nat/parity.lean
+++ b/library/data/nat/parity.lean
@ -185,7 +185,7 @@ lemma even_add_of_even_of_even {n m} : even n → even m → even (n+m) :=
 suppose even n, suppose even m,
 obtain k₁ (hk₁ : n = 2 * k₁), from exists_of_even `even n`,
 obtain k₂ (hk₂ : m = 2 * k₂), from exists_of_even `even m`,
-even_of_exists (exists.intro (k₁+k₂) (by rewrite [hk₁, hk₂, mul.left_distrib]))
+even_of_exists (exists.intro (k₁+k₂) (by rewrite [hk₁, hk₂, left_distrib]))

 lemma even_add_of_odd_of_odd {n m} : odd n → odd m → even (n+m) :=
 suppose odd n, suppose odd m,
--- a/library/data/nat/power.lean
+++ b/library/data/nat/power.lean
@ -45,7 +45,7 @@ theorem le_pow_self {x : ℕ} (H : x > 1) : ∀ i, i ≤ x^i
                succ j = j + 1         : rfl
                   ... ≤ x^j + 1       : add_le_add_right (le_pow_self j)
                   ... ≤ x^j + x^j     : add_le_add_left h₁
-                   ... = x^j * (1 + 1) : by rewrite [mul.left_distrib, *mul_one]
+                   ... = x^j * (1 + 1) : by rewrite [left_distrib, *mul_one]
                   ... = x^j * 2       : rfl
                   ... ≤ x^j * x       : mul_le_mul_left _ `x ≥ 2`
                   ... = x^(succ j)    : pow_succ'
--- a/library/data/nat/sub.lean
+++ b/library/data/nat/sub.lean
@ -6,7 +6,7 @@ Authors: Floris van Doorn, Jeremy Avigad
 Subtraction on the natural numbers, as well as min, max, and distance.
 -/
 import .order
-open eq.ops
+open eq.ops algebra

 namespace nat

@ -144,11 +144,11 @@ calc
          ... = n * m - n * k : {!mul.comm}

 theorem mul_self_sub_mul_self_eq (a b : nat) : a * a - b * b = (a + b) * (a - b) :=
-by rewrite [mul_sub_left_distrib, *mul.right_distrib, mul.comm b a, add.comm (a*a) (a*b), add_sub_add_left]
+by rewrite [mul_sub_left_distrib, *right_distrib, mul.comm b a, add.comm (a*a) (a*b), add_sub_add_left]

 theorem succ_mul_succ_eq (a : nat) : succ a * succ a = a*a + a + a + 1 :=
 calc succ a * succ a = (a+1)*(a+1)     : by rewrite [add_one]
-                ...  = a*a + a + a + 1 : by rewrite [mul.right_distrib, mul.left_distrib, one_mul, mul_one]
+                ...  = a*a + a + a + 1 : by rewrite [right_distrib, left_distrib, one_mul, mul_one]

 /- interaction with inequalities -/

@ -458,7 +458,7 @@ H ▸ !dist.triangle_inequality

 theorem dist_mul_right (n k m : ℕ) : dist (n * k) (m * k) = dist n m * k :=
 assert ∀ n m, dist n m = n - m + (m - n), from take n m, rfl,
-by rewrite [this, this n m, mul.right_distrib, *mul_sub_right_distrib]
+by rewrite [this, this n m, right_distrib, *mul_sub_right_distrib]

 theorem dist_mul_left (k n m : ℕ) : dist (k * n) (k * m) = k * dist n m :=
 begin rewrite [mul.comm k n, mul.comm k m, dist_mul_right, mul.comm] end
--- a/library/data/pnat.lean
+++ b/library/data/pnat.lean
@ -177,7 +177,7 @@ theorem one_mul (n : ℕ+) : pone * n = n :=
 begin
  apply pnat.eq,
  unfold pone,
-  rewrite [mul.def, ↑nat_of_pnat, one_mul]
+  rewrite [mul.def, ↑nat_of_pnat, algebra.one_mul]
 end

 theorem pone_le (n : ℕ+) : pone ≤ n :=
@ -256,13 +256,13 @@ begin
 end

 theorem mul.assoc (a b c : ℕ+) : a * b * c = a * (b * c) :=
-pnat.eq !nat.mul.assoc
+pnat.eq !mul.assoc

 theorem mul.comm (a b : ℕ+) : a * b = b * a :=
-pnat.eq !nat.mul.comm
+pnat.eq !mul.comm

 theorem add.assoc (a b c : ℕ+) : a + b + c = a + (b + c) :=
-pnat.eq !nat.add.assoc
+pnat.eq !add.assoc

 theorem mul_le_mul_left (p q : ℕ+) : q ≤ p * q :=
 begin
--- a/library/data/real/basic.lean
+++ b/library/data/real/basic.lean
@ -276,7 +276,7 @@ private theorem canon_bound {s : seq} (Hs : regular s) (n : ℕ+) : abs (s n)
      by rewrite [add.comm 1 (abs (s pone)), rat.add.comm 1, rat.add.assoc]
    ... ≤ of_nat (ubound (abs (s pone))) + (1 + 1) : algebra.add_le_add_right (!ubound_ge)
    ... = of_nat (ubound (abs (s pone)) + (1 + 1)) : of_nat_add
-    ... = of_nat (ubound (abs (s pone)) + 1 + 1)   : nat.add.assoc
+    ... = of_nat (ubound (abs (s pone)) + 1 + 1)   : algebra.add.assoc
    ... = rat_of_pnat (K s)                        : by esimp

 theorem bdd_of_regular {s : seq} (H : regular s) : ∃ b : ℚ, ∀ n : ℕ+, s n ≤ b :=
--- a/library/data/stream.lean
+++ b/library/data/stream.lean
@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 -/
 import data.nat data.list data.equiv
-open nat function option
+open nat function option algebra

 definition stream (A : Type) := nat → A

--- a/library/data/tuple.lean
+++ b/library/data/tuple.lean
@ -7,7 +7,7 @@ Tuples are lists of a fixed size.
 It is implemented as a subtype.
 -/
 import logic data.list data.fin
-open nat list subtype function
+open nat list subtype function algebra

 definition tuple [reducible] (A : Type) (n : nat) := {l : list A | length l = n}

--- a/library/theories/analysis/metric_space.lean
+++ b/library/theories/analysis/metric_space.lean
@ -105,7 +105,7 @@ exists.intro N

 proposition converges_to_seq_offset_left {X : ℕ → M} {y : M} (k : ℕ) (H : X ⟶ y in ℕ) :
  (λ n, X (k + n)) ⟶ y in ℕ :=
-have aux : (λ n, X (k + n)) = (λ n, X (n + k)), from funext (take n, by rewrite nat.add.comm),
+have aux : (λ n, X (k + n)) = (λ n, X (n + k)), from funext (take n, by rewrite add.comm),
 by+ rewrite aux; exact converges_to_seq_offset k H

 proposition converges_to_seq_offset_succ {X : ℕ → M} {y : M} (H : X ⟶ y in ℕ) :
@ -127,7 +127,7 @@ exists.intro (N + k)
 proposition converges_to_seq_of_converges_to_seq_offset_left
    {X : ℕ → M} {y : M} {k : ℕ} (H : (λ n, X (k + n)) ⟶ y in ℕ) :
  X ⟶ y in ℕ :=
-have aux : (λ n, X (k + n)) = (λ n, X (n + k)), from funext (take n, by rewrite nat.add.comm),
+have aux : (λ n, X (k + n)) = (λ n, X (n + k)), from funext (take n, by rewrite add.comm),
 by+ rewrite aux at H; exact converges_to_seq_of_converges_to_seq_offset H

 proposition converges_to_seq_of_converges_to_seq_offset_succ
--- a/library/theories/combinatorics/choose.lean
+++ b/library/theories/combinatorics/choose.lean
@ -85,12 +85,12 @@ begin
      cases k with k',
        {rewrite [*choose_self, one_mul, mul_one]},
        {have H : 1 < succ (succ k'), from succ_lt_succ !zero_lt_succ,
-          rewrite [one_mul, choose_zero_succ, choose_eq_zero_of_lt H, zero_mul]}},
+         krewrite [one_mul, choose_zero_succ, choose_eq_zero_of_lt H, zero_mul]}},
  intro k,
  cases k with k',
    {rewrite [choose_zero_right, choose_one_right]},
-  rewrite [choose_succ_succ (succ n), mul.right_distrib, -ih (succ k')],
-  rewrite [choose_succ_succ at {1}, mul.left_distrib, *succ_mul (succ n), mul_succ, -ih k'],
+  rewrite [choose_succ_succ (succ n), right_distrib, -ih (succ k')],
+  rewrite [choose_succ_succ at {1}, left_distrib, *succ_mul (succ n), mul_succ, -ih k'],
  rewrite [*add.assoc, add.left_comm (choose n _)]
 end

--- a/library/theories/group_theory/cyclic.lean
+++ b/library/theories/group_theory/cyclic.lean
@ -278,7 +278,7 @@ eq_of_veq begin rewrite [↑rotl, val_madd], esimp [mk_mod], rewrite [ mod_add_m
 lemma rotl_compose : ∀ {n : nat} {j k : nat}, (@rotl n j) ∘ (rotl k) = rotl (j + k)
 | 0        := take j k, funext take i, elim0 i
 | (succ n) :=  take j k, funext take i, eq.symm begin
-  rewrite [*rotl_succ', mul.left_distrib, -(@madd_mk_mod n (n*j)), madd_assoc],
+  rewrite [*rotl_succ', left_distrib, -(@madd_mk_mod n (n*j)), madd_assoc],
  end

 lemma rotr_rotl : ∀ {n : nat} (m : nat) {i : fin n}, rotr m (rotl m i) = i
--- a/library/theories/number_theory/prime_factorization.lean
+++ b/library/theories/number_theory/prime_factorization.lean
@ -120,7 +120,7 @@ private theorem mult_pow_mul {p n : ℕ} (i : ℕ) (pgt1 : p > 1) (npos : n > 0)
  mult p (p^i * n) = i + mult p n :=
 begin
  induction i with [i, ih],
-    {rewrite [pow_zero, one_mul, zero_add]},
+    {krewrite [pow_zero, one_mul, zero_add]},
  have p > 0, from lt.trans zero_lt_one pgt1,
  have psin_pos : p^(succ i) * n > 0, from mul_pos (!pow_pos_of_pos this) npos,
  have p ∣ p^(succ i) * n, by rewrite [pow_succ, mul.assoc]; apply dvd_mul_right,
@ -175,7 +175,7 @@ calc
 theorem mult_pow {p m : ℕ} (n : ℕ) (mpos : m > 0) (primep : prime p) : mult p (m^n) = n * mult p m :=
 begin
  induction n with n ih,
-    rewrite [pow_zero, mult_one_right, zero_mul],
+    krewrite [pow_zero, mult_one_right, zero_mul],
  rewrite [pow_succ, mult_mul primep mpos (!pow_pos_of_pos mpos), ih, succ_mul, add.comm]
 end

--- a/src/shell/CMakeLists.txt
+++ b/src/shell/CMakeLists.txt
@ -43,9 +43,6 @@ add_test(NAME "lean_eqn_macro"
 add_test(NAME "lean_print_notation"
         WORKING_DIRECTORY "${LEAN_SOURCE_DIR}/../tests/lean/extra"
         COMMAND bash "./test_single.sh" "${CMAKE_CURRENT_BINARY_DIR}/lean" "print_tests.lean")
-add_test(NAME "auto_completion_issue_422"
-         WORKING_DIRECTORY "${LEAN_SOURCE_DIR}/../tests/lean/extra"
-         COMMAND bash "./ac_bug.sh" "${CMAKE_CURRENT_BINARY_DIR}/lean")
 add_test(NAME "issue_597"
         WORKING_DIRECTORY "${LEAN_SOURCE_DIR}/../tests/lean/extra"
         COMMAND bash "./issue_597.sh" "${CMAKE_CURRENT_BINARY_DIR}/lean")
--- a/tests/lean/assert_tac2.lean
+++ b/tests/lean/assert_tac2.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 example (a b c : nat) : a = 2 → b = 3 → a + b + c = c + 5 :=
 begin
--- a/tests/lean/extra/ac_bug.sh
+++ b/tests/lean/extra/ac_bug.sh
@ -1,19 +0,0 @@
-#!/bin/sh
-# Regression test for issue https://github.com/leanprover/lean/issues/422
-set -e
-if [ $# -ne 1 ]; then
-    echo "Usage: ac_bug.sh [lean-executable-path]"
-    exit 1
-fi
-LEAN=$1
-export LEAN_PATH=../../../library:.
-for f in `ls ac_bug*.input`; do
-    echo "testing $f..."
-    "$LEAN" --server-trace "$f" > "$f.produced.out"
-    if grep "nat.mul.assoc" "$f.produced.out"; then
-        echo "FAILED"
-        exit 1
-    fi
-    rm -f -- "$f.produced.out"
-done
-echo "done"
--- a/tests/lean/extra/ac_bug1.input
+++ b/tests/lean/extra/ac_bug1.input
@ -1,9 +0,0 @@
-VISIT ac_bug.lean
-SYNC 4
-import data.nat
-namespace nat
-check mul.as
-exit
-WAIT
-FINDP 3
-mul.as
--- a/tests/lean/extra/ac_bug2.input
+++ b/tests/lean/extra/ac_bug2.input
@ -1,9 +0,0 @@
-VISIT ac_bug.lean
-SYNC 4
-import data.nat
-namespace nat
-check mul.as
-end nat
-WAIT
-FINDP 3
-mul.as
--- a/tests/lean/extra/ac_bug3.input
+++ b/tests/lean/extra/ac_bug3.input
@ -1,10 +0,0 @@
-VISIT ac_bug.lean
-SYNC 5
-import data.nat
-namespace nat
-check mul.as
-end
-nat
-WAIT
-FINDP 3
-mul.as
--- a/tests/lean/extra/ac_bug4.input
+++ b/tests/lean/extra/ac_bug4.input
@ -1,10 +0,0 @@
-VISIT ac_bug.lean
-SYNC 5
-import data.nat
-namespace nat
-check mul.as
-def a := 10
-end nat
-WAIT
-FINDP 3
-mul.as
--- a/tests/lean/extra/ac_bug5.input
+++ b/tests/lean/extra/ac_bug5.input
@ -1,9 +0,0 @@
-VISIT ac_bug.lean
-SYNC 4
-import data.nat
-namespace nat
-check mul.as
-def a := 10
-WAIT
-FINDP 3
-mul.as
--- a/tests/lean/extra/ac_bug6.input
+++ b/tests/lean/extra/ac_bug6.input
@ -1,10 +0,0 @@
-VISIT ac_bug.lean
-SYNC 5
-import data.nat
-namespace nat
-check mul.as
-def a := 10
-exit
-WAIT
-FINDP 3
-mul.as
--- a/tests/lean/extra/show_goal.lean
+++ b/tests/lean/extra/show_goal.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 example (a b c d : nat) : a + b = 0 → b = 0 → c + 1 + a = 1 → d = c - 1 → d = 0 :=
 begin
@ -14,7 +14,7 @@ begin
      rewrite aeq0 at h₃,
      rewrite add_zero at h₃,
      rewrite add_succ at h₃,
-      rewrite add_zero at h₃,
+      krewrite add_zero at h₃,
      injection h₃, exact a_1
    end,
    rewrite ceq at h₄,
--- a/tests/lean/interactive/consume_args.input
+++ b/tests/lean/interactive/consume_args.input
@ -1,7 +1,7 @@
 VISIT consume_args.lean
 SYNC 7
 import logic data.nat.basic
-open nat eq.ops
+open nat eq.ops algebra

 theorem tst (a b c : nat) : a + b + c = a + c + b :=
 calc a + b + c = a + (b + c) : !add.assoc
--- a/tests/lean/interactive/consume_args.input.expected.out
+++ b/tests/lean/interactive/consume_args.input.expected.out
@ -44,7 +44,7 @@ b
 a + c + b = a + (c + b)
 -- ACK
 -- IDENTIFIER|7|33
-nat.add.assoc
+algebra.add.assoc
 -- ACK
 -- TYPE|7|43
 a + c + b = a + (c + b) → a + (c + b) = a + c + b
--- a/tests/lean/run/592.lean
+++ b/tests/lean/run/592.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 definition foo (a b : nat) := a * b

--- a/tests/lean/run/695d.lean
+++ b/tests/lean/run/695d.lean
@ -3,5 +3,5 @@ open nat

 example (a b : nat) : 0 + a + 0 = a :=
 begin
-  rewrite [add_zero, zero_add,]
+  rewrite [nat.add_zero, nat.zero_add,]
 end
--- a/tests/lean/run/all_goals.lean
+++ b/tests/lean/run/all_goals.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 attribute nat.add [unfold 2]
 attribute nat.rec_on [unfold 2]
--- a/tests/lean/run/all_goals2.lean
+++ b/tests/lean/run/all_goals2.lean
@ -18,6 +18,6 @@ example (a b c : nat) : (a + 0 = 0 + a ∧ b + 0 = 0 + b) ∧ c = c :=

 begin
  apply and.intro,
-  apply and.intro ;; (state; rewrite zero_add),
+  apply and.intro ;; (state; rewrite nat.zero_add),
  apply rfl
 end
--- a/tests/lean/run/imp_curly.lean
+++ b/tests/lean/run/imp_curly.lean
@ -3,7 +3,7 @@ open nat

 theorem zero_left (n : ℕ) : 0 + n = n :=
 nat.induction_on n
-    !add_zero
+    !nat.add_zero
    (take m IH, show 0 + succ m = succ m, from
      calc
        0 + succ m = succ (0 + m) : add_succ
--- a/tests/lean/run/match_tac3.lean
+++ b/tests/lean/run/match_tac3.lean
@ -1,11 +1,11 @@
 import data.nat
-open nat
+open nat algebra

 theorem tst (a b : nat) (H : a = 0) : a + b = b :=
 begin
  revert H,
  match a with
-  | zero  := λ H, by rewrite zero_add
+  | zero  := λ H, by krewrite zero_add
  | (n+1) := λ H, nat.no_confusion H
  end
 end
--- a/tests/lean/run/mul_zero.lean
+++ b/tests/lean/run/mul_zero.lean
@ -2,4 +2,4 @@ import data.nat
 open nat

 example (x : ℕ) : 0 = x * 0 :=
-calc 0 = x * 0 : mul_zero
+calc 0 = x * 0 : nat.mul_zero
--- a/tests/lean/run/new_obtains.lean
+++ b/tests/lean/run/new_obtains.lean
@ -1,4 +1,5 @@
 import data.nat
+open algebra

 theorem tst2 (A B C D : Type) : (A × B) × (C × D) → C × B × A :=
 assume p : (A × B) × (C × D),
@ -62,5 +63,5 @@ obtain y (Hy : b = 2*y), from H₂,
 exists.intro
  (x+y)
  (calc a+b = 2*x + 2*y : by rewrite [Hx, Hy]
-        ... = 2*(x+y)   : by rewrite mul.left_distrib)
+        ... = 2*(x+y)   : by rewrite left_distrib)
 end hidden
--- a/tests/lean/run/parity.lean
+++ b/tests/lean/run/parity.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 namespace foo

@ -20,7 +20,7 @@ definition parity : Π (n : nat), Parity n
    end,
    begin
      change (Parity (2*k + 2*1)),
-      rewrite -mul.left_distrib,
+      rewrite -left_distrib,
      apply (Parity.even (k+1))
    end
  end
--- a/tests/lean/run/print_poly.lean
+++ b/tests/lean/run/print_poly.lean
@ -12,7 +12,7 @@ print nat.rec
 print classical.em
 print quot.lift
 print nat.of_num
-print nat.add.assoc
+print nat.add_assoc

 section
  parameter {A : Type}
--- a/tests/lean/run/refine3.lean
+++ b/tests/lean/run/refine3.lean
@ -1,5 +1,5 @@
 import data.nat.basic
-open nat
+open nat algebra

 theorem zero_left (n : ℕ) : 0 + n = n :=
 nat.induction_on n
--- a/tests/lean/run/rewrite10.lean
+++ b/tests/lean/run/rewrite10.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 section
  variables (a b c d e : nat)
@ -9,7 +9,7 @@ section
 end

 example (x y : ℕ) : (x + y) * (x + y) = x * x + y * x + x * y + y * y :=
-by rewrite [*mul.left_distrib, *mul.right_distrib, -add.assoc]
+by rewrite [*left_distrib, *right_distrib, -add.assoc]

 namespace tst
 definition even (a : nat) := ∃b, a = 2*b
@ -19,7 +19,7 @@ exists.elim H1 (fun (w1 : nat) (Hw1 : a = 2*w1),
 exists.elim H2 (fun (w2 : nat) (Hw2 : b = 2*w2),
  exists.intro (w1 + w2)
    begin
-      rewrite [Hw1, Hw2, mul.left_distrib]
+      rewrite [Hw1, Hw2, left_distrib]
    end))

 theorem T2 (a b c : nat) (H1 : a = b) (H2 : b = c + 1) : a ≠ 0 :=
--- a/tests/lean/run/rewrite12.lean
+++ b/tests/lean/run/rewrite12.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra
 variables (f : nat → nat → nat → nat) (a b c : nat)

 example (H₁ : a = b) (H₂ : f b a b = 0) : f a a a = 0 :=
--- a/tests/lean/run/rewrite8.lean
+++ b/tests/lean/run/rewrite8.lean
@ -3,4 +3,4 @@ open nat
 constant f : nat → nat

 theorem tst1 (x y : nat) (H1 : (λ z, z + 0) x = y) : f x = f y :=
-by rewrite [▸* at H1, add_zero at H1, H1]
+by rewrite [▸* at H1, nat.add_zero at H1, H1]
--- a/tests/lean/run/rewriter12.lean
+++ b/tests/lean/run/rewriter12.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra
 constant f : nat → nat

 example (x y : nat) (H1 : (λ z, z + 0) x = y) : f x = f y :=
--- a/tests/lean/run/rewriter14.lean
+++ b/tests/lean/run/rewriter14.lean
@ -8,5 +8,5 @@ attribute of_num [unfold 1]

 example (x y : nat) (H1 : f x 0 0 = 0) : x = 0 :=
 begin
-  rewrite [↑f at H1, 4>add_zero at H1, H1]
+  rewrite [↑f at H1, 4>nat.add_zero at H1, H1]
 end
--- a/tests/lean/run/rewriter15.lean
+++ b/tests/lean/run/rewriter15.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 set_option rewriter.syntactic true

--- a/tests/lean/run/rewriter16.lean
+++ b/tests/lean/run/rewriter16.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 theorem tst (x : nat) (H1 : x = 0) : x = 0 :=
 begin
--- a/tests/lean/run/rw_set1.lean
+++ b/tests/lean/run/rw_set1.lean
@ -1,15 +1,15 @@
 import data.nat

 namespace foo
-  attribute nat.add.assoc [simp]
-  print nat.add.assoc
+  attribute nat.add_assoc [simp]
+  print nat.add_assoc
 end foo

-print nat.add.assoc
+print nat.add_assoc

 namespace foo
-  print nat.add.assoc
-  attribute nat.add.comm [simp]
+  print nat.add_assoc
+  attribute nat.add_comm [simp]
  open nat
  print "---------"
  print [simp]
--- a/tests/lean/run/show2.lean
+++ b/tests/lean/run/show2.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 example : ∀ a b : nat, a + b = b + a :=
 show ∀ a b : nat, a + b = b + a
--- a/tests/lean/run/struct_infer.lean
+++ b/tests/lean/run/struct_infer.lean
@ -7,7 +7,7 @@ structure semigroup [class] (A : Type) :=
 mk {} :: (mul: A → A → A) (mul_assoc : associative mul)

 definition nat_semigroup [instance] : semigroup nat :=
-semigroup.mk nat.mul nat.mul.assoc
+semigroup.mk nat.mul nat.mul_assoc

 example (a b c : nat) : (a * b) * c = a * (b * c) :=
 semigroup.mul_assoc a b c
@ -15,7 +15,7 @@ semigroup.mul_assoc a b c
 structure semigroup2 (A : Type) :=
 mk () :: (mul: A → A → A) (mul_assoc : associative mul)

-definition s := semigroup2.mk nat nat.mul nat.mul.assoc
+definition s := semigroup2.mk nat nat.mul nat.mul_assoc

 example (a b c : nat) : (a * b) * c = a * (b * c) :=
 semigroup2.mul_assoc nat s a b c
--- a/tests/lean/run/struct_inst_exprs2.lean
+++ b/tests/lean/run/struct_inst_exprs2.lean
@ -12,7 +12,7 @@ structure s3 (A : Type) extends s1 A, s2 A :=

 definition v1 : s1 nat := {| s1, x := 10, y := 10, h := rfl |}
 definition v2 : s2 nat := {| s2, mul := nat.add, one := zero |}
-definition v3 : s3 nat := {| s3, mul_one := add_zero, v1, v2 |}
+definition v3 : s3 nat := {| s3, mul_one := nat.add_zero, v1, v2 |}

 example : s3.x v3 = 10 := rfl
 example : s3.y v3 = 10 := rfl
--- a/tests/lean/run/tactic31.lean
+++ b/tests/lean/run/tactic31.lean
@ -1,4 +1,5 @@
 import data.nat
+open algebra

 example (a b c : Prop) : a → b → c → a ∧ b ∧ c :=
 begin
--- a/tests/lean/run/unfold_tac_bug1.lean
+++ b/tests/lean/run/unfold_tac_bug1.lean
@ -18,7 +18,7 @@ private definition fib_fast_aux : nat → nat → nat → nat
 | (succ n) i j := fib_fast_aux n j (j+i)

 lemma fib_fast_aux_lemma : ∀ n m, fib_fast_aux n (fib m) (fib (succ m)) = fib (succ (n + m))
-| 0        m := by rewrite zero_add
+| 0        m := by rewrite nat.zero_add
 | (succ n) m :=
  begin
    have ih : fib_fast_aux n (fib (succ m)) (fib (succ (succ m))) = fib (succ (n + succ m)), from fib_fast_aux_lemma n (succ m),
--- a/tests/lean/rw_at_failure.lean
+++ b/tests/lean/rw_at_failure.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 example (a b : nat) : a = b → 0 + a = 0 + b :=
 begin
--- a/tests/lean/slow/list_elab2.lean
+++ b/tests/lean/slow/list_elab2.lean
@ -12,7 +12,7 @@
 import logic data.nat
 -- import congr

-open nat
+open nat algebra
 -- open congr
 open eq.ops eq

--- a/tests/lean/unfold.lean
+++ b/tests/lean/unfold.lean
@ -1,5 +1,5 @@
 import data.nat
-open nat
+open nat algebra

 definition f (a b : nat) := a + b