import Erdos1135.Terras.Probability.Bitstrings /-! # Terras/Everett Binomial-Tail Reduction This module starts the elementary tail side of the Terras/Everett density route. It proves that noncontracting parity words must lie in a simple upper tail, using only the rational bound `3 / 5 < log_3 2` encoded as exact power arithmetic. -/ namespace Erdos1135 namespace Terras set_option maxHeartbeats 400000 /-- The exact arithmetic threshold used to avoid logarithms: if the odd-count ratio is at most `3/5`, then the accelerated branch multiplier is strictly smaller than one. -/ theorem pow_three_lt_pow_two_of_five_mul_le_three_mul {r k : ℕ} (hk : 0 < k) (h : 5 * r ≤ 3 * k) : 3 ^ r < 2 ^ k := by by_cases hr : r = 0 · subst r exact Nat.one_lt_two_pow (Nat.ne_of_gt hk) · have hbase : (3 : ℕ) ^ 5 < 2 ^ 8 := by norm_num have hraise : ((3 : ℕ) ^ 5) ^ r < (2 ^ 8) ^ r := by exact Nat.pow_lt_pow_left hbase hr have hleft_raise : ((3 : ℕ) ^ 5) ^ r = 3 ^ (5 * r) := by rw [← Nat.pow_mul] have hright_raise : ((2 : ℕ) ^ 8) ^ r = 2 ^ (8 * r) := by rw [← Nat.pow_mul] have hraise' : (3 : ℕ) ^ (5 * r) < 2 ^ (8 * r) := by rwa [hleft_raise, hright_raise] at hraise have hright_le : 2 ^ (8 * r) ≤ 2 ^ (5 * k) := by have h8 : 8 * r ≤ 5 * k := by omega exact Nat.pow_le_pow_right (by norm_num : 0 < (2 : ℕ)) h8 have htmp : (3 : ℕ) ^ (5 * r) < 2 ^ (5 * k) := lt_of_lt_of_le hraise' hright_le have hleft_eq : ((3 : ℕ) ^ r) ^ 5 = 3 ^ (5 * r) := by rw [← Nat.pow_mul] congr 1 omega have hright_eq : ((2 : ℕ) ^ k) ^ 5 = 2 ^ (5 * k) := by rw [← Nat.pow_mul] congr 1 omega have hcomp : ((3 : ℕ) ^ r) ^ 5 < (2 ^ k) ^ 5 := by rw [hleft_eq, hright_eq] exact htmp exact (Nat.pow_lt_pow_iff_left (by omega : (5 : ℕ) ≠ 0)).mp hcomp theorem contractingOddCount_of_five_mul_le_three_mul {k r : ℕ} (hk : 0 < k) (hr : r ≤ k) (h : 5 * r ≤ 3 * k) : r ∈ contractingOddCounts k := by rw [mem_contractingOddCounts_iff] exact ⟨hr, pow_three_lt_pow_two_of_five_mul_le_three_mul hk h⟩ theorem contracting_of_five_mul_numOdd_le_three_mul {k : ℕ} (w : ParityWord k) (hk : 0 < k) (h : 5 * numOdd w ≤ 3 * k) : 3 ^ numOdd w < 2 ^ k := pow_three_lt_pow_two_of_five_mul_le_three_mul hk h /-- Odd-count values whose branch multiplier does not contract. -/ def noncontractingOddCounts (k : ℕ) : Finset ℕ := (Finset.range (k + 1)).filter fun r => ¬ 3 ^ r < 2 ^ k /-- A simple upper tail that contains every noncontracting odd-count value. -/ def highOddCounts (k : ℕ) : Finset ℕ := (Finset.range (k + 1)).filter fun r => 3 * k < 5 * r lemma binomial_weighted_sum_eq_five_pow (k : ℕ) : (∑ r ∈ Finset.range (k + 1), Nat.choose k r * 3 ^ r * 2 ^ (k - r)) = 5 ^ k := by calc (∑ r ∈ Finset.range (k + 1), Nat.choose k r * 3 ^ r * 2 ^ (k - r)) = ∑ r ∈ Finset.range (k + 1), 3 ^ r * 2 ^ (k - r) * k.choose r := by apply Finset.sum_congr rfl intro r hr ring_nf _ = (3 + 2 : ℕ) ^ k := by exact (add_pow (3 : ℕ) (2 : ℕ) k).symm _ = 5 ^ k := by norm_num lemma choose_mul_three_pow_mul_two_pow_le_five_pow (k r : ℕ) : Nat.choose k r * 3 ^ r * 2 ^ (k - r) ≤ 5 ^ k := by by_cases hr : r ≤ k · have hmem : r ∈ Finset.range (k + 1) := by rw [Finset.mem_range] omega have hterm : Nat.choose k r * 3 ^ r * 2 ^ (k - r) ≤ ∑ m ∈ Finset.range (k + 1), Nat.choose k m * 3 ^ m * 2 ^ (k - m) := by exact Finset.single_le_sum (s := Finset.range (k + 1)) (f := fun m => Nat.choose k m * 3 ^ m * 2 ^ (k - m)) (by intro m hm; exact Nat.zero_le _) hmem exact hterm.trans_eq (binomial_weighted_sum_eq_five_pow k) · have hzero : Nat.choose k r = 0 := Nat.choose_eq_zero_of_lt (by omega) simp [hzero] theorem highOddCounts_weighted_sum_le_five_pow (k : ℕ) : (∑ r ∈ highOddCounts k, Nat.choose k r * 3 ^ r * 2 ^ (k - r)) ≤ 5 ^ k := by have hsub : highOddCounts k ⊆ Finset.range (k + 1) := by intro r hr have hrange : r ≤ k ∧ 3 * k < 5 * r := by simpa [highOddCounts] using hr rw [Finset.mem_range] omega calc (∑ r ∈ highOddCounts k, Nat.choose k r * 3 ^ r * 2 ^ (k - r)) ≤ ∑ r ∈ Finset.range (k + 1), Nat.choose k r * 3 ^ r * 2 ^ (k - r) := by exact Finset.sum_le_sum_of_subset_of_nonneg hsub (by intro x hx hxnot; exact Nat.zero_le _) _ = 5 ^ k := binomial_weighted_sum_eq_five_pow k lemma highOddCount_weight_lower_bound {k r : ℕ} (hr : r ∈ highOddCounts k) : 3 ^ (3 * (k / 5)) * 2 ^ (k - 3 * (k / 5)) ≤ 3 ^ r * 2 ^ (k - r) := by have hr' : r ≤ k ∧ 3 * k < 5 * r := by simpa [highOddCounts] using hr let a := 3 * (k / 5) have h5q : 5 * (k / 5) ≤ k := Nat.mul_div_le k 5 have ha_le_r : a ≤ r := by dsimp [a] omega have hr_le_k : r ≤ k := hr'.1 let d := r - a have hr_eq : r = a + d := by dsimp [d] omega have hk_sub : k - a = d + (k - r) := by dsimp [d] omega calc 3 ^ a * 2 ^ (k - a) = 3 ^ a * 2 ^ (d + (k - r)) := by rw [hk_sub] _ = 3 ^ a * (2 ^ d * 2 ^ (k - r)) := by rw [Nat.pow_add] _ ≤ 3 ^ a * (3 ^ d * 2 ^ (k - r)) := by exact Nat.mul_le_mul_left _ (Nat.mul_le_mul_right _ (Nat.pow_le_pow_left (by norm_num : (2 : ℕ) ≤ 3) d)) _ = 3 ^ (a + d) * 2 ^ (k - r) := by rw [Nat.pow_add] ring _ = 3 ^ r * 2 ^ (k - r) := by rw [hr_eq] theorem highOddCounts_sum_choose_mul_weight_lower_le_five_pow (k : ℕ) : (3 ^ (3 * (k / 5)) * 2 ^ (k - 3 * (k / 5))) * (∑ r ∈ highOddCounts k, Nat.choose k r) ≤ 5 ^ k := by let W := 3 ^ (3 * (k / 5)) * 2 ^ (k - 3 * (k / 5)) have hweighted : W * (∑ r ∈ highOddCounts k, Nat.choose k r) ≤ ∑ r ∈ highOddCounts k, Nat.choose k r * 3 ^ r * 2 ^ (k - r) := by rw [Finset.mul_sum] apply Finset.sum_le_sum intro r hr have hW : W ≤ 3 ^ r * 2 ^ (k - r) := by simpa [W] using highOddCount_weight_lower_bound hr calc W * Nat.choose k r ≤ (3 ^ r * 2 ^ (k - r)) * Nat.choose k r := Nat.mul_le_mul_right _ hW _ = Nat.choose k r * 3 ^ r * 2 ^ (k - r) := by ring exact hweighted.trans (highOddCounts_weighted_sum_le_five_pow k) lemma mem_noncontractingOddCounts_iff {k r : ℕ} : r ∈ noncontractingOddCounts k ↔ r ≤ k ∧ ¬ 3 ^ r < 2 ^ k := by simp [noncontractingOddCounts] lemma mem_highOddCounts_iff {k r : ℕ} : r ∈ highOddCounts k ↔ r ≤ k ∧ 3 * k < 5 * r := by simp [highOddCounts] lemma numOdd_mem_noncontractingOddCounts_iff {k : ℕ} (w : ParityWord k) : numOdd w ∈ noncontractingOddCounts k ↔ ¬ 3 ^ numOdd w < 2 ^ k := by rw [mem_noncontractingOddCounts_iff] exact ⟨fun h => h.2, fun h => ⟨numOdd_le w, h⟩⟩ lemma three_mul_lt_five_mul_of_not_contracting {k r : ℕ} (hk : 0 < k) (hnot : ¬ 3 ^ r < 2 ^ k) : 3 * k < 5 * r := by by_contra hle have hlin : 5 * r ≤ 3 * k := by omega exact hnot (pow_three_lt_pow_two_of_five_mul_le_three_mul hk hlin) lemma noncontractingOddCounts_subset_highOddCounts {k : ℕ} (hk : 0 < k) : noncontractingOddCounts k ⊆ highOddCounts k := by intro r hr rw [mem_noncontractingOddCounts_iff] at hr rw [mem_highOddCounts_iff] exact ⟨hr.1, three_mul_lt_five_mul_of_not_contracting hk hr.2⟩ noncomputable def noncontractingWordsEquivSigmaOddCount (k : ℕ) : {w : ParityWord k // ¬ 3 ^ numOdd w < 2 ^ k} ≃ Sigma (fun r : {r : ℕ // r ∈ noncontractingOddCounts k} => {w : ParityWord k // numOdd w = r.1}) where toFun w := ⟨⟨numOdd w.1, (numOdd_mem_noncontractingOddCounts_iff w.1).mpr w.2⟩, ⟨w.1, rfl⟩⟩ invFun rw := ⟨rw.2.1, by rw [rw.2.2] exact (mem_noncontractingOddCounts_iff.mp rw.1.2).2⟩ left_inv w := by ext i rfl right_inv rw := by cases rw with | mk r w => cases r with | mk r hr => cases w with | mk w hw => simp at hw ⊢ constructor · exact hw · subst r simp theorem card_noncontracting_words_eq_sum_choose (k : ℕ) : Fintype.card {w : ParityWord k // ¬ 3 ^ numOdd w < 2 ^ k} = ∑ r ∈ noncontractingOddCounts k, Nat.choose k r := by classical rw [Fintype.card_congr (noncontractingWordsEquivSigmaOddCount k)] rw [Fintype.card_sigma] simp [card_words_with_numOdd] rw [Finset.sum_attach] theorem card_noncontracting_words_le_high_odd_tail {k : ℕ} (hk : 0 < k) : Fintype.card {w : ParityWord k // ¬ 3 ^ numOdd w < 2 ^ k} ≤ ∑ r ∈ highOddCounts k, Nat.choose k r := by rw [card_noncontracting_words_eq_sum_choose] exact Finset.sum_le_sum_of_subset_of_nonneg (noncontractingOddCounts_subset_highOddCounts hk) (by intro x hx hxnot; exact Nat.zero_le _) end Terras end Erdos1135