wip

wip

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

GoldbachTm/Tm27/Content.lean

@@ -0,0 +1,16 @@
+import GoldbachTm.Tm27.TuringMachine27
+import GoldbachTm.Tm27.Search0
+import Mathlib.Data.Nat.Prime.Defs
+
+namespace Tm27
+
+theorem lemma_26 (n : ℕ) (i : ℕ)
+(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
+sorry
+
+
+end Tm27

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

GoldbachTm/Tm27/PBP.lean

@@ -0,0 +1,94 @@
+-- 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
+
+end Tm27