From 41b928a54622b4deba0580776a5cfb1ec8250cec Mon Sep 17 00:00:00 2001
From: Mario Carneiro <di.gama@gmail.com>
Date: Wed, 31 May 2017 03:47:16 -0400
Subject: [PATCH] feat(init/data/nat/bitwise): properties of bitwise ops

---
 library/init/data/nat/bitwise.lean | 167 ++++++++++++++++++++++++++++-
 library/init/data/nat/lemmas.lean  |  17 +++
 library/library.md                 |   2 +-
 3 files changed, 180 insertions(+), 6 deletions(-)

diff --git a/library/init/data/nat/bitwise.lean b/library/init/data/nat/bitwise.lean
index d943049c3d..ad354b42f9 100644
--- a/library/init/data/nat/bitwise.lean
+++ b/library/init/data/nat/bitwise.lean
@@ -23,30 +23,187 @@ namespace nat
 
   def test_bit (m n : ℕ) : bool := bodd (shiftr m n)
 
-  def binary_rec {α : Type u} (f : bool → ℕ → α → α) (z : α) : ℕ → α
-  | n := if n0 : n = 0 then z else let n' := shiftr n 1 in
+  def bit (b : bool) : ℕ → ℕ := cond b bit1 bit0
+
+  lemma bit0_val (n : nat) : bit0 n = 2 * n := (two_mul _).symm
+
+  lemma bit1_val (n : nat) : bit1 n = 2 * n + 1 := congr_arg succ (bit0_val _)
+
+  lemma bit_val (b n) : bit b n = 2 * n + cond b 1 0 :=
+  by { cases b, apply bit0_val, apply bit1_val }
+
+  lemma mod_two_of_bodd (n : nat) : n % 2 = cond (bodd n) 1 0 :=
+  match by apply_instance : ∀ d, n % 2 = cond (@to_bool (n % 2 = 1) d) 1 0 with
+  | is_true  h := h
+  | is_false h := (mod_two_eq_zero_or_one _).resolve_right h
+  end
+
+  lemma bit_decomp (n : nat) : bit (bodd n) (shiftr n 1) = n :=
+  (bit_val _ _).trans $ (add_comm _ _).trans $
+  eq.trans (by rw mod_two_of_bodd; refl) (mod_add_div n 2)
+
+  lemma bit_cases_on {C : nat → Sort u} (n) (h : ∀ b n, C (bit b n)) : C n :=
+  by rw -bit_decomp n; apply h
+
+  lemma bodd_bit (b n) : bodd (bit b n) = b :=
+  begin
+    rw bit_val, dsimp [bodd],
+    rw [add_comm, add_mul_mod_self_left, mod_eq_of_lt];
+    cases b; exact dec_trivial
+  end
+
+  lemma shiftr1_bit (b n) : shiftr (bit b n) 1 = n :=
+  begin
+    rw bit_val, dsimp [shiftr],
+    rw [add_comm, add_mul_div_left, div_eq_of_lt, zero_add];
+    cases b; exact dec_trivial
+  end
+
+  def shiftl_add (m n) : ∀ k, shiftl m (n + k) = shiftl (shiftl m n) k
+  | 0     := rfl
+  | (k+1) := congr_arg ((*) 2) (shiftl_add k)
+
+  def shiftr_add (m n) : ∀ k, shiftr m (n + k) = shiftr (shiftr m n) k
+  | 0     := rfl
+  | (k+1) := congr_arg (/ 2) (shiftr_add k)
+
+  def shiftl_eq_mul_pow (m) : ∀ n, shiftl m n = m * 2 ^ n
+  | 0     := (mul_one _).symm
+  | (k+1) := (congr_arg ((*) 2) (shiftl_eq_mul_pow k)).trans $ by simp [pow_succ]
+
+  def one_shiftl (n) : shiftl 1 n = 2 ^ n :=
+  (shiftl_eq_mul_pow _ _).trans (one_mul _)
+
+  def zero_shiftl (n) : shiftl 0 n = 0 :=
+  (shiftl_eq_mul_pow _ _).trans (zero_mul _)
+
+  def shiftr_eq_div_pow (m) : ∀ n, shiftr m n = m / 2 ^ n
+  | 0     := (nat.div_one _).symm
+  | (k+1) := (congr_arg (/ 2) (shiftr_eq_div_pow k)).trans $
+             by dsimp; rw [nat.div_div_eq_div_mul]; refl
+
+  def zero_shiftr (n) : shiftr 0 n = 0 :=
+  (shiftr_eq_div_pow _ _).trans (nat.zero_div _)
+
+  def test_bit_zero (b n) : test_bit (bit b n) 0 = b := bodd_bit _ _
+
+  def test_bit_succ (m b n) : test_bit (bit b n) (succ m) = test_bit n m :=
+  have bodd (shiftr (shiftr (bit b n) 1) m) = bodd (shiftr n m), by rw shiftr1_bit,
+  by rw [-shiftr_add, add_comm] at this; exact this
+
+  def binary_rec {C : nat → Sort u} (f : ∀ b n, C n → C (bit b n)) (z : C 0) : Π n, C n
+  | n := if n0 : n = 0 then by rw n0; exact z else let n' := shiftr n 1 in
     have n' < n, from (div_lt_iff_lt_mul _ _ dec_trivial).2 $
     by note := nat.mul_lt_mul_of_pos_left (dec_trivial : 1 < 2)
          (lt_of_le_of_ne (zero_le _) (ne.symm n0));
        rwa mul_one at this,
+    by rw [-show bit (bodd n) n' = n, from bit_decomp n]; exact 
     f (bodd n) n' (binary_rec n')
 
   def size : ℕ → ℕ := binary_rec (λ_ _, succ) 0
 
   def bits : ℕ → list bool := binary_rec (λb _ IH, b :: IH) []
 
-  def bit (b : bool) : ℕ → ℕ := cond b bit1 bit0
-
   def bitwise (f : bool → bool → bool) : ℕ → ℕ → ℕ :=
   binary_rec
     (λa m Ia, binary_rec
       (λb n _, bit (f a b) (Ia n))
       (cond (f tt ff) (bit a m) 0))
-    (λb, cond (f ff tt) b 0)
+    (λn, cond (f ff tt) n 0)
 
   def lor   : ℕ → ℕ → ℕ := bitwise bor
   def land  : ℕ → ℕ → ℕ := bitwise band
   def ldiff : ℕ → ℕ → ℕ := bitwise (λ a b, a && bnot b)
   def lxor  : ℕ → ℕ → ℕ := bitwise bxor
 
+  set_option type_context.unfold_lemmas true
+  lemma binary_rec_eq {C : nat → Sort u} {f : ∀ b n, C n → C (bit b n)} {z}
+    (h : f ff 0 z = z) (b n) :
+    binary_rec f z (bit b n) = f b n (binary_rec f z n) :=
+  begin
+    rw [binary_rec.equations._eqn_1],
+    cases (by apply_instance : decidable (bit b n = 0)) with b0 b0; dsimp [dite],
+    { generalize (binary_rec._main._pack._proof_2 (bit b n)) e,
+      rw [bodd_bit, shiftr1_bit], intro e, refl },
+    { generalize (binary_rec._main._pack._proof_1 (bit b n) b0) e,
+      note bf := bodd_bit b n, note n0 := shiftr1_bit b n,
+      rw b0 at bf n0, rw [-show ff = b, from bf, -show 0 = n, from n0], intro e,
+      exact h.symm },
+  end
+  
+  lemma binary_rec_zero {C : nat → Sort u} (f : ∀ b n, C n → C (bit b n)) (z) :
+    binary_rec f z 0 = z := rfl
+
+  lemma bitwise_bit_aux {f : bool → bool → bool} (h : f ff ff = ff) :
+    @binary_rec (λ_, ℕ)
+      (λ b n _, bit (f ff b) (cond (f ff tt) n 0))
+      (cond (f tt ff) (bit ff 0) 0) =
+    λ (n : ℕ), cond (f ff tt) n 0 :=
+  begin
+    apply funext, intro n,
+    apply bit_cases_on n, intros b n, rw [binary_rec_eq],
+    { cases b; try {rw h}; ginduction f ff tt with fft; dsimp [cond]; refl },
+    { rw [h, show cond (f ff tt) 0 0 = 0, by cases f ff tt; refl,
+             show cond (f tt ff) (bit ff 0) 0 = 0, by cases f tt ff; refl]; refl }
+  end
+
+  lemma bitwise_zero_left (f : bool → bool → bool) (n) :
+    bitwise f 0 n = cond (f ff tt) n 0 :=
+  by unfold bitwise; rw [binary_rec_zero]
+
+  lemma bitwise_zero_right (f : bool → bool → bool) (h : f ff ff = ff) (m) :
+    bitwise f m 0 = cond (f tt ff) m 0 :=
+  by unfold bitwise; apply bit_cases_on m; intros;
+     rw [binary_rec_eq, binary_rec_zero]; exact bitwise_bit_aux h
+
+  lemma bitwise_zero (f : bool → bool → bool) :
+    bitwise f 0 0 = 0 :=
+  by rw bitwise_zero_left; cases f ff tt; refl
+
+  lemma bitwise_bit {f : bool → bool → bool} (h : f ff ff = ff) (a m b n) :
+    bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n) :=
+  begin
+    unfold bitwise,
+    rw [binary_rec_eq, binary_rec_eq],
+    { ginduction f tt ff with ftf; dsimp [cond],
+      rw [show f a ff = ff, by cases a; assumption],
+      apply @congr_arg _ _ _ 0 (bit ff), tactic.swap,
+      rw [show f a ff = a, by cases a; assumption],
+      apply congr_arg (bit a),
+      all_goals {
+        apply bit_cases_on m, intros a m,
+        rw [binary_rec_eq, binary_rec_zero],
+        rw [-bitwise_bit_aux h, ftf], refl } },
+    { exact bitwise_bit_aux h }
+  end
+
+  lemma lor_bit : ∀ (a m b n),
+    lor (bit a m) (bit b n) = bit (a || b) (lor m n) := bitwise_bit rfl
+  lemma land_bit : ∀ (a m b n),
+    land (bit a m) (bit b n) = bit (a && b) (land m n) := bitwise_bit rfl
+  lemma ldiff_bit : ∀ (a m b n),
+    ldiff (bit a m) (bit b n) = bit (a && bnot b) (ldiff m n) := bitwise_bit rfl
+  lemma lxor_bit : ∀ (a m b n),
+    lxor (bit a m) (bit b n) = bit (bxor a b) (lxor m n) := bitwise_bit rfl
+
+  def test_bit_bitwise {f : bool → bool → bool} (h : f ff ff = ff) (m n k) :
+    test_bit (bitwise f m n) k = f (test_bit m k) (test_bit n k) :=
+  begin
+    revert m n; induction k with k IH; intros m n;
+    apply bit_cases_on m; intros a m';
+    apply bit_cases_on n; intros b n';
+    rw bitwise_bit h,
+    { simp [test_bit_zero] },
+    { simp [test_bit_succ, IH] }
+  end
+
+  lemma test_bit_lor : ∀ (m n k),
+    test_bit (lor m n) k = test_bit m k || test_bit n k := test_bit_bitwise rfl
+  lemma test_bit_land : ∀ (m n k),
+    test_bit (land m n) k = test_bit m k && test_bit n k := test_bit_bitwise rfl
+  lemma test_bit_ldiff : ∀ (m n k),
+    test_bit (ldiff m n) k = test_bit m k && bnot (test_bit n k) := test_bit_bitwise rfl
+  lemma test_bit_lxor : ∀ (m n k),
+    test_bit (lxor m n) k = bxor (test_bit m k) (test_bit n k) := test_bit_bitwise rfl
+
 end nat
diff --git a/library/init/data/nat/lemmas.lean b/library/init/data/nat/lemmas.lean
index 4a7fb9e525..289e49499d 100644
--- a/library/init/data/nat/lemmas.lean
+++ b/library/init/data/nat/lemmas.lean
@@ -1184,6 +1184,23 @@ protected theorem div_eq_of_eq_mul_right {m n k : ℕ} (H1 : n > 0) (H2 : m = n
   m / n = k :=
 by rw [H2, nat.mul_div_cancel_left _ H1]
 
+protected theorem div_div_eq_div_mul (m n k : ℕ) : m / n / k = m / (n * k) :=
+begin
+  cases eq_zero_or_pos k with k0 kpos, {rw [k0, mul_zero, nat.div_zero, nat.div_zero]},
+  cases eq_zero_or_pos n with n0 npos, {rw [n0, zero_mul, nat.div_zero, nat.zero_div]},
+  apply le_antisymm,
+  { apply (le_div_iff_mul_le _ _ (mul_pos npos kpos)).2,
+    rw [mul_comm n k, -mul_assoc],
+    apply (le_div_iff_mul_le _ _ npos).1,
+    apply (le_div_iff_mul_le _ _ kpos).1,
+    refl },
+  { apply (le_div_iff_mul_le _ _ kpos).2,
+    apply (le_div_iff_mul_le _ _ npos).2,
+    rw [mul_assoc, mul_comm n k],
+    apply (le_div_iff_mul_le _ _ (mul_pos kpos npos)).1,
+    refl }
+end
+
 /- pow -/
 
 @[simp] lemma pow_one (b : ℕ) : b^1 = b := by simp [pow_succ]
diff --git a/library/library.md b/library/library.md
index ff718b80cb..bec3226bf1 100644
--- a/library/library.md
+++ b/library/library.md
@@ -44,6 +44,6 @@ sparingly. For example:
 * Hilbert choice is used to define the multiplicative inverse on the reals, and
   also to define function inverses classically.
 
-You can use `print axioms foo` to see which axioms `foo` depends
+You can use `#print axioms foo` to see which axioms `foo` depends
 on. Many of the theories in the `theories` folder are unreservedly
 classical.