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‎CollatzFunction/Tm8/Main.lean‎

Lines changed: 99 additions & 81 deletions
Original file line numberDiff line numberDiff line change
@@ -16,94 +16,112 @@ lemma G_even (n i: ℕ) (h_even: Even n) (init_cfg: Cfg)
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nth_cfg_is_n (i+(n^2+3*n)/2+3) (collatz n) init_cfg
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:= by
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obtain ⟨l, h⟩ := h
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cases n with
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| zero => unfold collatz nth_cfg_is_n
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match n with
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| 0 => use (zero :: l)
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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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forward h
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ring_nf at *
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simp [h]
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| 1 => tauto
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| Nat.succ (Nat.succ n) =>
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obtain ⟨k, h_even⟩ := h_even
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cases k with
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| zero => omega
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| succ k =>
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have h_even : n = 2 * k := by omega
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subst n
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forward h
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forward h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate _ one))] at h
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apply B_even at h
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. forward h
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rw [add_comm 1] at h
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rw [List.replicate_succ] at h
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simp at h
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forward h
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ring_nf at *
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have g : 3 + i + ((k * 2).succ.succ * 3 + (k * 2).succ.succ ^ 2) / 2 = 8 + i + k ^ 2 * 4 / 2 + k * 14 / 2 := by
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simp
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forward h
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forward h
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forward h
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ring_nf at *
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simp [h]
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use (zero :: l)
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| succ n => forward h
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cases n with
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| zero => ring_nf at *; tauto
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| succ n => obtain ⟨k, h_even⟩ := h_even
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cases k with
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| zero => omega
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| succ k =>
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have h_even : n = 2 * k := by omega
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subst n
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forward h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate _ one))] at h
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apply B_even at h
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. forward h
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rw [add_comm 1] at h
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rw [List.replicate_succ] at h
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simp at h
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forward h
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ring_nf at *
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have g :(3 + i + (10 + k * 14 + k ^ 2 * 4) / 2) = (8 + i + k ^ 2 * 4 / 2 + k * 14 / 2) := by omega
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rw [g]
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use (List.replicate k one ++ zero :: zero :: l)
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simp [h]
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simp [collatz]
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rw [List.replicate_succ]
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. use k; omega
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ring_nf
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calc
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_ = 3 + i + 5 + k * 7 + 2 * k ^ 2 := by omega
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_ = 8 + i + k ^ 2 * 2 + k * 7 := by ring_nf
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_ = _ := by omega
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rw [g]
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use (List.replicate k one ++ zero :: zero :: l)
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simp [h]
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simp [collatz]
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ring_nf
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simp
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rw [List.replicate_succ]
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. use k; omega
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5362

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lemma G_odd (n i: ℕ) (h_odd: Odd n) (init_cfg: Cfg)
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(h: nth_cfg_is_n i n init_cfg) :
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nth_cfg_is_n (i+(3 * n^2+4*n+1)/2) (collatz n) init_cfg
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:= by
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obtain ⟨l, h⟩ := h
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cases n with
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| zero => ring_nf at *; tauto
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| succ n => forward h
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cases n with
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| zero => ring_nf at *
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forward h
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forward h
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forward h
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use l
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simp [h]
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simp [collatz]
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| succ n => obtain ⟨k, h_odd⟩ := h_odd
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cases k with
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| zero => omega
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| succ k =>
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have h_odd : n = 2 * k+1 := by omega
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subst n
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forward h
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rw [← List.replicate_succ] at h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate _ one))] at h
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apply B_odd at h
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. forward h
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have g : (2 +k*2) /2 = k + 1:= by omega
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rw [g] 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 G_to_J at h
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forward h
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ring_nf at *
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have g : (i + (40 + k * 44 + k ^ 2 * 12) / 2) = (16 + i + k * 13 + k ^ 2 * 4 + (8 + k * 18 + k ^ 2 * 4) / 2) := by omega
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rw [g]
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use l
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simp [h]
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simp [collatz]
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rw [← List.replicate_one]
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rw [List.replicate_append_replicate]
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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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apply congr
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any_goals rfl
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omega
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. use k; omega
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match n with
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| 0 => tauto
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| 1 => forward h
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ring_nf at *
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forward h
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forward h
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forward h
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use l
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simp [h]
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simp [collatz]
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| Nat.succ (Nat.succ n) =>
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forward h
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obtain ⟨k, h_odd⟩ := h_odd
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cases k with
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| zero => omega
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| succ k =>
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have h_odd : n = 2 * k+1 := by omega
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subst n
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forward h
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rw [← List.replicate_succ] at h
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rw [← (ListBlank_empty_eq_single_zero (List.replicate _ one))] at h
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apply B_odd at h
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. forward h
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have g : (2 +k*2) /2 = k + 1:= by omega
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rw [g] 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 G_to_J at h
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forward h
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ring_nf at *
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have g : i + (1 + (1 + k * 2).succ.succ * 4 + (1 + k * 2).succ.succ ^ 2 * 3) / 2 = 16 + i + k * 13 + k ^ 2 * 4 + (8 + k * 18 + k ^ 2 * 4) / 2 := by
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simp
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ring_nf
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calc
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_ = i + 20 + 22 * k + 6 * k^2 := by omega
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_ = 16 + i + k * 13 + k ^ 2 * 4 + 4 + k * 9 + k ^ 2 * 2 := by ring_nf
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_ = _ := by omega
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rw [g]
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use l
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simp [h]
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simp [collatz]
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ring_nf
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simp
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rw [← List.replicate_one]
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rw [List.replicate_append_replicate]
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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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apply congr
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any_goals rfl
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omega
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. use k; omega
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-- n -> collatz n
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theorem G_collatz (n i: ℕ) (init_cfg: Cfg)

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