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F_collatz
listblank helper
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‎CollatzFunction/ListBlank.lean‎

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@@ -31,3 +31,17 @@ match head_h : l.head with
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let _termination_proof : (l |> to_list_rev |> List.length) > (l.tail |> to_list_rev |> List.length):= by simp [h]
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1 + count1 l.tail
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termination_by (l |> to_list_rev |> List.length)
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open Γ
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-- a more descriptive name for the theorem
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theorem ListBlank_empty_eq_single_zero: ∀ l, Turing.ListBlank.mk (l ++ [zero]) = Turing.ListBlank.mk l := by
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intros l
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simp! [Turing.ListBlank.mk]
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rw [Quotient.eq'']
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right
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use 1
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tauto

‎CollatzFunction/NonHalt/Main.lean‎

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import CollatzFunction.NonHalt.TuringMachine9
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import CollatzFunction.NonHalt.Even
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import CollatzFunction.NonHalt.Odd
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import CollatzFunction.NonHalt.Search0
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import CollatzFunction.NonHalt.Search1
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import CollatzFunction.NonHalt.Miscellaneous
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import CollatzFunction.NonHalt.Transition
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import CollatzFunction.Collatz
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import CollatzFunction.ListBlank
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namespace NonHalt
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open Lean Meta Elab Tactic Std Term TmState Γ
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lemma B_collatz (n: ℕ): ∀ (i: ℕ)(l r: List Γ),
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nth_cfg i = some ⟨B, ⟨one,
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Turing.ListBlank.mk (zero :: l),
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Turing.ListBlank.mk (List.replicate n one ++ zero :: r)⟩⟩ →
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/-
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-- if n >= 3, n -> collatz n
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lemma B_collatz (n: ℕ) (_ : n > 0) (i: ℕ)
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(h : nth_cfg i = some ⟨B, ⟨one,
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Turing.ListBlank.mk [],
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Turing.ListBlank.mk (List.replicate n one)⟩⟩) :
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∃ j>i, nth_cfg j = some ⟨B, ⟨one,
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Turing.ListBlank.mk (zero :: l),
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Turing.ListBlank.mk (List.replicate (collatz (n+2)) one ++ zero :: r)⟩⟩
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Turing.ListBlank.mk [],
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Turing.ListBlank.mk (List.replicate ((collatz (n+2))-2) one)⟩⟩
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:= by
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sorry
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unfold collatz
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split_ifs <;> rename_i g
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. rw [← ListBlank_empty_eq_single_zero []] at h
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simp at h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate n one))] at h
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apply B_even at h
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obtain ⟨j, _, h⟩ := h
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forward h
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rw [add_comm 1 (n/2)] at h
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rw [List.replicate_succ] at h
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simp at h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate (n/2) one))] at h
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apply E_zeroing at h
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forward h
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apply recF at h
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forward h
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forward h
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use (6 + j + n / 2 * 2)
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simp [h]
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repeat any_goals apply And.intro
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. omega
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. cases n with
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| zero => omega
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| succ n => cases n with
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| zero => omega
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| succ n => ring_nf
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have h : (2+n)/2 = n/2 + 1:= by omega
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rw [h]
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rw [List.replicate_succ]
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simp
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. simp! [Turing.ListBlank.mk]
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rw [Quotient.eq'']
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right
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use (n/2+1+1+1+1)
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rw [← List.cons_append]
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rw [← List.cons_append]
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rw [← List.cons_append]
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rw [← List.replicate_succ]
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rw [← List.replicate_succ]
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rw [← List.replicate_succ]
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rw [List.append_cons]
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rw [← List.replicate_succ']
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simp
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tauto
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. rw [← Nat.even_iff] at g
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obtain ⟨a, g⟩ := g
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cases a with
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| zero => omega
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| succ a => use a; omega
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. rw [← ListBlank_empty_eq_single_zero []] at h
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simp at h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate n one))] at h
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apply B_odd at h
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obtain ⟨j, _, h⟩ := h
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forward h
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cases n with
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| zero => omega
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| succ n =>
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ring_nf at h
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have k : (2+n)/2 = n/2 + 1:= by omega
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rw [k] at h
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rw [List.replicate_succ] at h
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simp at h
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rw [← List.replicate_succ] at h
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apply copy_half at h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate _ one))] at h
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apply lemma_G_to_H at h
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apply lemma_H_to_J at h
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forward h
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forward h
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forward h
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rw [← List.replicate_succ'] at h
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rw [List.replicate_succ] at h
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simp at h
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rw [← List.replicate_succ'] at h
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ring_nf at h
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use (12 + j + n / 2 * 13 + (n / 2) ^ 2 * 4)
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simp [h]
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repeat any_goals apply And.intro
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. omega
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. simp! [Turing.ListBlank.mk]
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rw [Quotient.eq'']
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right
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use 2
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tauto
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. ring_nf
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apply congr
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any_goals rfl
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apply congr
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any_goals rfl
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apply congr
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any_goals rfl
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omega
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. have g : (n+2) % 2 = 1 := by omega
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rw [← Nat.odd_iff] at g
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obtain ⟨a, g⟩ := g
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cases a with
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| zero => omega
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| succ a => use a; omega
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-/
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-- if n ≥ 1, n -> collatz n
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lemma F_collatz (n: ℕ) (_ : n ≥ 1) (i: ℕ)
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(h : nth_cfg i = some ⟨F, ⟨one,
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Turing.ListBlank.mk [],
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Turing.ListBlank.mk (List.replicate (n-1) one)⟩⟩) :
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∃ j>i, nth_cfg j = some ⟨F, ⟨one,
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Turing.ListBlank.mk [],
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Turing.ListBlank.mk (List.replicate ((collatz n)-1) one)⟩⟩
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:= by
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cases n with
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| zero => omega
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| succ n => forward h
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cases n with
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| zero => simp
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forward h
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unfold collatz
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simp
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forward h
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forward h
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use (4+i)
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simp [h]
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simp! [Turing.ListBlank.mk]
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rw [Quotient.eq'']
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right
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use 1
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tauto
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| succ n =>
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rw [List.replicate_succ] at h
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simp at h
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unfold collatz
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split_ifs <;> rename_i g
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. rw [← (ListBlank_empty_eq_single_zero (List.replicate n one))] at h
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apply B_even at h
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obtain ⟨j, _, h⟩ := h
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forward h
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rw [add_comm 1 (n/2)] at h
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rw [List.replicate_succ] at h
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simp at h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate (n/2) one))] at h
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apply E_zeroing at h
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forward h
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apply recF at h
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forward h
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use (5 + j + n / 2 * 2)
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ring_nf at *
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simp [h]
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repeat any_goals apply And.intro
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. omega
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. rw [← List.replicate_succ']
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ring_nf
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simp! [Turing.ListBlank.mk]
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rw [Quotient.eq'']
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right
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use 3 + n/2
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rw [← List.replicate_succ]
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rw [← List.replicate_succ]
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simp
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ring_nf
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tauto
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. rw [← Nat.even_iff] at g
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obtain ⟨a, g⟩ := g
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cases a with
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| zero => omega
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| succ a => use a; omega
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. cases n with
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| zero => omega
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| succ n =>
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rw [← (ListBlank_empty_eq_single_zero (List.replicate (n+1) one))] at h
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apply B_odd at h
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obtain ⟨j, _, h⟩ := h
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forward h
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have k : (2+n)/2 = n/2 + 1:= by omega
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rw [k] at h
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rw [List.replicate_succ] at h
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simp at h
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rw [← List.replicate_succ] at h
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apply copy_half at h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate _ one))] at h
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apply lemma_G_to_H at h
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apply lemma_H_to_J at h
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forward h
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forward h
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use (11 + j + n / 2 * 13 + (n / 2) ^ 2 * 4)
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simp [h]
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repeat any_goals apply And.intro
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. omega
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. simp! [Turing.ListBlank.mk]
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rw [Quotient.eq'']
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right
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use 1
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tauto
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. apply congr
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any_goals rfl
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rw [← List.replicate_succ']
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apply congr
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any_goals rfl
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apply congr
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any_goals rfl
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omega
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. have g : (n+3) % 2 = 1 := by omega
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rw [← Nat.odd_iff] at g
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obtain ⟨a, g⟩ := g
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cases a with
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| zero => omega
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| succ a => use a; omega
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end NonHalt

‎CollatzFunction/NonHalt/Miscellaneous.lean‎

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Original file line numberDiff line numberDiff line change
@@ -6,8 +6,12 @@ namespace NonHalt
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open Lean Meta Elab Tactic Std Term TmState
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theorem E_zeroing (k : ℕ): ∀ (i : ℕ) (l r : List Γ),
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nth_cfg i = some ⟨E, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ →
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nth_cfg (i + k + 1) = some ⟨E, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (k+1) Γ.zero ++ r)⟩⟩ := by
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nth_cfg i = some ⟨E, ⟨Γ.one,
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Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l),
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Turing.ListBlank.mk r⟩⟩ →
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nth_cfg (i + k + 1) = some ⟨E, ⟨Γ.zero,
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Turing.ListBlank.mk l,
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Turing.ListBlank.mk (List.replicate (k+1) Γ.zero ++ r)⟩⟩ := by
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induction k with intros i l r h
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| zero => forward h
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ring_nf at *

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