From a023128738345ce3cf486cbb5b8d799e99f749ea Mon Sep 17 00:00:00 2001
From: Leonardo de Moura <leonardo@microsoft.com>
Date: Tue, 10 Apr 2018 13:11:40 -0700
Subject: [PATCH] chore(*): reduce corelib

---
 library/init/data/int/bitwise.lean |  73 -------
 library/init/data/int/default.lean |   1 -
 library/init/data/nat/bitwise.lean | 314 -----------------------------
 library/init/data/nat/default.lean |   1 -
 4 files changed, 389 deletions(-)
 delete mode 100644 library/init/data/int/bitwise.lean
 delete mode 100644 library/init/data/nat/bitwise.lean

diff --git a/library/init/data/int/bitwise.lean b/library/init/data/int/bitwise.lean
deleted file mode 100644
index 3cb979f62d..0000000000
--- a/library/init/data/int/bitwise.lean
+++ /dev/null
@@ -1,73 +0,0 @@
-/-
-Copyright (c) 2017 Mario Carneiro. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Author: Mario Carneiro
--/
-
-prelude
-import init.data.int.basic init.data.nat.bitwise
-
-universe u
-
-namespace int
-
-  def div2 : ℤ → ℤ
-  | (of_nat n) := n.div2
-  | -[1+ n] := -[1+ n.div2]
-
-  def bodd : ℤ → bool
-  | (of_nat n) := n.bodd
-  | -[1+ n] := bnot (n.bodd)
-
-  def bit (b : bool) : ℤ → ℤ := cond b bit1 bit0
-
-  def test_bit : ℤ → ℕ → bool
-  | (m : ℕ) n := nat.test_bit m n
-  | -[1+ m] n := bnot (nat.test_bit m n)
-
-  def nat_bitwise (f : bool → bool → bool) (m n : ℕ) : ℤ :=
-  cond (f ff ff) -[1+ nat.bitwise (λx y, bnot (f x y)) m n] (nat.bitwise f m n)
-
-  def bitwise (f : bool → bool → bool) : ℤ → ℤ → ℤ
-  | (m : ℕ) (n : ℕ) := nat_bitwise f m n
-  | (m : ℕ) -[1+ n] := nat_bitwise (λ x y, f x (bnot y)) m n
-  | -[1+ m] (n : ℕ) := nat_bitwise (λ x y, f (bnot x) y) m n
-  | -[1+ m] -[1+ n] := nat_bitwise (λ x y, f (bnot x) (bnot y)) m n
-
-  def lnot : ℤ → ℤ
-  | (m : ℕ) := -[1+ m]
-  | -[1+ m] := m
-
-  def lor : ℤ → ℤ → ℤ
-  | (m : ℕ) (n : ℕ) := nat.lor m n
-  | (m : ℕ) -[1+ n] := -[1+ nat.ldiff n m]
-  | -[1+ m] (n : ℕ) := -[1+ nat.ldiff m n]
-  | -[1+ m] -[1+ n] := -[1+ nat.land m n]
-
-  def land : ℤ → ℤ → ℤ
-  | (m : ℕ) (n : ℕ) := nat.land m n
-  | (m : ℕ) -[1+ n] := nat.ldiff m n
-  | -[1+ m] (n : ℕ) := nat.ldiff n m
-  | -[1+ m] -[1+ n] := -[1+ nat.lor m n]
-
-  def ldiff : ℤ → ℤ → ℤ
-  | (m : ℕ) (n : ℕ) := nat.ldiff m n
-  | (m : ℕ) -[1+ n] := nat.land m n
-  | -[1+ m] (n : ℕ) := -[1+ nat.lor m n]
-  | -[1+ m] -[1+ n] := nat.ldiff n m
-
-  def lxor : ℤ → ℤ → ℤ
-  | (m : ℕ) (n : ℕ) := nat.lxor m n
-  | (m : ℕ) -[1+ n] := -[1+ nat.lxor m n]
-  | -[1+ m] (n : ℕ) := -[1+ nat.lxor m n]
-  | -[1+ m] -[1+ n] := nat.lxor m n
-
-  def shiftl : ℤ → ℤ → ℤ
-  | (m : ℕ) (n : ℕ) := nat.shiftl m n
-  | (m : ℕ) -[1+ n] := nat.shiftr m (nat.succ n)
-  | -[1+ m] (n : ℕ) := -[1+ nat.shiftl' tt m n]
-  | -[1+ m] -[1+ n] := -[1+ nat.shiftr m (nat.succ n)]
-
-  def shiftr (m n : ℤ) : ℤ := shiftl m (-n)
-
-end int
diff --git a/library/init/data/int/default.lean b/library/init/data/int/default.lean
index 931f9d77ce..07e5cc3227 100644
--- a/library/init/data/int/default.lean
+++ b/library/init/data/int/default.lean
@@ -5,4 +5,3 @@ Authors: Leonardo de Moura
 -/
 prelude
 import init.data.int.basic init.data.int.order init.data.int.comp_lemmas
-import init.data.int.bitwise
diff --git a/library/init/data/nat/bitwise.lean b/library/init/data/nat/bitwise.lean
deleted file mode 100644
index 6a3072d33c..0000000000
--- a/library/init/data/nat/bitwise.lean
+++ /dev/null
@@ -1,314 +0,0 @@
-/-
-Copyright (c) 2017 Microsoft Corporation. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Author: Mario Carneiro
--/
-
-prelude
-import init.data.nat.lemmas init.meta.well_founded_tactics
-
-universe u
-
-namespace nat
-
-def bodd_div2 : ℕ → bool × ℕ
-| 0        := (ff, 0)
-| (succ n) :=
-    match bodd_div2 n with
-    | (ff, m) := (tt, m)
-    | (tt, m) := (ff, succ m)
-    end
-
-def div2 (n : ℕ) : ℕ := (bodd_div2 n).2
-
-def bodd (n : ℕ) : bool := (bodd_div2 n).1
-
-@[simp] lemma bodd_zero : bodd 0 = ff := rfl
-@[simp] lemma bodd_one : bodd 1 = tt := rfl
-@[simp] lemma bodd_two : bodd 2 = ff := rfl
-
-@[simp] lemma bodd_succ (n : ℕ) : bodd (succ n) = bnot (bodd n) :=
-by unfold bodd bodd_div2; cases bodd_div2 n; cases fst; refl
-
-@[simp] lemma bodd_add (m n : ℕ) : bodd (m + n) = bxor (bodd m) (bodd n) :=
-begin
-    induction n with n IH,
-    { simp, cases bodd m; refl },
-    { simp [IH], cases bodd m; cases bodd n; refl }
-end
-
-@[simp] lemma bodd_mul (m n : ℕ) : bodd (m * n) = bodd m && bodd n :=
-begin
-    induction n with n IH,
-    { simp, cases bodd m; refl },
-    { simp [mul_succ, IH], cases bodd m; cases bodd n; refl }
-end
-
-lemma mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 :=
-begin
-    have := congr_arg bodd (mod_add_div n 2),
-    simp [bnot] at this,
-    rw [show ∀ b, ff && b = ff, by intros; cases b; refl,
-        show ∀ b, bxor b ff = b, by intros; cases b; refl] at this,
-    rw [← this],
-    cases mod_two_eq_zero_or_one n with h h; rw h; refl
-end
-
-@[simp] lemma div2_zero : div2 0 = 0 := rfl
-@[simp] lemma div2_one : div2 1 = 0 := rfl
-@[simp] lemma div2_two : div2 2 = 1 := rfl
-
-@[simp] lemma div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) :=
-by unfold bodd div2 bodd_div2; cases bodd_div2 n; cases fst; refl
-
-local attribute [simp] add_comm add_assoc add_left_comm mul_comm mul_assoc mul_left_comm
-
-theorem bodd_add_div2 : ∀ n, cond (bodd n) 1 0 + 2 * div2 n = n
-| 0        := rfl
-| (succ n) := begin
-    simp,
-    refine eq.trans _ (congr_arg succ (bodd_add_div2 n)),
-    cases bodd n; simp [cond, bnot],
-    { rw add_comm; refl },
-    { rw [succ_mul, add_comm 1] }
-end
-
-theorem div2_val (n) : div2 n = n / 2 :=
-by refine eq_of_mul_eq_mul_left dec_trivial
-      (nat.add_left_cancel (eq.trans _ (mod_add_div n 2).symm));
-     rw [mod_two_of_bodd, bodd_add_div2]
-
-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 bit_decomp (n : nat) : bit (bodd n) (div2 n) = n :=
-(bit_val _ _).trans $ (add_comm _ _).trans $ bodd_add_div2 _
-
-def bit_cases_on {C : nat → Sort u} (n) (h : ∀ b n, C (bit b n)) : C n :=
-by rw [← bit_decomp n]; apply h
-
-@[simp] lemma bit_zero : bit ff 0 = 0 := rfl
-
-def shiftl' (b : bool) (m : ℕ) : ℕ → ℕ
-| 0     := m
-| (n+1) := bit b (shiftl' n)
-
-def shiftl : ℕ → ℕ → ℕ := shiftl' ff
-
-@[simp] theorem shiftl_zero (m) : shiftl m 0 = m := rfl
-@[simp] theorem shiftl_succ (m n) : shiftl m (n + 1) = bit0 (shiftl m n) := rfl
-
-def shiftr : ℕ → ℕ → ℕ
-| m 0     := m
-| m (n+1) := div2 (shiftr m n)
-
-def test_bit (m n : ℕ) : bool := bodd (shiftr m n)
-
-def binary_rec {C : nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit b n)) : Π n, C n
-| n := if n0 : n = 0 then by rw n0; exact z else let n' := div2 n in
-    have n' < n, begin
-      change div2 n < n, rw div2_val,
-      apply (div_lt_iff_lt_mul _ _ (succ_pos 1)).2,
-      have := nat.mul_lt_mul_of_pos_left (lt_succ_self 1)
-        (lt_of_le_of_ne (zero_le _) (ne.symm n0)),
-      rwa mul_one at this
-    end,
-    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 0 (λ_ _, succ)
-
-def bits : ℕ → list bool := binary_rec [] (λb _ IH, b :: IH)
-
-def bitwise (f : bool → bool → bool) : ℕ → ℕ → ℕ :=
-binary_rec
-    (λn, cond (f ff tt) n 0)
-    (λa m Ia, binary_rec
-      (cond (f tt ff) (bit a m) 0)
-      (λb n _, bit (f a b) (Ia n)))
-
-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
-
-@[simp] lemma binary_rec_zero {C : nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit b n)) :
-    binary_rec z f 0 = z :=
-by {rw [binary_rec], refl}
-
-/- bitwise ops -/
-
-lemma bodd_bit (b n) : bodd (bit b n) = b :=
-by rw bit_val; simp; cases b; cases bodd n; refl
-
-lemma div2_bit (b n) : div2 (bit b n) = n :=
-by rw [bit_val, div2_val, add_comm, add_mul_div_left, div_eq_of_lt, zero_add];
-   cases b; exact dec_trivial
-
-lemma shiftl'_add (b m n) : ∀ k, shiftl' b m (n + k) = shiftl' b (shiftl' b m n) k
-| 0     := rfl
-| (k+1) := congr_arg (bit b) (shiftl'_add k)
-
-lemma shiftl_add : ∀ m n k, shiftl m (n + k) = shiftl (shiftl m n) k := shiftl'_add _
-
-lemma shiftr_add (m n) : ∀ k, shiftr m (n + k) = shiftr (shiftr m n) k
-| 0     := rfl
-| (k+1) := congr_arg div2 (shiftr_add k)
-
-lemma shiftl'_sub (b m) : ∀ {n k}, k ≤ n → shiftl' b m (n - k) = shiftr (shiftl' b m n) k
-| n     0     h := rfl
-| (n+1) (k+1) h := begin
-  simp [shiftl'], rw [add_comm, shiftr_add],
-  simp [shiftr, div2_bit],
-  apply shiftl'_sub (nat.le_of_succ_le_succ h)
-end
-
-lemma shiftl_sub : ∀ m {n k}, k ≤ n → shiftl m (n - k) = shiftr (shiftl m n) k := shiftl'_sub _
-
-lemma shiftl_eq_mul_pow (m) : ∀ n, shiftl m n = m * 2 ^ n
-| 0     := (mul_one _).symm
-| (k+1) := show bit0 (shiftl m k) = m * (2^k * 2), by rw [bit0_val, shiftl_eq_mul_pow]; simp
-
-lemma shiftl'_tt_eq_mul_pow (m) : ∀ n, shiftl' tt m n + 1 = (m + 1) * 2 ^ n
-| 0     := by simp [shiftl, shiftl']
-| (k+1) := begin
-  change bit1 (shiftl' tt m k) + 1 = (m + 1) * (2^k * 2),
-  rw bit1_val,
-  change 2 * (shiftl' tt m k + 1) = _,
-  rw shiftl'_tt_eq_mul_pow; simp
-end
-
-lemma one_shiftl (n) : shiftl 1 n = 2 ^ n :=
-(shiftl_eq_mul_pow _ _).trans (one_mul _)
-
-@[simp] lemma zero_shiftl (n) : shiftl 0 n = 0 :=
-(shiftl_eq_mul_pow _ _).trans (zero_mul _)
-
-lemma shiftr_eq_div_pow (m) : ∀ n, shiftr m n = m / 2 ^ n
-| 0     := (nat.div_one _).symm
-| (k+1) := (congr_arg div2 (shiftr_eq_div_pow k)).trans $
-           by rw [div2_val, nat.div_div_eq_div_mul]; refl
-
-@[simp] lemma zero_shiftr (n) : shiftr 0 n = 0 :=
-(shiftr_eq_div_pow _ _).trans (nat.zero_div _)
-
-@[simp] lemma test_bit_zero (b n) : test_bit (bit b n) 0 = b := bodd_bit _ _
-
-lemma 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 dsimp [shiftr]; rw div2_bit,
-by rw [← shiftr_add, add_comm] at this; exact this
-
-lemma binary_rec_eq {C : nat → Sort u} {z : C 0} {f : ∀ b n, C n → C (bit b n)}
-  (h : f ff 0 z = z) (b n) :
-  binary_rec z f (bit b n) = f b n (binary_rec z f n) :=
-begin
-  rw [binary_rec],
-  with_cases { by_cases bit b n = 0 },
-  case pos : h' {
-    simp [dif_pos h'],
-    generalize : binary_rec._main._pack._proof_1 (bit b n) h' = e,
-    revert e,
-    have bf := bodd_bit b n,
-    have n0 := div2_bit b n,
-    rw h' at bf n0,
-    simp at bf n0,
-    rw [← bf, ← n0, binary_rec_zero],
-    intros, exact h.symm },
-  case neg : h' {
-    simp [dif_neg h'],
-    generalize : binary_rec._main._pack._proof_2 (bit b n) = e,
-    revert e,
-    rw [bodd_bit, div2_bit],
-    intros, refl}
-end
-
-lemma bitwise_bit_aux {f : bool → bool → bool} (h : f ff ff = ff) :
-  @binary_rec (λ_, ℕ)
-    (cond (f tt ff) (bit ff 0) 0)
-    (λ b n _, bit (f ff b) (cond (f ff tt) n 0)) =
-  λ (n : ℕ), cond (f ff tt) n 0 :=
-begin
-  funext n,
-  apply bit_cases_on n, intros b n, rw [binary_rec_eq],
-  { cases b; try {rw h}; induction fft : f ff tt; simp [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
-
-@[simp] 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]
-
-@[simp] 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
-
-@[simp] lemma bitwise_zero (f : bool → bool → bool) :
-  bitwise f 0 0 = 0 :=
-by rw bitwise_zero_left; cases f ff tt; refl
-
-@[simp] 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],
-  { induction ftf : f tt ff; 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
-
-theorem bitwise_swap {f : bool → bool → bool} (h : f ff ff = ff) :
-  bitwise (function.swap f) = function.swap (bitwise f) :=
-begin
-  funext m n, revert n,
-  dsimp [function.swap],
-  apply binary_rec _ (λ a m' IH, _) m; intro n,
-  { rw [bitwise_zero_left, bitwise_zero_right], exact h },
-  apply bit_cases_on n; intros b n',
-  rw [bitwise_bit, bitwise_bit, IH]; exact h
-end
-
-@[simp] lemma lor_bit : ∀ (a m b n),
-  lor (bit a m) (bit b n) = bit (a || b) (lor m n) := bitwise_bit rfl
-@[simp] lemma land_bit : ∀ (a m b n),
-  land (bit a m) (bit b n) = bit (a && b) (land m n) := bitwise_bit rfl
-@[simp] lemma ldiff_bit : ∀ (a m b n),
-  ldiff (bit a m) (bit b n) = bit (a && bnot b) (ldiff m n) := bitwise_bit rfl
-@[simp] lemma lxor_bit : ∀ (a m b n),
-  lxor (bit a m) (bit b n) = bit (bxor a b) (lxor m n) := bitwise_bit rfl
-
-@[simp] lemma 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
-
-@[simp] 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
-@[simp] 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
-@[simp] 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
-@[simp] 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/default.lean b/library/init/data/nat/default.lean
index 00050a9423..ac93e59093 100644
--- a/library/init/data/nat/default.lean
+++ b/library/init/data/nat/default.lean
@@ -5,4 +5,3 @@ Authors: Leonardo de Moura
 -/
 prelude
 import init.data.nat.basic init.data.nat.div init.data.nat.lemmas
-       init.data.nat.bitwise