Squashed commit of the following:

Batteries/Data/Nat.lean

@@ -1,3 +1,4 @@
 import Batteries.Data.Nat.Basic
+import Batteries.Data.Nat.BinaryRec
 import Batteries.Data.Nat.Gcd
 import Batteries.Data.Nat.Lemmas

Batteries/Data/Nat/BinaryRec.lean

@@ -0,0 +1,134 @@
+/-
+Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Praneeth Kolichala, Yuyang Zhao
+-/
+
+/-!
+# Binary recursion on `Nat`
+
+This file defines binary recursion on `Nat`.
+
+## Main results
+* `Nat.binaryRec`: A recursion principle for `bit` representations of natural numbers.
+* `Nat.binaryRec'`: The same as `binaryRec`, but the induction step can assume that if `n=0`,
+  the bit being appended is `true`.
+* `Nat.binaryRecFromOne`: The same as `binaryRec`, but special casing both 0 and 1 as base cases.
+-/
+
+namespace Nat
+
+/-- `bit b` appends the digit `b` to the little end of the binary representation of
+  its natural number input. -/
+def bit (b : Bool) (n : Nat) : Nat :=
+  cond b (n + n + 1) (n + n)
+
+theorem shiftRight_one (n) : n >>> 1 = n / 2 := rfl
+
+theorem bit_testBit_zero_shiftRight_one (n : Nat) : bit (n.testBit 0) (n >>> 1) = n := by
+  simp only [bit, testBit_zero]
+  cases mod_two_eq_zero_or_one n with | _ h =>
+    simpa [h, shiftRight_one] using Eq.trans (by simp [h, Nat.two_mul]) (Nat.div_add_mod n 2)
+
+theorem bit_eq_zero_iff {n : Nat} {b : Bool} : bit b n = 0 ↔ n = 0 ∧ b = false := by
+  cases n <;> cases b <;> simp [bit, ← Nat.add_assoc]
+
+/-- For a predicate `C : Nat → Sort u`, if instances can be
+  constructed for natural numbers of the form `bit b n`,
+  they can be constructed for any given natural number. -/
+@[inline]
+def bitCasesOn {C : Nat → Sort u} (n) (h : ∀ b n, C (bit b n)) : C n :=
+  -- `1 &&& n != 0` is faster than `n.testBit 0`. This may change when we have faster `testBit`.
+  let x := h (1 &&& n != 0) (n >>> 1)
+  -- `congrArg C _` is `rfl` in non-dependent case
+  congrArg C n.bit_testBit_zero_shiftRight_one ▸ x
+
+/-- A recursion principle for `bit` representations of natural numbers.
+  For a predicate `C : Nat → Sort u`, if instances can be
+  constructed for natural numbers of the form `bit b n`,
+  they can be constructed for all natural numbers. -/
+@[elab_as_elim, specialize]
+def binaryRec {C : Nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit b n)) (n : Nat) : C n :=
+  if n0 : n = 0 then congrArg C n0 ▸ z
+  else
+    let x := f (1 &&& n != 0) (n >>> 1) (binaryRec z f (n >>> 1))
+    congrArg C n.bit_testBit_zero_shiftRight_one ▸ x
+decreasing_by exact bitwise_rec_lemma n0
+
+/-- The same as `binaryRec`, but the induction step can assume that if `n=0`,
+  the bit being appended is `true`-/
+@[elab_as_elim, specialize]
+def binaryRec' {C : Nat → Sort u} (z : C 0)
+    (f : ∀ b n, (n = 0 → b = true) → C n → C (bit b n)) : ∀ n, C n :=
+  binaryRec z fun b n ih =>
+    if h : n = 0 → b = true then f b n h ih
+    else
+      have : bit b n = 0 := by
+        rw [bit_eq_zero_iff]
+        cases n <;> cases b <;> simp at h <;> simp [h]
+      congrArg C this ▸ z
+
+/-- The same as `binaryRec`, but special casing both 0 and 1 as base cases -/
+@[elab_as_elim, specialize]
+def binaryRecFromOne {C : Nat → Sort u} (z₀ : C 0) (z₁ : C 1)
+    (f : ∀ b n, n ≠ 0 → C n → C (bit b n)) : ∀ n, C n :=
+  binaryRec' z₀ fun b n h ih =>
+    if h' : n = 0 then
+      have : bit b n = bit true 0 := by
+        rw [h', h h']
+      congrArg C this ▸ z₁
+    else f b n h' ih
+
+theorem bit_val (b n) : bit b n = 2 * n + b.toNat := by
+  rw [Nat.mul_comm]
+  induction b with
+  | false => exact congrArg (· + n) n.zero_add.symm
+  | true => exact congrArg (· + n + 1) n.zero_add.symm
+
+@[simp]
+theorem bit_div_two (b n) : bit b n / 2 = n := by
+  rw [bit_val, Nat.add_comm, add_mul_div_left, div_eq_of_lt, Nat.zero_add]
+  · cases b <;> decide
+  · decide
+
+@[simp]
+theorem bit_mod_two (b n) : bit b n % 2 = b.toNat := by
+  cases b <;> simp [bit_val, mul_add_mod]
+
+@[simp] theorem bit_shiftRight_one (b n) : bit b n >>> 1 = n :=
+  bit_div_two b n
+
+theorem bit_testBit_zero (b n) : (bit b n).testBit 0 = b := by
+  simp
+
+@[simp]
+theorem bitCasesOn_bit {C : Nat → Sort u} (h : ∀ b n, C (bit b n)) (b : Bool) (n : Nat) :
+    bitCasesOn (bit b n) h = h b n := by
+  change congrArg C (bit b n).bit_testBit_zero_shiftRight_one ▸ h _ _ = h b n
+  generalize congrArg C (bit b n).bit_testBit_zero_shiftRight_one = e; revert e
+  rw [bit_testBit_zero, bit_shiftRight_one]
+  intros; rfl
+
+unseal binaryRec in
+@[simp]
+theorem binaryRec_zero {C : Nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit b n)) :
+    binaryRec z f 0 = z :=
+  rfl
+
+theorem binaryRec_eq {C : Nat → Sort u} {z : C 0} {f : ∀ b n, C n → C (bit b n)} (b n)
+    (h : f false 0 z = z ∨ (n = 0 → b = true)) :
+    binaryRec z f (bit b n) = f b n (binaryRec z f n) := by
+  by_cases h' : bit b n = 0
+  case pos =>
+    obtain ⟨rfl, rfl⟩ := bit_eq_zero_iff.mp h'
+    simp only [forall_const, or_false] at h
+    unfold binaryRec
+    exact h.symm
+  case neg =>
+    rw [binaryRec, dif_neg h']
+    change congrArg C (bit b n).bit_testBit_zero_shiftRight_one ▸ f _ _ _ = _
+    generalize congrArg C (bit b n).bit_testBit_zero_shiftRight_one = e; revert e
+    rw [bit_testBit_zero, bit_shiftRight_one]
+    intros; rfl
+
+end Nat