feat(library/algebra/ordered_ring,library/data/int/): add sign and theorems about abs, make int an instance, port theorems

library/algebra/order.lean

@@ -333,6 +333,22 @@ section
     (assume H : ¬ a ≤ b,
       (inr (assume H1 : a = b, H (H1 ▸ !le.refl))))
 
+  -- testing equality first may result in more definitional equalities
+  definition lt.cases {B : Type} (a b : A) (t_lt t_eq t_gt : B) : B :=
+  if a = b then t_eq else (if a < b then t_lt else t_gt)
+
+  theorem lt.cases_of_eq {B : Type} {a b : A} {t_lt t_eq t_gt : B} (H : a = b) :
+    lt.cases a b t_lt t_eq t_gt = t_eq := if_pos H
+
+  theorem lt.cases_of_lt {B : Type} {a b : A} {t_lt t_eq t_gt : B} (H : a < b) :
+    lt.cases a b t_lt t_eq t_gt = t_lt :=
+  if_neg (ne_of_lt H) ⬝ if_pos H
+
+  theorem lt.cases_of_gt {B : Type} {a b : A} {t_lt t_eq t_gt : B} (H : a > b) :
+    lt.cases a b t_lt t_eq t_gt = t_gt :=
+  if_neg (ne.symm (ne_of_lt H)) ⬝ if_neg (lt.asymm H)
+
+/-
   definition lt.by_cases' {a b : A} {P : Type}
     (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
   if H4 : a < b then H1 H4 else
@@ -355,6 +371,7 @@ section
   have H5 [visible] : ¬ a < b, from lt.asymm H4,
   have H6 [visible] : a ≠ b, from (assume H7: a = b, lt.irrefl b (H7 ▸ H4)),
   dif_neg H6 ▸ dif_neg H5
+-/
 end
 
 end algebra

library/algebra/ordered_group.lean

@@ -464,7 +464,7 @@ section
   theorem abs_pos_of_ne_zero (H : a ≠ 0) : |a| > 0 :=
   or.elim (lt_or_gt_of_ne H) abs_pos_of_neg abs_pos_of_pos
 
-  theorem abs_sub : |a - b| = |b - a| :=
+  theorem abs_sub (a b : A) : |a - b| = |b - a| :=
   calc
     |a - b| = |-(b - a)| : neg_sub
         ... = |b - a|    : abs_neg
@@ -481,7 +481,7 @@ section
   abs.by_cases H1 H2
 
   -- the triangle inequality
-  theorem abs_add_le_abs_add_abs : |a + b| ≤ |a| + |b| :=
+  theorem abs_add_le_abs_add_abs (a b : A) : |a + b| ≤ |a| + |b| :=
   have aux1 : ∀{a b}, a + b ≥ 0 → a ≥ 0 → |a + b| ≤ |a| + |b|,
   proof
     take a b,
@@ -535,14 +535,13 @@ section
               ... = |a| + |b|   : abs_neg)
   qed
 
-  theorem abs_sub_abs_le_abs_sub : |a| - |b| ≤ |a - b| :=
+  theorem abs_sub_abs_le_abs_sub (a b : A) : |a| - |b| ≤ |a - b| :=
   have H1 : |a| - |b| + |b| ≤ |a - b| + |b|, from
     calc
       |a| - |b| + |b| = |a| : sub_add_cancel
         ... = |a - b + b|   : sub_add_cancel
         ... ≤ |a - b| + |b| : algebra.abs_add_le_abs_add_abs,
   algebra.le_of_add_le_add_right H1
-
 end
 
 end algebra

library/algebra/ordered_ring.lean

@@ -233,13 +233,16 @@ lt.by_cases
     lt.by_cases
       (assume Hb : 0 < b, absurd (H ▸ show 0 < a * b, from mul_pos Ha Hb) (lt.irrefl 0))
       (assume Hb : 0 = b, or.inr (Hb⁻¹))
-      (assume Hb : 0 > b, absurd (H ▸ show 0 > a * b, from mul_neg_of_pos_of_neg Ha Hb) (lt.irrefl 0)))
+      (assume Hb : 0 > b,
+        absurd (H ▸ show 0 > a * b, from mul_neg_of_pos_of_neg Ha Hb) (lt.irrefl 0)))
   (assume Ha : 0 = a, or.inl (Ha⁻¹))
   (assume Ha : 0 > a,
     lt.by_cases
-      (assume Hb : 0 < b, absurd (H ▸ show 0 > a * b, from mul_neg_of_neg_of_pos Ha Hb) (lt.irrefl 0))
+      (assume Hb : 0 < b,
+        absurd (H ▸ show 0 > a * b, from mul_neg_of_neg_of_pos Ha Hb) (lt.irrefl 0))
       (assume Hb : 0 = b, or.inr (Hb⁻¹))
-      (assume Hb : 0 > b, absurd (H ▸ show 0 < a * b, from mul_pos_of_neg_of_neg Ha Hb) (lt.irrefl 0)))
+      (assume Hb : 0 > b,
+        absurd (H ▸ show 0 < a * b, from mul_pos_of_neg_of_neg Ha Hb) (lt.irrefl 0)))
 
 -- Linearity implies no zero divisors. Doesn't need commutativity.
 definition linear_ordered_comm_ring.to_integral_domain [instance] [coercion]
@@ -282,15 +285,178 @@ section
         (assume Hb : b < 0, or.inr (and.intro Ha Hb)))
 end
 
-/-
+/- TODO: Isabelle's library has all kinds of cancelation rules for the simplifier.
-  Still left to do:
+   Search on mult_le_cancel_right1 in Rings.thy. -/
 
-  Isabelle's library has all kinds of cancelation rules for the simplifier, search on
+structure decidable_linear_ordered_comm_ring [class] (A : Type) extends linear_ordered_comm_ring A,
-    mult_le_cancel_right1 in Rings.thy.
+    decidable_linear_ordered_comm_group A
 
-  Properties of abs, sgn, and dvd.
+section
+  variable [s : decidable_linear_ordered_comm_ring A]
+  variables {a b c : A}
+  include s
 
-  Multiplication and one, starting with mult_right_le_one_le.
+  definition sign (a : A) : A := lt.cases a 0 (-1) 0 1
--/
+
+  theorem sign_of_neg (H : a < 0) : sign a = -1 := lt.cases_of_lt H
+
+  theorem sign_zero : sign 0 = 0 := lt.cases_of_eq rfl
+
+  theorem sign_of_pos (H : a > 0) : sign a = 1 := lt.cases_of_gt H
+
+  theorem sign_one : sign 1 = 1 := sign_of_pos zero_lt_one
+
+  theorem sign_neg_one : sign (-1) = -1 := sign_of_neg (neg_neg_of_pos zero_lt_one)
+
+  theorem sign_sign (a : A) : sign (sign a) = sign a :=
+  lt.by_cases
+    (assume H : a > 0,
+      calc
+        sign (sign a) = sign 1 : {sign_of_pos H}
+                  ... = 1      : sign_one
+                  ... = sign a : sign_of_pos H)
+    (assume H : 0 = a,
+      calc
+        sign (sign a) = sign (sign 0) : H
+                  ... = sign 0        : sign_zero
+                  ... = sign a        : H)
+    (assume H : a < 0,
+      calc
+        sign (sign a) = sign (-1)     : {sign_of_neg H}
+                  ... = -1            : sign_neg_one
+                  ... = sign a        : sign_of_neg H)
+
+  theorem pos_of_sign_eq_one (H : sign a = 1) : a > 0 :=
+  lt.by_cases
+    (assume H1 : 0 < a, H1)
+    (assume H1 : 0 = a, absurd (sign_zero⁻¹ ⬝ (H1⁻¹ ▸ H)) zero_ne_one)
+    (assume H1 : 0 > a,
+      have H2 : -1 = 1, from (sign_of_neg H1)⁻¹ ⬝ H,
+      absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
+
+  theorem eq_zero_of_sign_eq_zero (H : sign a = 0) : a = 0 :=
+  lt.by_cases
+    (assume H1 : 0 < a, absurd (H⁻¹ ⬝ sign_of_pos H1) zero_ne_one)
+    (assume H1 : 0 = a, H1⁻¹)
+    (assume H1 : 0 > a,
+      have H2 : 0 = -1, from H⁻¹ ⬝ sign_of_neg H1,
+      have H3 : 1 = 0, from eq_neg_of_eq_neg H2 ⬝ neg_zero,
+      absurd (H3⁻¹) zero_ne_one)
+
+  theorem neg_of_sign_eq_neg_one (H : sign a = -1) : a < 0 :=
+  lt.by_cases
+    (assume H1 : 0 < a,
+      have H2 : -1 = 1, from H⁻¹ ⬝ (sign_of_pos H1),
+      absurd ((eq_zero_of_neg_eq H2)⁻¹) zero_ne_one)
+    (assume H1 : 0 = a,
+      have H2 : 0 = -1, from (H1 ▸ sign_zero)⁻¹ ⬝ H,
+      have H3 : 1 = 0, from eq_neg_of_eq_neg H2 ⬝ neg_zero,
+      absurd (H3⁻¹) zero_ne_one)
+    (assume H1 : 0 > a, H1)
+
+  theorem sign_neg (a : A) : sign (-a) = -(sign a) :=
+  lt.by_cases
+    (assume H1 : 0 < a,
+      calc
+        sign (-a) = -1        : sign_of_neg (neg_neg_of_pos H1)
+              ... = -(sign a) : {(sign_of_pos H1)⁻¹})
+    (assume H1 : 0 = a,
+      calc
+        sign (-a) = sign (-0) : H1
+              ... = sign 0    : {neg_zero} -- TODO: why do we need {}?
+              ... = 0         : sign_zero
+              ... = -0        : neg_zero
+              ... = -(sign 0) : sign_zero
+              ... = -(sign a) : H1)
+    (assume H1 : 0 > a,
+      calc
+        sign (-a) = 1         : sign_of_pos (neg_pos_of_neg H1)
+              ... = -(-1)     : neg_neg
+              ... = -(sign a) : {(sign_of_neg H1)⁻¹})
+
+  -- hopefully, will be quick with the simplifier
+  theorem sign_mul (a b : A) : sign (a * b) = sign a * sign b := sorry
+
+  theorem abs_eq_sign_mul (a : A) : |a| = sign a * a :=
+  lt.by_cases
+    (assume H1 : 0 < a,
+      calc
+        |a| = a          : abs_of_pos H1
+        ... = 1 * a      : one_mul
+        ... = sign a * a : {(sign_of_pos H1)⁻¹})
+    (assume H1 : 0 = a,
+      calc
+        |a| = |0|        : H1
+        ... = 0          : abs_zero
+        ... = 0 * a      : zero_mul
+        ... = sign 0 * a : sign_zero
+        ... = sign a * a : H1)
+    (assume H1 : a < 0,
+      calc
+        |a| = -a : abs_of_neg H1
+          ... = -1 * a : neg_eq_neg_one_mul
+          ... = sign a * a : {(sign_of_neg H1)⁻¹})
+
+  theorem eq_sign_mul_abs (a : A) : a = sign a * |a| :=
+  lt.by_cases
+    (assume H1 : 0 < a,
+      calc
+        a = |a|              : abs_of_pos H1
+          ... = 1 * |a|      : one_mul
+          ... = sign a * |a| : {(sign_of_pos H1)⁻¹})
+    (assume H1 : 0 = a,
+      calc
+        a = 0        : H1
+          ... = 0 * |a|      : zero_mul
+          ... = sign 0 * |a| : sign_zero
+          ... = sign a * |a| : H1)
+    (assume H1 : a < 0,
+      calc
+        a = -(-a)            : neg_neg
+          ... = -|a|         : {(abs_of_neg H1)⁻¹}
+          ... = -1 * |a|     : neg_eq_neg_one_mul
+          ... = sign a * |a| : {(sign_of_neg H1)⁻¹})
+
+  theorem abs_dvd_iff_dvd (a b : A) : |a| | b ↔ a | b :=
+  abs.by_cases !iff.refl !neg_dvd_iff_dvd
+
+  theorem dvd_abs_iff (a b : A) : a | |b| ↔ a | b :=
+  abs.by_cases !iff.refl !dvd_neg_iff_dvd
+
+  theorem abs_mul (a b : A) : |a * b| = |a| * |b| :=
+  or.elim (le.total 0 a)
+    (assume H1 : 0 ≤ a,
+      or.elim (le.total 0 b)
+        (assume H2 : 0 ≤ b,
+          calc
+            |a * b| = a * b     : abs_of_nonneg (mul_nonneg H1 H2)
+                ... = |a| * b   : {(abs_of_nonneg H1)⁻¹}
+                ... = |a| * |b| : {(abs_of_nonneg H2)⁻¹})
+        (assume H2 : b ≤ 0,
+          calc
+            |a * b| = -(a * b)  : abs_of_nonpos (mul_nonpos_of_nonneg_of_nonpos H1 H2)
+                ... = a * -b    : neg_mul_eq_mul_neg
+                ... = |a| * -b  : {(abs_of_nonneg H1)⁻¹}
+                ... = |a| * |b| : {(abs_of_nonpos H2)⁻¹}))
+    (assume H1 : a ≤ 0,
+      or.elim (le.total 0 b)
+        (assume H2 : 0 ≤ b,
+          calc
+            |a * b| = -(a * b)  : abs_of_nonpos (mul_nonpos_of_nonpos_of_nonneg H1 H2)
+                ... = -a * b    : neg_mul_eq_neg_mul
+                ... = |a| * b   : {(abs_of_nonpos H1)⁻¹}
+                ... = |a| * |b| : {(abs_of_nonneg H2)⁻¹})
+        (assume H2 : b ≤ 0,
+          calc
+            |a * b| = a * b     : abs_of_nonneg (mul_nonneg_of_nonpos_of_nonpos H1 H2)
+                ... = -a * -b   : neg_mul_neg
+                ... = |a| * -b  : {(abs_of_nonpos H1)⁻¹}
+                ... = |a| * |b| : {(abs_of_nonpos H2)⁻¹}))
+
+  theorem abs_mul_self (a : A) : |a| * |a| = a * a :=
+  abs.by_cases rfl !neg_mul_neg
+end
+
+/- TODO: Multiplication and one, starting with mult_right_le_one_le. -/
 
 end algebra

library/algebra/ring.lean

@@ -183,13 +183,13 @@ section
         ... = 0 : zero_mul)
 
   theorem neg_mul_eq_mul_neg : -(a * b) = a * -b :=
-  neg_eq_of_add_eq_zero
+   neg_eq_of_add_eq_zero
     (calc
       a * b + a * -b = a * (b + -b) : left_distrib
         ... = a * 0 : add.right_inv
         ... = 0 : mul_zero)
 
-  theorem neg_mul_neg_eq : -a * -b = a * b :=
+  theorem neg_mul_neg : -a * -b = a * b :=
   calc
      -a * -b = -(a * -b) : !neg_mul_eq_neg_mul⁻¹
        ... = - -(a * b) : neg_mul_eq_mul_neg
@@ -197,6 +197,11 @@ section
 
   theorem neg_mul_comm : -a * b = a * -b := !neg_mul_eq_neg_mul⁻¹ ⬝ !neg_mul_eq_mul_neg
 
+  theorem neg_eq_neg_one_mul : -a = -1 * a :=
+  calc
+    -a = -(1 * a)  : one_mul a ▸ rfl
+      ... = -1 * a : neg_mul_eq_neg_mul
+
   theorem mul_sub_left_distrib : a * (b - c) = a * b - a * c :=
   calc
     a * (b - c) = a * b + a * -c : left_distrib
@@ -270,7 +275,7 @@ section
         (take c, assume H' : a * c = b,
           dvd.intro
             (calc
-              -a * -c = a * c : neg_mul_neg_eq
+              -a * -c = a * c : neg_mul_neg
                 ... = b : H')))
 
   theorem dvd_sub (H₁ : a | b) (H₂ : a | c) : a | (b - c) :=

library/data/int/basic.lean

@@ -426,7 +426,7 @@ 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
 
 theorem padd_padd_pneg (p q : ℕ × ℕ) : padd (padd p q) (pneg q) ≡ p :=
-show pr1 p + pr1 q + pr2 q + pr2 p = pr2 p + pr2 q + pr1 q + pr1 p, by simp
+show pr1 p + pr1 q + pr2 q + pr2 p = pr2 p + pr2 q + pr1 q + pr1 p, from by simp
 
 theorem add.left_inv (a : ℤ) : -a + a = 0 :=
 have H : repr (-a + a) ≡ repr 0, from
@@ -718,8 +718,9 @@ section port_algebra
   theorem mul_zero : ∀a : ℤ, a * 0 = 0 := algebra.mul_zero
   theorem neg_mul_eq_neg_mul : ∀a b : ℤ, -(a * b) = -a * b := algebra.neg_mul_eq_neg_mul
   theorem neg_mul_eq_mul_neg : ∀a b : ℤ, -(a * b) = a * -b := algebra.neg_mul_eq_mul_neg
-  theorem neg_mul_neg_eq : ∀a b : ℤ, -a * -b = a * b := algebra.neg_mul_neg_eq
+  theorem neg_mul_neg : ∀a b : ℤ, -a * -b = a * b := algebra.neg_mul_neg
   theorem neg_mul_comm : ∀a b : ℤ, -a * b = a * -b := algebra.neg_mul_comm
+  theorem neg_eq_neg_one_mul : ∀a : ℤ, -a = -1 * a := algebra.neg_eq_neg_one_mul
   theorem mul_sub_left_distrib : ∀a b c : ℤ, a * (b - c) = a * b - a * c :=
     algebra.mul_sub_left_distrib
   theorem mul_sub_right_distrib : ∀a b c : ℤ, (a - b) * c = a * c - b * c :=

library/data/int/order.lean

@@ -219,6 +219,13 @@ section
     add.comm mul mul.assoc (of_num 1) one_mul mul_one mul.left_distrib mul.right_distrib
     zero_ne_one le le.refl @le.trans @le.antisymm lt lt_iff_le_and_ne @add_le_add_left
     @mul_nonneg @mul_pos le_iff_lt_or_eq le.total mul.comm
+
+
+  protected definition decidable_linear_ordered_comm_ring [instance] :
+    algebra.decidable_linear_ordered_comm_ring int :=
+  ⦃algebra.decidable_linear_ordered_comm_ring,
+    int.linear_ordered_comm_ring,
+    decidable_lt := decidable_lt⦄
 end
 
 /- instantiate ordered ring theorems to int -/
@@ -414,6 +421,36 @@ section port_algebra
   theorem sub_lt_sub_of_lt_of_le : ∀{a b c d : ℤ}, a < b → c ≤ d → a - d < b - c :=
     @algebra.sub_lt_sub_of_lt_of_le _ _
 
+  theorem eq_zero_of_neg_eq : ∀{a : ℤ}, -a = a → a = 0 := @algebra.eq_zero_of_neg_eq _ _
+  definition abs : ℤ → ℤ := algebra.abs
+  notation `|` a `|` := abs a
+  theorem abs_of_nonneg : ∀{a : ℤ}, a ≥ 0 → |a| = a := @algebra.abs_of_nonneg _ _
+  theorem abs_of_pos : ∀{a : ℤ}, a > 0 → |a| = a := @algebra.abs_of_pos _ _
+  theorem abs_of_neg : ∀{a : ℤ}, a < 0 → |a| = -a := @algebra.abs_of_neg _ _
+  theorem abs_zero : |0| = 0 := algebra.abs_zero
+  theorem abs_of_nonpos : ∀{a : ℤ}, a ≤ 0 → |a| = -a := @algebra.abs_of_nonpos _ _
+  theorem abs_neg : ∀a : ℤ, |-a| = |a| := algebra.abs_neg
+  theorem abs_nonneg : ∀a : ℤ, | a | ≥ 0 := algebra.abs_nonneg
+  theorem abs_abs : ∀a : ℤ, | |a| | = |a| := algebra.abs_abs
+  theorem le_abs_self : ∀a : ℤ, a ≤ |a| := algebra.le_abs_self
+  theorem neg_le_abs_self : ∀a : ℤ, -a ≤ |a| := algebra.neg_le_abs_self
+  theorem eq_zero_of_abs_eq_zero : ∀{a : ℤ}, |a| = 0 → a = 0 := @algebra.eq_zero_of_abs_eq_zero _ _
+  theorem abs_eq_zero_iff_eq_zero : ∀a : ℤ, |a| = 0 ↔ a = 0 := algebra.abs_eq_zero_iff_eq_zero
+  theorem abs_pos_of_pos : ∀{a : ℤ}, a > 0 → |a| > 0 := @algebra.abs_pos_of_pos _ _
+  theorem abs_pos_of_neg : ∀{a : ℤ}, a < 0 → |a| > 0 := @algebra.abs_pos_of_neg _ _
+  theorem abs_pos_of_ne_zero : ∀a : ℤ, a ≠ 0 → |a| > 0 := @algebra.abs_pos_of_ne_zero _ _
+  theorem abs_sub : ∀a b : ℤ, |a - b| = |b - a| := algebra.abs_sub
+  theorem abs.by_cases : ∀{P : ℤ → Prop}, ∀{a : ℤ}, P a → P (-a) → P (|a|) :=
+    @algebra.abs.by_cases _ _
+  theorem abs_le_of_le_of_neg_le : ∀{a b : ℤ}, a ≤ b → -a ≤ b → |a| ≤ b :=
+    @algebra.abs_le_of_le_of_neg_le _ _
+  theorem abs_lt_of_lt_of_neg_lt : ∀{a b : ℤ}, a < b → -a < b → |a| < b :=
+    @algebra.abs_lt_of_lt_of_neg_lt _ _
+  theorem abs_add_le_abs_add_abs : ∀a b : ℤ, |a + b| ≤ |a| + |b| :=
+    algebra.abs_add_le_abs_add_abs
+  theorem abs_sub_abs_le_abs_sub : ∀a b : ℤ, |a| - |b| ≤ |a - b| :=
+    algebra.abs_sub_abs_le_abs_sub
+
   theorem mul_le_mul_of_nonneg_left : ∀{a b c : ℤ}, a ≤ b → 0 ≤ c → c * a ≤ c * b :=
     @algebra.mul_le_mul_of_nonneg_left _ _
   theorem mul_le_mul_of_nonneg_right : ∀{a b c : ℤ}, a ≤ b → 0 ≤ c → a * c ≤ b * c :=
@@ -465,6 +502,27 @@ section port_algebra
   theorem zero_lt_one : #int 0 < 1 := trivial
   theorem pos_and_pos_or_neg_and_neg_of_mul_pos : ∀{a b : ℤ}, a * b > 0 →
     (a > 0 ∧ b > 0) ∨ (a < 0 ∧ b < 0) := @algebra.pos_and_pos_or_neg_and_neg_of_mul_pos _ _
+
+  definition sign : ∀a : ℤ, ℤ := algebra.sign
+  theorem sign_of_neg : ∀{a : ℤ}, a < 0 → sign a = -1 := @algebra.sign_of_neg _ _
+  theorem sign_zero : sign 0 = 0 := algebra.sign_zero
+  theorem sign_of_pos : ∀{a : ℤ}, a > 0 → sign a = 1 := @algebra.sign_of_pos _ _
+  theorem sign_one : sign 1 = 1 := algebra.sign_one
+  theorem sign_neg_one : sign (-1) = -1 := algebra.sign_neg_one
+  theorem sign_sign : ∀a : ℤ, sign (sign a) = sign a := algebra.sign_sign
+  theorem pos_of_sign_eq_one : ∀{a : ℤ}, sign a = 1 → a > 0 := @algebra.pos_of_sign_eq_one _ _
+  theorem eq_zero_of_sign_eq_zero : ∀{a : ℤ}, sign a = 0 → a = 0 :=
+    @algebra.eq_zero_of_sign_eq_zero _ _
+  theorem neg_of_sign_eq_neg_one : ∀{a : ℤ}, sign a = -1 → a < 0 :=
+    @algebra.neg_of_sign_eq_neg_one _ _
+  theorem sign_neg : ∀a : ℤ, sign (-a) = -(sign a) := algebra.sign_neg
+  theorem sign_mul : ∀a b : ℤ, sign (a * b) = sign a * sign b := algebra.sign_mul
+  theorem abs_eq_sign_mul : ∀a : ℤ, |a| = sign a * a := algebra.abs_eq_sign_mul
+  theorem eq_sign_mul_abs : ∀a : ℤ, a = sign a * |a| := algebra.eq_sign_mul_abs
+  theorem abs_dvd_iff_dvd : ∀a b : ℤ, |a| | b ↔ a | b := algebra.abs_dvd_iff_dvd
+  theorem dvd_abs_iff : ∀a b : ℤ, a | |b| ↔ a | b := algebra.dvd_abs_iff
+  theorem abs_mul : ∀a b : ℤ, |a * b| = |a| * |b| := algebra.abs_mul
+  theorem abs_mul_self : ∀a : ℤ, |a| * |a| = a * a := algebra.abs_mul_self
 end port_algebra
 
 /- more facts specific to int -/