feat(data/nat/hyperoperation): Defined hyperoperation and added relat…

…ed lemmas (#18116)

Added the hyperoperation sequence as defined here: https://en.wikipedia.org/wiki/Hyperoperation
Proved main lemmas.

src/data/nat/hyperoperation.lean

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+/-
+Copyright (c) 2023 Mark Andrew Gerads. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Mark Andrew Gerads, Junyan Xu, Eric Wieser
+-/
+import tactic.ring
+import data.nat.parity
+
+/-!
+# Hyperoperation sequence
+
+This file defines the Hyperoperation sequence.
+`hyperoperation 0 m k = k + 1`
+`hyperoperation 1 m k = m + k`
+`hyperoperation 2 m k = m * k`
+`hyperoperation 3 m k = m ^ k`
+`hyperoperation (n + 3) m 0 = 1`
+`hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k)`
+
+## References
+
+* <https://en.wikipedia.org/wiki/Hyperoperation>
+
+## Tags
+
+hyperoperation
+-/
+
+/--
+Implementation of the hyperoperation sequence
+where `hyperoperation n m k` is the `n`th hyperoperation between `m` and `k`.
+-/
+def hyperoperation : ℕ → ℕ → ℕ → ℕ
+| 0 _ k := k + 1
+| 1 m 0 := m
+| 2 _ 0 := 0
+| (n + 3) _ 0 := 1
+| (n + 1) m (k + 1) := hyperoperation n m (hyperoperation (n + 1) m k)
+
+-- Basic hyperoperation lemmas
+
+@[simp] lemma hyperoperation_zero (m : ℕ) : hyperoperation 0 m = nat.succ :=
+funext $ λ k, by rw [hyperoperation, nat.succ_eq_add_one]
+
+lemma hyperoperation_ge_three_eq_one (n m : ℕ) : hyperoperation (n + 3) m 0 = 1 :=
+by rw hyperoperation
+
+lemma hyperoperation_recursion (n m k : ℕ) :
+  hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k) :=
+by obtain (_|_|_) := n; rw hyperoperation
+
+-- Interesting hyperoperation lemmas
+
+@[simp] lemma hyperoperation_one : hyperoperation 1 = (+) :=
+begin
+  ext m k,
+  induction k with bn bih,
+  { rw [nat_add_zero m, hyperoperation], },
+  { rw [hyperoperation_recursion, bih, hyperoperation_zero],
+    exact nat.add_assoc m bn 1, },
+end
+
+@[simp] lemma hyperoperation_two : hyperoperation 2 = (*) :=
+begin
+  ext m k,
+  induction k with bn bih,
+  { rw hyperoperation,
+    exact (nat.mul_zero m).symm, },
+  { rw [hyperoperation_recursion, hyperoperation_one, bih],
+    ring, },
+end
+
+@[simp] lemma hyperoperation_three : hyperoperation 3 = (^) :=
+begin
+  ext m k,
+  induction k with bn bih,
+  { rw hyperoperation_ge_three_eq_one,
+    exact (pow_zero m).symm, },
+  { rw [hyperoperation_recursion, hyperoperation_two, bih],
+    exact (pow_succ m bn).symm, },
+end
+
+lemma hyperoperation_ge_two_eq_self (n m : ℕ) : hyperoperation (n + 2) m 1 = m :=
+begin
+  induction n with nn nih,
+  { rw hyperoperation_two,
+    ring, },
+  { rw [hyperoperation_recursion, hyperoperation_ge_three_eq_one, nih], },
+end
+
+lemma hyperoperation_two_two_eq_four (n : ℕ) : hyperoperation (n + 1) 2 2 = 4 :=
+begin
+  induction n with nn nih,
+  { rw hyperoperation_one, },
+  { rw [hyperoperation_recursion, hyperoperation_ge_two_eq_self, nih], },
+end
+
+lemma hyperoperation_ge_three_one (n : ℕ) : ∀ (k : ℕ), hyperoperation (n + 3) 1 k = 1 :=
+begin
+  induction n with nn nih,
+  { intros k,
+    rw [hyperoperation_three, one_pow], },
+  { intros k,
+    cases k,
+    { rw hyperoperation_ge_three_eq_one, },
+    { rw [hyperoperation_recursion, nih], }, },
+end
+
+lemma hyperoperation_ge_four_zero (n k : ℕ) :
+  hyperoperation (n + 4) 0 k = if (even k) then 1 else 0 :=
+begin
+  induction k with kk kih,
+  { rw hyperoperation_ge_three_eq_one,
+    simp only [even_zero, if_true], },
+  { rw hyperoperation_recursion,
+    rw kih,
+    simp_rw nat.even_add_one,
+    split_ifs,
+    { exact hyperoperation_ge_two_eq_self (n + 1) 0, },
+    { exact hyperoperation_ge_three_eq_one n 0, }, },
+end