| import OAI.NumberTheory.TwoPoint.Bounds.ActualWordStateIdentity |
| import OAI.NumberTheory.TwoPoint.Bounds.ShiftedIntegerPaths |
|
|
| namespace OAI |
|
|
| /-! The actual closed masked word is its finite-state coefficient times |
| one indicator for survival of all its vertices. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| lemma wordVertexSite_closed {R : ℕ} (h : ℕ) (step : Fin R → SignedStep) (n : ℤ) |
| (hclosed : wordDisplacement h (List.ofFn step) = 0) : |
| wordVertexSite h step n (Fin.last R) = wordVertexSite h step n 0 := by |
| simp [wordVertexSite, List.take_of_length_le (by simp : (List.ofFn step).length ≤ R), hclosed] |
|
|
| lemma closed_word_vertex_iff {R : ℕ} (hR : 0 < R) (h : ℕ) |
| (step : Fin R → SignedStep) (n : ℤ) |
| (hclosed : wordDisplacement h (List.ofFn step) = 0) (keep : ℤ → Prop) : |
| (∀ i : Fin R, keep (wordVertexSite h step n i.castSucc)) ↔ |
| ∀ i : Fin (R + 1), keep (wordVertexSite h step n i) := by |
| constructor |
| · intro hi i |
| by_cases hb : i.val < R |
| · have he : (⟨i.val, hb⟩ : Fin R).castSucc = i := Fin.ext rfl |
| simpa only [he] using hi ⟨i.val, hb⟩ |
| · have he : i = Fin.last R := Fin.ext (by simp only [Fin.val_last]; omega) |
| rw [he, wordVertexSite_closed h step n hclosed] |
| exact hi ⟨0, hR⟩ |
| · intro hi i |
| exact hi i.castSucc |
|
|
| lemma closed_word_mask {R : ℕ} (hR : 0 < R) (h : ℕ) |
| (step : Fin R → SignedStep) (n : ℤ) |
| (hclosed : wordDisplacement h (List.ofFn step) = 0) (keep : ℤ → Prop) : |
| scalarWalkProduct h (fun _ z => vertexIndicator keep z) n (List.ofFn step) = |
| if ∀ i : Fin (R + 1), keep (wordVertexSite h step n i) then 1 else 0 := by |
| rw [scalarWalkProduct_ofFn] |
| change (∏ i : Fin R, vertexIndicator keep (wordVertexSite h step n i.castSucc)) = _ |
| by_cases hv : ∀ i : Fin R, keep (wordVertexSite h step n i.castSucc) |
| · have hv' := (closed_word_vertex_iff hR h step n hclosed keep).mp hv |
| simp [vertexIndicator, hv, hv'] |
| · have hv' : ¬∀ i : Fin (R + 1), keep (wordVertexSite h step n i) := |
| fun hh => hv ((closed_word_vertex_iff hR h step n hclosed keep).mpr hh) |
| rw [ite_eq_right hv'] |
| obtain ⟨i, hi⟩ := not_forall.mp hv |
| exact prod_eq_zero (mem_univ i) (by simp [vertexIndicator, hi]) |
|
|
| lemma closed_actual_word_state {J R m : ℕ} {P : Fin J → Finset ℕ} |
| (w : ColumnPrimeAssignment J R P) (forward : Fin R → Bool) (padding : Fin R → ℕ) |
| (hprime : ∀ j, ∀ p ∈ P j, p.Prime) |
| (hdisjoint : ∀ j l, l ≠ j → Disjoint (P j) (P l)) |
| (Qp Q : Finset ℕ) (hQp : ∀ p ∈ Qp, p.Prime) (hQ : Q ⊆ retainedPrimeDivisors Qp) |
| (e : Fin m ≃ Qp) (eligible : ℕ → ℕ → Prop) (L K : ℝ) (h : ℕ) (n : ℤ) |
| (hR : 0 < R) (hclosed : wordDisplacement h (columnTupleWord w forward padding) = 0) |
| (keep : ℤ → Prop) : |
| let step := fun i => SignedStep.mk (forward i) (columnTuple w i) (padding i) |
| scalarWalkProduct h |
| (maskedSignedIntegerWeight Q actualPaddingCoefficient eligible (actualPaddingVertex Qp) |
| (fun d => centeredTuple d.primeFactors) L K (fun _ => actualPaddingDegreeCut Qp L) h keep) |
| n (List.ofFn step) = |
| if ∀ i : Fin (R + 1), keep (wordVertexSite h step n i) then |
| actualWordCoefficient Qp Q e (fun t => eligible t.tuple) L K step (fun i j => (w j i).val) |
| (fun i => paddingActiveState Qp e (wordVertexSite h step n i)) (wordCenteredBits w h step n) |
| else 0 := by |
| dsimp only |
| rw [show maskedSignedIntegerWeight Q actualPaddingCoefficient eligible (actualPaddingVertex Qp) |
| (fun d => centeredTuple d.primeFactors) L K (fun _ => actualPaddingDegreeCut Qp L) h keep = |
| fun t z => vertexIndicator keep z * |
| signedIntegerWeight Q actualPaddingCoefficient (eligible t.tuple) (actualPaddingVertex Qp) |
| (centeredTuple t.tuple.primeFactors) L K (actualPaddingDegreeCut Qp L) h t z * |
| vertexIndicator keep (z + t.displacement h) from rfl] |
| rw [closed_masked_scalar_word hR h _ n hclosed _ _ |
| (fun z => by unfold vertexIndicator; split_ifs <;> simp)] |
| rw [closed_signed_word_factorization Q actualPaddingCoefficient (fun t => eligible t.tuple) |
| (actualPaddingVertex Qp) (fun t => centeredTuple t.tuple.primeFactors) L K |
| (fun _ => actualPaddingDegreeCut Qp L) h n |
| (List.ofFn (fun i => SignedStep.mk (forward i) (columnTuple w i) (padding i))) |
| (actualPaddingVertex_ne_zero Qp) |
| (fun t q _ z => centeredTuple_padding_periodic t.tuple.primeFactors h t.tuple |
| (fun p hp => (Nat.mem_primeFactors.mp hp).2.1) q z) hclosed] |
| rw [← actualWordCoefficient_integer_identity w forward padding hprime hdisjoint Qp Q hQp hQ e] |
| rw [closed_word_mask hR h _ n hclosed keep] |
| split_ifs <;> simp |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
