From df8665eaa6b2b6a1176561c14fd577f28ab758e0 Mon Sep 17 00:00:00 2001 From: lengyijun Date: Wed, 16 Oct 2024 11:50:06 +0800 Subject: [PATCH] tm27 finish --- GoldbachTm.lean | 7 + GoldbachTm/Tm27/Content.lean | 126 ++++++++ GoldbachTm/Tm27/Miscellaneous.lean | 116 +++++++ GoldbachTm/Tm27/PBP.lean | 315 ++++++++++++++++++ GoldbachTm/Tm27/Prime.lean | 465 +++++++++++++++++++++++++++ GoldbachTm/Tm27/Search0.lean | 129 ++++++++ GoldbachTm/Tm27/Transition.lean | 68 ++++ GoldbachTm/Tm27/TuringMachine27.lean | 78 +++++ archieve/Busy beaver - Wikipedia.pdf | Bin 0 -> 1506633 bytes 9 files changed, 1304 insertions(+) create mode 100644 GoldbachTm/Tm27/Content.lean create mode 100644 GoldbachTm/Tm27/Miscellaneous.lean create mode 100644 GoldbachTm/Tm27/PBP.lean create mode 100644 GoldbachTm/Tm27/Prime.lean create mode 100644 GoldbachTm/Tm27/Search0.lean create mode 100644 GoldbachTm/Tm27/Transition.lean create mode 100644 archieve/Busy beaver - Wikipedia.pdf diff --git a/GoldbachTm.lean b/GoldbachTm.lean index c732ab9..a36ede7 100644 --- a/GoldbachTm.lean +++ b/GoldbachTm.lean @@ -4,6 +4,13 @@ import GoldbachTm.Basic import GoldbachTm.Format import GoldbachTm.ListBlank import GoldbachTm.Tm27.TuringMachine27 +import GoldbachTm.Tm27.Search0 +import GoldbachTm.Tm27.Prime +import GoldbachTm.Tm27.PBP +import GoldbachTm.Tm27.Content +import GoldbachTm.Tm27.Transition +import GoldbachTm.Tm27.Miscellaneous + import GoldbachTm.Tm31.TuringMachine31 import GoldbachTm.Tm31.Search0 import GoldbachTm.Tm31.Content diff --git a/GoldbachTm/Tm27/Content.lean b/GoldbachTm/Tm27/Content.lean new file mode 100644 index 0000000..329e125 --- /dev/null +++ b/GoldbachTm/Tm27/Content.lean @@ -0,0 +1,126 @@ +import GoldbachTm.Tm27.TuringMachine27 +import GoldbachTm.Tm27.Search0 +import GoldbachTm.Tm27.PBP +import Mathlib.Data.Nat.Prime.Defs + +namespace Tm27 + +theorem lemma_26 (n : ℕ) (i : ℕ) +(even_n : Even (n+2)) +(g : +nth_cfg i = some ⟨⟨26, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (n+4) Γ.one), Turing.ListBlank.mk []⟩⟩ ) +( hpp : goldbach (n+4)) : +∃ j>i, nth_cfg j = some ⟨⟨26, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (n+4+2) Γ.one), Turing.ListBlank.mk []⟩⟩ +:= by +forward g g i +repeat rw [← List.replicate_succ] at g +apply (leap_18 _ _ 0) at g +any_goals omega +any_goals assumption +refine (?_ ∘ g) ?_ +. intros g + obtain ⟨k, _, h⟩ := g + use k + constructor + . omega + . simp [h] +. obtain ⟨x, y, _, hx, hy⟩ := hpp + by_cases x ≤ y + . use! x, y + repeat any_goals apply And.intro + any_goals assumption + apply Nat.Prime.two_le at hx + omega + . use! y, x + repeat any_goals apply And.intro + any_goals assumption + any_goals omega + apply Nat.Prime.two_le at hy + omega + +lemma never_halt_step (n : ℕ) : +(∀ i < n, goldbach (2*i+4)) -> +∃ j, nth_cfg j = some ⟨⟨26, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (2*n+4) Γ.one), Turing.ListBlank.mk []⟩⟩ + := by +induction n with +| zero => +intros _ +use 45 +simp [cfg45] +tauto +| succ n => +rename_i induction_step +intros h +refine (?_ ∘ induction_step) ?_ +. intros g + obtain ⟨j, g⟩ := g + specialize h n (by omega) + apply lemma_26 at g + . specialize g h + obtain ⟨k, g⟩ := g + use k + simp [g] + ring_nf + . use (n+1) + ring_nf +. intros i hi + apply h i (by omega) + +theorem never_halt (never_none : ∀ i, nth_cfg i ≠ none) (n : ℕ): +goldbach (2*n + 4) +:= by +induction' n using Nat.strongRecOn with n IH +apply never_halt_step at IH +obtain ⟨j, g⟩ := IH +by_contra! h +forward g g j +repeat rw [← List.replicate_succ] at g +apply (leap_18_halt _ _ 0) at g +any_goals omega +refine (?_ ∘ g) ?_ +. intros h + obtain ⟨k, h⟩ := h + apply never_none _ h +. intros x y _ + apply h ⟨x, y, ?_⟩ + ring_nf at * + tauto + + +theorem halt_lemma_rev' (h : ∀ n, goldbach (2*n+4)) : +∀ i, nth_cfg i ≠ none := by +apply propagating_induction (fun i => nth_cfg i ≠ none) (fun i n => nth_cfg i = some ⟨⟨26, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (2*n+4) Γ.one), Turing.ListBlank.mk []⟩⟩) 45 +. simp [cfg45]; tauto +. intros i n g + apply (lemma_26 (2*n)) at g + . specialize g (h _) + obtain ⟨j, g⟩ := g + use j + simp! [g] + . use (n+1) + ring_nf +. intros i n g j hij _ + have h₂ : ∀ k, nth_cfg (j+k) = none := by + intro k + induction k with + | zero => tauto + | succ k => rename_i h₁ + forward h₁ h₁ (j+k) + rw [← h₁] + ring_nf + specialize h₂ (i-j) + have h₃ : j + (i-j) = i := by omega + rw [h₃] at h₂ + rw [h₂] at g + tauto + +theorem halt_lemma_rev : +(∃ i, nth_cfg i = none) → (∃ n, ¬ goldbach (2*n+4)) +:= by +intros h +by_contra! g +apply halt_lemma_rev' at g +tauto + + +end Tm27 diff --git a/GoldbachTm/Tm27/Miscellaneous.lean b/GoldbachTm/Tm27/Miscellaneous.lean new file mode 100644 index 0000000..c32bc5e --- /dev/null +++ b/GoldbachTm/Tm27/Miscellaneous.lean @@ -0,0 +1,116 @@ +-- some lemmas tm31 doesn't contain + +import GoldbachTm.Tm27.TuringMachine27 +import GoldbachTm.Tm27.Transition + +namespace Tm27 + +theorem lemma7 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨7, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨7, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (k+1) Γ.zero ++ r)⟩⟩ := by +induction k with (intros i l r h; simp_all) +| zero => forward h h i + rw [← h] + ring_nf +| succ k => forward h h i + rename_i induction_step + apply induction_step at h + ring_nf at * + simp! [h] + rw [List.append_cons] + rw [← List.replicate_succ'] + ring_nf + +theorem lemma8 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨8, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ → +nth_cfg (i + k + 2) = some ⟨⟨9, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (Γ.zero :: l), Turing.ListBlank.mk (List.replicate k Γ.zero ++ r)⟩⟩ := by +induction k with (intros i l r h; simp_all) +| zero => forward h h i + forward h h (1+i) + rw [← h] + ring_nf +| succ k => forward h h i + rename_i induction_step + apply induction_step at h + ring_nf at * + simp! [h] + rw [List.append_cons] + rw [← List.replicate_succ'] + ring_nf + +theorem lemma5 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨5, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ → +nth_cfg (i + k + 2) = some ⟨⟨4, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (Γ.zero :: l), Turing.ListBlank.mk (List.replicate k Γ.zero ++ r)⟩⟩ := by +induction k with (intros i l r h; simp_all) +| zero => forward h h i + forward h h (1+i) + rw [← h] + ring_nf +| succ k => forward h h i + rename_i induction_step + apply induction_step at h + ring_nf at * + simp! [h] + rw [List.append_cons] + rw [← List.replicate_succ'] + ring_nf + +theorem lemma10 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨10, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate k Γ.zero ++ List.cons Γ.one r) ⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨10, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ l), Turing.ListBlank.mk r⟩⟩ := by +induction k with (intros i l r h; simp_all) +| zero => forward h h i + rw [← h] + ring_nf +| succ k => rename_i induction_step + specialize induction_step (i+1) (List.cons Γ.one l) r + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +theorem lemma_6_to_7 (i : ℕ) (l1: ℕ) (l r : List Γ) +(h : +nth_cfg i = some ⟨⟨6, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate l1 Γ.one ++ List.cons Γ.zero l), + Turing.ListBlank.mk r, + ⟩⟩) : +∃ j>i, nth_cfg j = some ⟨⟨7, by omega⟩, ⟨Γ.zero, + Turing.ListBlank.mk l, + Turing.ListBlank.mk (List.replicate (l1+1) Γ.zero ++ r), + ⟩⟩ +:= by +forward h h i +cases l1 with +| zero => use (1+i) + simp [h] +| succ l1 => simp! at h + apply lemma7 at h + use (1+i+l1 + 1) + simp [h] + constructor + . omega + . rw [List.append_cons, ← List.replicate_succ'] + +theorem lemma_9_to_10 (i : ℕ) (r1: ℕ) (l r : List Γ) +(h : +nth_cfg i = some ⟨⟨9, by omega⟩, ⟨Γ.zero, + Turing.ListBlank.mk l, + Turing.ListBlank.mk (List.replicate r1 Γ.zero ++ List.cons Γ.one r), + ⟩⟩) : +∃ j>i, nth_cfg j = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate r1 Γ.one ++ Γ.zero :: l), + Turing.ListBlank.mk r, + ⟩⟩ +:= by +forward h h i +cases r1 with simp_all +| zero => use (1+i) + simp [h] +| succ l1 => simp! at h + apply lemma10 at h + use (1+i+l1 + 1) + simp [h] + omega + +end Tm27 diff --git a/GoldbachTm/Tm27/PBP.lean b/GoldbachTm/Tm27/PBP.lean new file mode 100644 index 0000000..5fabbbe --- /dev/null +++ b/GoldbachTm/Tm27/PBP.lean @@ -0,0 +1,315 @@ +-- PDP is short for "prime board prime" +import GoldbachTm.Tm27.TuringMachine27 +import GoldbachTm.Tm27.Search0 +import GoldbachTm.Tm27.Prime +import Mathlib.Data.Nat.Prime.Defs + +namespace Tm27 + +-- l1 : count of 1 on the left +-- r1 : count of 1 on the right +theorem lemma_18 (i l1 r1: ℕ) +(hp : ¬ Nat.Prime (l1+1) \/ ¬ Nat.Prime (r1+1)) +(g : +nth_cfg i = some ⟨⟨18, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate (l1+1) Γ.one), Turing.ListBlank.mk (List.replicate (r1+1) Γ.one)⟩⟩ +): +∃ j>i, nth_cfg j = some ⟨⟨18, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate l1 Γ.one), Turing.ListBlank.mk (List.replicate (r1+2) Γ.one)⟩⟩ := by +forward g g i +forward g g (1+i) +forward g g (2+i) +by_cases hr1 : Nat.Prime (r1+1) +. refine (?_ ∘ (prime_21 (3+i) r1 (Γ.one :: List.replicate l1 Γ.one) [])) ?_ + . intros g + specialize g hr1 + obtain ⟨j, _, g⟩ := g + forward g g j + refine (?_ ∘ (lemma_22_to_24 (1+j) l1 [] (Γ.zero :: Γ.one :: (List.replicate r1 Γ.one ++ [Γ.zero])))) ?_ + . intros g + obtain ⟨k, g⟩ := g + forward g g (k+(1+j)) + have h : ¬ Nat.Prime (l1+1) := by tauto + apply n_prime_17 at h + pick_goal 5 + . rw [g] + simp + repeat any_goals apply And.intro + all_goals rfl + obtain ⟨m, _, h⟩ := h + forward h h m + forward h h (1+m) + forward h h (2+m) + apply rec26 at h + forward h h (3+m+l1+1) + use (5+m+l1) + constructor + . omega + . simp [h] + repeat any_goals apply And.intro + all_goals simp! [Turing.ListBlank.mk] + all_goals rw [Quotient.eq''] + all_goals right + . use 2 + tauto + . use 1 + simp + tauto + . simp! [g, Turing.ListBlank.mk] + rw [Quotient.eq''] + left + use 1 + tauto + . simp! [g, Turing.ListBlank.mk] + rw [Quotient.eq''] + left + use 1 + tauto +. apply n_prime_17 at hr1 + pick_goal 5 + . rw [g] + simp + repeat any_goals apply And.intro + . rfl + . simp! [Turing.ListBlank.mk] + rw [Quotient.eq''] + left + use 1 + tauto + obtain ⟨j, _, g⟩ := hr1 + forward g g j + use (1+j) + constructor + . omega + . simp! [g] + simp! [Turing.ListBlank.mk] + rw [Quotient.eq''] + right + use 1 + rw [← List.cons_append] + rw [← List.replicate_succ] + rw [← List.cons_append] + rw [← List.replicate_succ] + ring_nf + tauto + +-- +2 +theorem both_prime (i l1 r1: ℕ) +(hpl : Nat.Prime (l1+1)) +(hpr : Nat.Prime (r1+1)) +(even_sum : Even (l1+r1)) +(g : +nth_cfg i = some ⟨⟨18, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate (l1+1) Γ.one), Turing.ListBlank.mk (List.replicate (r1+1) Γ.one)⟩⟩ +) : +∃ j>i, nth_cfg j = some ⟨⟨26, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (l1+r1+4) Γ.one), Turing.ListBlank.mk []⟩⟩ := by +forward g g i +forward g g (1+i) +forward g g (2+i) +apply prime_21 at hpr +pick_goal 5 +. rw [g] + simp + constructor + . rfl + . simp! [Turing.ListBlank.mk] + rw [Quotient.eq''] + left + use 1 + tauto +clear g +obtain ⟨j, _, g⟩ := hpr +forward g g j +refine (?_ ∘ (lemma_22_to_24 (1+j) l1 [] (Γ.zero :: Γ.one :: (List.replicate r1 Γ.one ++ [Γ.zero])))) ?_ +. intros h + obtain ⟨k, h⟩ := h + forward h h (k+(1+j)) + apply prime_21 at hpl + pick_goal 5 + . rw [h] + simp + constructor <;> rfl + obtain ⟨m, _, g⟩ := hpl + forward g g m + forward g g (1+m) + forward g g (2+m) + apply rec23 at g + forward g g (3+m+l1+1) + rw [← List.cons_append] at g + rw [← List.replicate_succ] at g + rw [← List.cons_append] at g + rw [← List.replicate_succ] at g + rw [List.append_cons] at g + rw [← List.replicate_succ'] at g + apply rec24 at g + forward g g (5+m+l1+(l1+1)+1) + rw [← List.cons_append] at g + rw [← List.replicate_succ] at g + rw [← List.cons_append] at g + rw [← List.replicate_succ] at g + rw [List.append_cons] at g + rw [← List.replicate_succ'] at g + rw [← List.append_assoc] at g + rw [← List.replicate_add] at g + apply n_prime_17 at g + refine (?_ ∘ g) ?_ + . clear g + intros g + obtain ⟨n, _, g⟩ := g + forward g g n + forward g g (1+n) + forward g g (2+n) + rw [← List.cons_append] at g + rw [← List.replicate_succ] at g + apply rec26 at g + use (3 + n + (2 + l1 + r1 + 1) + 1) + constructor + . omega + . rw [g] + simp [Turing.ListBlank.mk] + rw [Quotient.eq''] + right + use 2 + ring_nf + tauto + . apply (@Nat.not_prime_of_dvd_of_lt 2) + any_goals omega + ring_nf + rw [add_assoc] + apply Nat.dvd_add + . omega + . apply Even.two_dvd + assumption +. rw [g] + simp! [Turing.ListBlank.mk] + rw [Quotient.eq''] + left + use 1 + tauto + + +theorem leap_18 (l1 : ℕ) : ∀ (i r1: ℕ), +l1 + r1 ≥ 2 → +Even (l1+r1) → +nth_cfg i = some ⟨⟨18, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate (l1+1) Γ.one), Turing.ListBlank.mk (List.replicate (r1+1) Γ.one)⟩⟩ → +(∃ x y, x + y = l1 + r1 + 2 /\ Nat.Prime x /\ Nat.Prime y /\ (r1+1) ≤ x /\ x ≤ y) → +∃ j>i, nth_cfg j = some ⟨⟨26, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (l1+r1+4) Γ.one), Turing.ListBlank.mk []⟩⟩ +:= by +induction l1 with +| zero => omega +| succ l1 => + rename_i induction_step + intros i r1 h _ g hpp + by_cases hp : ¬ Nat.Prime (l1+2) \/ ¬ Nat.Prime (r1+1) + . -- induction step + apply lemma_18 at g + any_goals tauto + obtain ⟨j, _, g⟩ := g + apply induction_step at g + all_goals ring_nf at * + any_goals assumption + refine (?_ ∘ g) ?_ + . intros g + obtain ⟨m, g₂⟩ := g + use m + ring_nf at * + simp [g₂] + omega + . obtain ⟨x, y, _, _, _, _⟩ := hpp + use! x, y + repeat any_goals apply And.intro + any_goals assumption + any_goals omega + by_cases x = r1+1 + any_goals omega + subst x + exfalso + cases hp with + | inl hp => apply hp + have : y = l1+2 := by omega + subst y + ring_nf at * + assumption + | inr hp => apply hp + ring_nf at * + assumption + . -- both prime + apply both_prime at g + all_goals tauto + + +theorem leap_18_halt (l1 : ℕ) : ∀ (i r1: ℕ), +l1 + r1 ≥ 2 → +nth_cfg i = some ⟨⟨18, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate (l1+1) Γ.one), Turing.ListBlank.mk (List.replicate (r1+1) Γ.one)⟩⟩ → +(∀ x y, x + y = l1 + r1 + 2 /\ Nat.Prime x /\ Nat.Prime y → False) → +∃ j, nth_cfg j = none +:= by +induction l1 with +| zero => + intros i r1 h g hpp + have hr1 : ∃ r2, r2 + 2 = r1 := by use (r1-2); omega + obtain ⟨r2, _⟩ := hr1 + subst r1 + forward g g i + forward g g (1+i) + forward g g (2+i) + rw [← List.replicate_succ] at g + rw [← List.replicate_succ] at g + by_cases hp : Nat.Prime (r2+3) + . apply prime_21 at hp + pick_goal 5 + . rw [g] + simp + constructor + . rfl + . simp! [Turing.ListBlank.mk] + rw [Quotient.eq''] + left + use 1 + tauto + obtain ⟨j, _, h⟩ := hp + forward h h j + forward h h (1+j) + forward h h (2+j) + forward h h (3+j) + forward h h (4+j) + forward h h (5+j) + forward h h (6+j) + forward h h (7+j) + forward h h (8+j) + forward h h (9+j) + forward h h (10+j) + forward h h (11+j) + forward h h (12+j) + use (13+j) + . apply n_prime_17 at hp + pick_goal 5 + . rw [g] + simp + constructor + . rfl + . simp! [Turing.ListBlank.mk] + rw [Quotient.eq''] + left + use 1 + tauto + obtain ⟨j, _, h⟩ := hp + forward h h j + forward h h (1+j) + forward h h (2+j) + use (3+j) +| succ l1 => + intros i r1 h g hpp + apply lemma_18 at g + . rename_i induction_step + obtain ⟨j, _, g⟩ := g + apply induction_step at g + any_goals omega + apply g + ring_nf at hpp + ring_nf + assumption + . specialize hpp (l1+2) (r1+1) + ring_nf at hpp + simp at hpp + ring_nf + tauto + +end Tm27 diff --git a/GoldbachTm/Tm27/Prime.lean b/GoldbachTm/Tm27/Prime.lean new file mode 100644 index 0000000..b4281df --- /dev/null +++ b/GoldbachTm/Tm27/Prime.lean @@ -0,0 +1,465 @@ +import GoldbachTm.Tm27.TuringMachine27 +import GoldbachTm.Tm27.Transition +import GoldbachTm.Tm27.Miscellaneous + +namespace Tm27 + +-- c1++, c2++ +-- l 0 [la 11] 0 [lb 11] 0 [ra 11111] 0 [(rb+1) 1] 0 r +-- c1 ^ c2 +-- +-- l 0 [la' 1] 0 [lb' 1] 0 [(ra+1) 1] 0 [rb 11111] 0 r +-- c1 ^ c2 +private lemma step_10_pointer (i la lb ra rb: ℕ) (l r : List Γ) : +nth_cfg i = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate lb Γ.one ++ List.cons Γ.zero (List.replicate la Γ.one ++ List.cons Γ.zero l)), + Turing.ListBlank.mk (List.replicate ra Γ.one ++ List.cons Γ.zero (List.replicate (rb+1) Γ.one ++ List.cons Γ.zero r))⟩⟩ → + (ra + 2) ≡ (la + 1) [MOD (la + lb + 1)] → +∃ j>i, ∃ la' lb', nth_cfg j = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate lb' Γ.one ++ List.cons Γ.zero (List.replicate la' Γ.one ++ List.cons Γ.zero l)), + Turing.ListBlank.mk (List.replicate (ra+1) Γ.one ++ List.cons Γ.zero (List.replicate rb Γ.one ++ List.cons Γ.zero r))⟩⟩ + /\ (ra + 3) ≡ (la' + 1) [MOD (la + lb + 1)] + /\ la + lb = la' + lb' +:= by +intros g hm +apply lemma_10_to_11 at g +obtain ⟨j, g⟩ := g +forward g g (j+i) +forward g g (1+j+i) +apply rec13 at g +forward g g (2+j+i+ra+1) +cases lb with simp_all +| zero => forward g g (4+j+i+ra) + cases la with simp_all + | zero => forward g g (5+j+i+ra) + forward g g (6+j+i+ra) + use (7+j+i+ra) + constructor + . omega + . simp [g] + use! 0, 0 + simp! + apply Nat.modEq_one + | succ la => simp! at g + apply lemma8 at g + forward g g (5+j+i+ra+la+2) + rw [List.append_cons] at g + rw [← List.replicate_succ'] at g + simp! at g + apply lemma10 at g + use (8+j+i+ra+la+la+1) + constructor + . omega + . use! 0, (la+1) + rw [g] + simp! + apply Nat.ModEq.add_right 1 + apply Nat.ModEq.trans hm + rw [Nat.modEq_zero_iff_dvd] +| succ lb => simp! at g + apply lemma_6_to_7 at g + obtain ⟨k, _, g⟩ := g + forward g g k + apply lemma_9_to_10 at g + obtain ⟨n, _, g⟩ := g + use n + constructor + . omega + . use! (la+1), lb + simp! [g] + constructor + . apply Nat.ModEq.add_right 1 hm + . omega + +-- c1++, c2++ +-- l 0 [la 11] 0 [lb 11] 0 [ra 11111111] 0 [rb 1] 0 r +-- c1 ^ c2 +-- +-- l 0 [la' 1] 0 [lb' 1] 0 [(ra+rb) 111] 0 0 r +-- c1 ^ c2 +theorem end_10 (rb : ℕ): ∀ (i la lb ra : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate lb Γ.one ++ List.cons Γ.zero (List.replicate la Γ.one ++ List.cons Γ.zero l)), + Turing.ListBlank.mk (List.replicate ra Γ.one ++ List.cons Γ.zero (List.replicate rb Γ.one ++ List.cons Γ.zero r))⟩⟩ → + (ra + 2) ≡ (la + 1) [MOD (la + lb + 1)] → +∃ j≥i, ∃ la' lb', nth_cfg j = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate lb' Γ.one ++ List.cons Γ.zero (List.replicate la' Γ.one ++ List.cons Γ.zero l)), + Turing.ListBlank.mk (List.replicate (ra+rb) Γ.one ++ List.cons Γ.zero (List.cons Γ.zero r))⟩⟩ + /\ (ra + rb + 2) ≡ (la' + 1) [MOD (la + lb + 1)] + /\ la + lb = la' + lb' +:= by +induction rb with intros i la lb ra l r g hm +| zero => use! i + constructor + . omega + . use! la, lb + simp [g, hm] +| succ rb => rename_i induction_step + apply step_10_pointer at g + specialize g hm + obtain ⟨j, _, la', lb', g, h₁, h₂⟩ := g + apply induction_step at g + rw [← h₂] at g + ring_nf at * + specialize g h₁ + obtain ⟨k, _, la'', lb'', g, h₃, h₄⟩ := g + use! k + constructor + . omega + . use! la'', lb'' + ring_nf at * + repeat any_goals constructor + all_goals tauto + + +-- l 0 1 0 [lb 11] 0 0 [(rb+1) 1] 0 r +-- c1 ^ c2 +-- +-- l 0 1 0 [(lb+1) 1] 0 0 [rb 11111] 0 r +-- c1 ^ c2 +theorem step_10_n_dvd (i lb rb: ℕ) (l r : List Γ) : +nth_cfg i = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate lb Γ.one ++ Γ.zero :: Γ.one :: Γ.zero :: l), + Turing.ListBlank.mk (List.cons Γ.zero (List.replicate (rb+1) Γ.one ++ List.cons Γ.zero r))⟩⟩ → + ¬ (lb + 2) ∣ (rb + 3) → +∃ j>i, nth_cfg j = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate (lb+1) Γ.one ++ Γ.zero :: Γ.one :: Γ.zero :: l), + Turing.ListBlank.mk (List.cons Γ.zero (List.replicate rb Γ.one ++ List.cons Γ.zero r))⟩⟩ +:= by +intros g ne_dvd +have h := end_10 (rb+1) i 1 lb 0 l r +simp at h +specialize h g rfl +clear g +obtain ⟨j, _, la', lb', g, h₁, h₂⟩ := h +apply lemma_10_to_11 at g +obtain ⟨k, g⟩ := g +forward g g (k+j) +forward g g (1+k+j) +rw [← List.cons_append, ← List.replicate.eq_2] at g +apply rec14 at g +forward g g (2+k+j+rb.succ+1) +cases lb' with (simp! [*, -nth_cfg] at g; simp_all) +| zero => subst la' + exfalso + apply ne_dvd + rw [← Nat.modEq_zero_iff_dvd] + ring_nf at * + apply Nat.ModEq.trans h₁ + rw [Nat.modEq_zero_iff_dvd] +| succ lb' => apply lemma_15_to_19 at g + obtain ⟨m, g⟩ := g + forward g g (m + (5 + k + j + rb)) + apply rec20 at g + forward g g (6 + m + k + j + rb + lb' + 1) + forward g g (8 + m + k + j + rb + lb') + forward g g (9 + m + k + j + rb + lb') + forward g g (10 + m + k + j + rb + lb') + rw [← List.cons_append] at g + rw [← List.replicate.eq_2] at g + rw [ List.append_cons ] at g + rw [← List.replicate_succ'] at g + rw [← List.append_assoc] at g + rw [← List.replicate_add] at g + apply lemma5 at g + forward g g (11 + m + k + j + rb + lb' + (lb'.succ + 1 + la') + 2) + rw [List.replicate_add] at g + rw [List.replicate_add] at g + simp! [*, -nth_cfg] at g + forward g g (16 + m + k + j + rb + lb' * 2 + la') + rw [List.replicate_add] at g + rw [List.replicate_add] at g + simp! [*, -nth_cfg] at g + apply lemma10 at g + use (17 + m + k + j + rb + lb' * 2 + la' + (1 + lb' + la' - 1) + 1) + constructor + . omega + . rw [g] + simp! + repeat any_goals first | omega | rfl | apply congr + +-- l 0 1 0 [lb 11] 0 0 [rb 1] 0 r +-- c1 ^ c2 +-- 10 +-- +-- l 0 1 1 [lb 11] 1 1 [rb 1] 0 r +-- ^ +-- 17 +theorem step_10_dvd (i lb rb: ℕ) (l r : List Γ) : +nth_cfg i = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate lb Γ.one ++ Γ.zero :: Γ.one :: Γ.zero :: l), + Turing.ListBlank.mk (List.cons Γ.zero (List.replicate rb Γ.one ++ List.cons Γ.zero r))⟩⟩ → + (lb + 2) ∣ (rb + 2) → +∃ j>i, nth_cfg j = some ⟨⟨17, by omega⟩, ⟨Γ.zero, + Turing.ListBlank.mk l, + Turing.ListBlank.mk (List.replicate (lb+rb+4) Γ.one ++ List.cons Γ.zero r)⟩⟩ +:= by +intros g dvd +have h := end_10 rb i 1 lb 0 l r +simp at h +specialize h g rfl +clear g +obtain ⟨j, _, la', lb', g, h₁, h₂⟩ := h +apply lemma_10_to_11 at g +obtain ⟨k, g⟩ := g +forward g g (k+j) +forward g g (1+k+j) +apply rec14 at g +forward g g (2+k+j+rb+1) +rw [← List.cons_append] at g +cases lb' with (simp! [*, -nth_cfg] at g; simp_all) +| zero => subst la' + forward g g (4+k+j+rb) + forward g g (5+k+j+rb) + apply rec17 at g + use (6 + k + j + rb + (1+lb)+1) + constructor + . omega + . simp [g] + rw [List.append_cons, ← List.replicate_succ'] + rw [List.append_cons, ← List.replicate_succ'] + rw [← List.append_assoc, ← List.replicate_add] + ring_nf +| succ lb' => have : lb = la'+ lb' := by omega + subst lb + clear h₂ + ring_nf at * + rw [Nat.ModEq.comm] at h₁ + rw [← Nat.modEq_zero_iff_dvd] at dvd + have h := Nat.ModEq.trans h₁ dvd + rw [Nat.modEq_zero_iff_dvd] at h + apply Nat.le_of_dvd at h <;> omega + +-- l 0 1 0 [lb 11] 0 0 [rb 1] 0 r +-- c1 ^ c2 +-- +-- l 0 1 0 [(lb+rb) 1] 0 0 0 r +-- c1 ^ c2 +theorem leap_10_prime (rb : ℕ) : ∀ (i lb: ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate lb Γ.one ++ Γ.zero :: Γ.one :: Γ.zero :: l), + Turing.ListBlank.mk (List.cons Γ.zero (List.replicate rb Γ.one ++ List.cons Γ.zero r))⟩⟩ → +Nat.Prime (lb+rb+4) → +∃ j≥i, nth_cfg j = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate (lb+rb) Γ.one ++ Γ.zero :: Γ.one :: Γ.zero :: l), + Turing.ListBlank.mk (Γ.zero :: Γ.zero :: r)⟩⟩ +:= by +induction rb with intros i lb l r g p +| zero => use i + constructor + . omega + . simp! [g] +| succ rb => apply step_10_n_dvd at g + by_cases h : (lb + 2 ∣ rb + 3) + . have g : (lb + 2 ∣ lb + 2) := Nat.dvd_refl _ + have g := Nat.dvd_add g h + ring_nf at p g + rw [Nat.Prime.dvd_iff_eq] at g + any_goals omega + all_goals assumption + . specialize g h + rename_i induction_step + obtain ⟨j, _, g⟩ := g + apply induction_step at g + ring_nf at g p + specialize g p + obtain ⟨k, _, g⟩ := g + use k + constructor + . omega + . simp! [g] + rw [← List.cons_append] + rw [← List.replicate_succ] + ring_nf + +theorem prime_21 (i r1: ℕ) (l r : List Γ) +(g : +nth_cfg i = some ⟨⟨0, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (Γ.zero :: l), Turing.ListBlank.mk (List.replicate r1 Γ.one ++ List.cons Γ.zero r)⟩⟩) +(p : Nat.Prime (r1+1)) : +∃ j>i, nth_cfg j = some ⟨⟨21, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (r1+1) Γ.one ++ List.cons Γ.zero r)⟩⟩ := +match h : r1 with +| Nat.zero => by subst r1 + exfalso + apply Nat.not_prime_one p +| Nat.succ Nat.zero => by + subst r1 + forward g g i + forward g g (1+i) + forward g g (2+i) + forward g g (3+i) + forward g g (4+i) + use (5+i) + simp [g] +| Nat.succ (Nat.succ Nat.zero) => by + subst r1 + forward g g i + forward g g (1+i) + forward g g (2+i) + forward g g (3+i) + forward g g (4+i) + forward g g (5+i) + forward g g (6+i) + use (7+i) + simp [g] +| Nat.succ (Nat.succ (Nat.succ r1)) => by + clear h + forward g g i + forward g g (1+i) + forward g g (2+i) + forward g g (3+i) + forward g g (4+i) + forward g g (5+i) + forward g g (6+i) + forward g g (7+i) + forward g g (8+i) + forward g g (9+i) + apply (leap_10_prime _ _ 0) at g + have h : r1.succ.succ.succ + 1 = 0 + r1 +4 := by omega + rw [h] at p + specialize g p + obtain ⟨j, _, g⟩ := g + forward g g j + forward g g (1+j) + forward g g (2+j) + forward g g (3+j) + forward g g (4+j) + cases r1 with + | zero => exfalso + have g : ¬ Nat.Prime 4 := by decide + apply g p + | succ r1 => apply lemma_15_to_19 at g + obtain ⟨k, g⟩ := g + forward g g (k+(5+j)) + apply rec20 at g + forward g g (6 + k + j + r1 + 1) + forward g g (8 + k + j + r1) + forward g g (9 + k + j + r1) + forward g g (10 + k + j + r1) + rw [List.append_cons, ← List.replicate_succ'] at g + rw [List.append_cons, ← List.replicate_succ'] at g + rw [← List.cons_append] at g + rw [← List.replicate_succ] at g + apply rec21 at g + use (11 + k + j + r1 + (r1 + 1 + 1 + 1) + 1) + constructor + . omega + . rw [g] + simp + rw [List.append_cons, ← List.replicate_succ'] + + +-- l 0 1 0 [lb 11] 0 0 [rb 1] 0 r +-- c1 ^ c2 +-- 10 +-- +-- l 0 1 1 [lb 11] 1 1 [rb 1] 0 r +-- ^ +-- 17 +theorem leap_10_dvd (rb : ℕ) : ∀ (i lb: ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate lb Γ.one ++ Γ.zero :: Γ.one :: Γ.zero :: l), + Turing.ListBlank.mk (List.cons Γ.zero (List.replicate rb Γ.one ++ List.cons Γ.zero r))⟩⟩ → +(∃ divisor, divisor ≥ (lb+2) /\ divisor < lb + rb + 4 /\ divisor ∣ (lb + rb + 4)) → +∃ j>i, nth_cfg j = some ⟨⟨17, by omega⟩, ⟨Γ.zero, + Turing.ListBlank.mk l, + Turing.ListBlank.mk (List.replicate (lb+rb+4) Γ.one ++ List.cons Γ.zero r)⟩⟩ +:= by +induction rb with intros i lb l r g hd +| zero => by_cases h : (lb+2) ∣ 2 + . apply step_10_dvd at g + ring_nf at * + apply g h + . obtain ⟨divisor, _, _, h₃⟩ := hd + have g : divisor = lb + 2 \/ divisor = lb + 3 := by omega + have h₁ : divisor ∣ divisor := Nat.dvd_refl _ + ring_nf at * + cases g <;> subst divisor + . have g := Nat.dvd_sub (by omega) h₃ h₁ + have h₁ : 4+lb-(2+lb) = 2 := by omega + rw [h₁] at g + specialize h g + tauto + . have g := Nat.dvd_sub (by omega) h₃ h₁ + have h : 4+lb-(3+lb) = 1 := by omega + rw [h] at g + apply Nat.eq_one_of_dvd_one at g + omega +| succ rb => by_cases h : (lb+2) ∣ rb+3 + . apply step_10_dvd at g + apply g h + . apply step_10_n_dvd at g + specialize g h + rename_i induction_step + obtain ⟨j, _, g⟩ := g + apply induction_step at g + refine (?_ ∘ g) ?_ + . intros g + obtain ⟨k, _, g⟩ := g + use k + constructor + . omega + . simp [g] + ring_nf + . obtain ⟨divisor, h₁, h₂, h₃⟩ := hd + use divisor + ring_nf at * + repeat any_goals apply And.intro + any_goals omega + by_cases h : divisor = 2 + lb + . subst divisor + exfalso + apply h + have h := Nat.dvd_sub (by omega) h₃ (Nat.dvd_refl _) + have g : (5 + lb + rb - (2+lb)) = 3 + rb := by omega + rw [g] at h + assumption + . omega + +theorem n_prime_17 (i r1: ℕ) (l r : List Γ) +(g : +nth_cfg i = some ⟨⟨0, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (Γ.zero :: l), Turing.ListBlank.mk (List.replicate r1 Γ.one ++ List.cons Γ.zero r)⟩⟩) +(hp : ¬ Nat.Prime (r1+1)) : +∃ j>i, nth_cfg j = some ⟨⟨17, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (r1+1) Γ.one ++ List.cons Γ.zero r)⟩⟩ := +match r1 with +| Nat.zero => by simp_all + forward g g i + forward g g (1+i) + forward g g (2+i) + use (3+i) + rw [← g] + simp +| Nat.succ Nat.zero => by exfalso + apply hp + decide +| Nat.succ (Nat.succ Nat.zero) => by exfalso + apply hp + decide +| Nat.succ (Nat.succ (Nat.succ r1)) => by + forward g g i + forward g g (1+i) + forward g g (2+i) + forward g g (3+i) + forward g g (4+i) + forward g g (5+i) + forward g g (6+i) + forward g g (7+i) + forward g g (8+i) + forward g g (9+i) + apply (leap_10_dvd _ _ 0) at g + refine (?_ ∘ g) ?_ + . intros h + obtain ⟨j, _, h⟩ := h + use j + constructor + . omega + . simp [h] + . rw [Nat.not_prime_iff_exists_dvd_lt] at hp <;> try omega + obtain ⟨p, h, g, hp⟩ := hp + use p + repeat any_goals apply And.intro + any_goals omega + have g : r1.succ.succ.succ +1 = 0+r1+4 := by omega + rw [g] at h + exact h + +end Tm27 diff --git a/GoldbachTm/Tm27/Search0.lean b/GoldbachTm/Tm27/Search0.lean new file mode 100644 index 0000000..4184c80 --- /dev/null +++ b/GoldbachTm/Tm27/Search0.lean @@ -0,0 +1,129 @@ +-- theorem of recursive states +-- all these states' usage is to search 0 +import GoldbachTm.Tm27.TuringMachine27 + +namespace Tm27 + +-- left +theorem rec13 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨13, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨13, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ r)⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) l (List.cons Γ.one r) + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +theorem rec14 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨14, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨14, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ r)⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) l (List.cons Γ.one r) + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +theorem rec17 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨17, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨17, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ r)⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) l (List.cons Γ.one r) + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +theorem rec19 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨19, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨19, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ r)⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) l (List.cons Γ.one r) + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +theorem rec21 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨21, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨21, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ r)⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) l (List.cons Γ.one r) + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +theorem rec24 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨24, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨24, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ r)⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) l (List.cons Γ.one r) + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +--right +theorem rec11 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨11, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero r) ⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨11, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ l), Turing.ListBlank.mk r⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) (List.cons Γ.one l) r + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +theorem rec20 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨20, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero r) ⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨20, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ l), Turing.ListBlank.mk r⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) (List.cons Γ.one l) r + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +theorem rec23 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨23, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero r) ⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨23, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ l), Turing.ListBlank.mk r⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) (List.cons Γ.one l) r + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +theorem rec26 (k : ℕ): ∀ (i : ℕ) (l r : List Γ), +nth_cfg i = some ⟨⟨26, by omega⟩, ⟨Γ.one, Turing.ListBlank.mk l, Turing.ListBlank.mk (List.replicate k Γ.one ++ List.cons Γ.zero r) ⟩⟩ → +nth_cfg (i + k + 1) = some ⟨⟨26, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate (k+1) Γ.one ++ l), Turing.ListBlank.mk r⟩⟩ := by +induction k with intros i l r h +| zero => simp [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] +| succ k => rename_i induction_step + specialize induction_step (i+1) (List.cons Γ.one l) r + have g : i + (k+1) +1 = i + 1 + k + 1 := by omega + rw [g, induction_step] + . simp [List.replicate_succ' (k+1)] + . simp! [nth_cfg, h, step, machine, Turing.Tape.write, Turing.Tape.move] + +end Tm27 diff --git a/GoldbachTm/Tm27/Transition.lean b/GoldbachTm/Tm27/Transition.lean new file mode 100644 index 0000000..cb80cfd --- /dev/null +++ b/GoldbachTm/Tm27/Transition.lean @@ -0,0 +1,68 @@ +import GoldbachTm.Tm27.TuringMachine27 +import GoldbachTm.Tm27.Search0 + +namespace Tm27 + +theorem lemma_10_to_11 (i : ℕ) (r1: ℕ) (l r : List Γ) +(h : +nth_cfg i = some ⟨⟨10, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk l, + Turing.ListBlank.mk (List.replicate r1 Γ.one ++ List.cons Γ.zero r)⟩⟩) : + +∃ j, nth_cfg (j + i) = some ⟨⟨11, by omega⟩, ⟨Γ.zero, Turing.ListBlank.mk (List.replicate r1 Γ.one ++ List.cons Γ.zero l), Turing.ListBlank.mk r⟩⟩ +:= by +forward h h i +cases r1 with +| zero => use 1 + simp [h] +| succ r1 => apply rec11 at h + use (r1 + 2) + rw [← h] + ring_nf + +theorem lemma_15_to_19 (i : ℕ) (l1: ℕ) (l r : List Γ) +(h : +nth_cfg i = some ⟨⟨15, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate l1 Γ.one ++ List.cons Γ.zero l), + Turing.ListBlank.mk r, + ⟩⟩) : +∃ j, nth_cfg (j + i) = some ⟨⟨19, by omega⟩, ⟨Γ.zero, + Turing.ListBlank.mk l, + Turing.ListBlank.mk (List.replicate (l1+1) Γ.one ++ r), + ⟩⟩ +:= by +forward h h i +cases l1 with simp! [*, -nth_cfg] at h +| zero => use 1 + simp [h] +| succ r1 => apply rec19 at h + use (r1+2) + ring_nf at * + simp [h] + rw [List.append_cons, ← List.replicate_succ'] + ring_nf + +theorem lemma_22_to_24 (i : ℕ) (l1: ℕ) (l r : List Γ) +(h : +nth_cfg i = some ⟨⟨22, by omega⟩, ⟨Γ.one, + Turing.ListBlank.mk (List.replicate l1 Γ.one ++ List.cons Γ.zero l), + Turing.ListBlank.mk r, + ⟩⟩) : + +∃ j, nth_cfg (j + i) = some ⟨⟨24, by omega⟩, ⟨Γ.zero, + Turing.ListBlank.mk l, + Turing.ListBlank.mk (List.replicate (l1+1) Γ.one ++ r), + ⟩⟩ +:= by +forward h h i +cases l1 with +| zero => use 1 + simp [h] +| succ l1 => apply rec24 at h + use (l1 + 2) + ring_nf at * + simp [h] + rw [List.append_cons, ← List.replicate_succ'] + ring_nf + +end Tm27 diff --git a/GoldbachTm/Tm27/TuringMachine27.lean b/GoldbachTm/Tm27/TuringMachine27.lean index f7f8015..73d9112 100644 --- a/GoldbachTm/Tm27/TuringMachine27.lean +++ b/GoldbachTm/Tm27/TuringMachine27.lean @@ -1,5 +1,6 @@ -- inspired by https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Computability/TuringMachine.lean import Mathlib.Computability.TuringMachine +import Mathlib.Data.Real.Sqrt import GoldbachTm.Basic import GoldbachTm.Format import GoldbachTm.ListBlank @@ -91,4 +92,81 @@ def machine : Machine | ⟨26, _⟩, Γ.one => some ⟨⟨26, by omega⟩, ⟨Turing.Dir.right, Γ.one⟩⟩ | ⟨_+27, _⟩, _ => by omega -- https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Pattern.20matching.20on.20Fin.20isn't.20exhaustive.20for.20large.20matches/near/428048252 +def nth_cfg : (n : Nat) -> Option Cfg +| 0 => init [] +| Nat.succ n => match (nth_cfg n) with + | none => none + | some cfg => step machine cfg + + +-- g1 = g2 +macro "forward" g1:ident g2:Lean.binderIdent i:term: tactic => `(tactic| ( +have h : nth_cfg ($i + 1) = nth_cfg ($i + 1) := rfl +nth_rewrite 2 [nth_cfg] at h +simp [*, step, Option.map, machine, Turing.Tape.write, Turing.Tape.move] at h +try simp! [*, -nth_cfg] at h +try ring_nf at h +clear $g1 +rename_i $g2 +)) + +theorem cfg45 : nth_cfg 45 = some ⟨26, + { head := default, left := Turing.ListBlank.mk (List.replicate 4 Γ.one), right := Turing.ListBlank.mk [] } ⟩ := by +have h : nth_cfg 0 = init [] := by simp! +simp [init, Turing.Tape.mk₁, Turing.Tape.mk₂, Turing.Tape.mk'] at h +forward h h 0 +forward h h 1 +forward h h 2 +forward h h 3 +forward h h 4 +forward h h 5 +forward h h 6 +forward h h 7 +forward h h 8 +forward h h 9 +forward h h 10 +forward h h 11 +forward h h 12 +forward h h 13 +forward h h 14 +forward h h 15 +forward h h 16 +forward h h 17 +forward h h 18 +forward h h 19 +forward h h 20 +forward h h 21 +forward h h 22 +forward h h 23 +forward h h 24 +forward h h 25 +forward h h 26 +forward h h 27 +forward h h 28 +forward h h 29 +forward h h 30 +forward h h 31 +forward h h 32 +forward h h 33 +forward h h 34 +forward h h 35 +forward h h 36 +forward h h 37 +forward h h 38 +forward h h 39 +forward h h 40 +forward h h 41 +forward h h 42 +forward h h 43 +forward h h 44 +simp [h] +constructor +. tauto +. simp! [Turing.ListBlank.mk] + rw [Quotient.eq''] + right + use 2 + tauto + + end Tm27 diff --git a/archieve/Busy beaver - Wikipedia.pdf b/archieve/Busy beaver - Wikipedia.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aba930020555855cfac7e4b9b9be474732006ced GIT binary patch literal 1506633 zcmaHSWl)~MvL+VXArRc%9fG^NdvJI6gy8NHAh=s_|A62c2=4Cg4*QaGx6c09x|p9<0?*O6w|Mw*V0`O+`rte%}`!6?4jEGs_nMCA?nG_tH z?Tl>yyNLb2i$sOr7yW0coG39fJd>Qvzu%hgf0vgdW`}1|6()9db~F3`@5TAwy%dPq z;F%<>Y+Zr7GD+A1ml88GaWDlwlryupaJ3|6<>3?%Aa-#Dq+kc{ndukTA=C2(Gla-S z7_(0}6d4%@UvK)=GN7xlQcF)aSfiL-K5{-gu4wvJ>dqU(Pw78)Pb1ox=fy$KxpH$D}Nwn8FT!X_ z+>BlS{YBc&$O7QIrj;qcc6JUxtWsuH7M8BW9PGgVMICG%oK+o-Ow5RNnWRNUMT}g` zOo`tmsiF@^*~-;L$;?^Q!Oqda-pt;WnEQX1DjPQ&7yJJ!*W)ZWUlX&Iw-M42=FwRz z%B)&gQUgs?@DA^efSa+%_ZgS(-aLfL^sljpfM>B(fK|Gxx5;~cU zPQJ*k3pI6ha1lfQhn>gAN3)TXN@6lHGCX5lknw}c#>lysgJDaNzuDM$azmR2z5XxX zRlyXdYbL*~{=Uvu$BU`p{Z8oY+WUjj$ZSS*gXa76gOPFr{m)ysW_739gx*dEYp*_D 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