| import Mathlib |
| import OAI.Computability.MaxCut.Encoding.VerticesBits |
| import OAI.Computability.MaxCut.Estimates.VecEquivSymmVal |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace OptimalMaxCut.RawGrid |
| open scoped BigOperators |
| open Finset MaxCutGames.Foundations.Target MaxCutGames.Foundations.Hastad |
| open LongCode LongCode.InstanceAdapter CounterMachine.Expr |
| attribute [local instance] Classical.propDecidable |
|
|
| theorem adjacency_bits {q : ℕ} (g : Instance q) (t : ℚ) |
| (ht : t ∈ Set.Icc (-1:ℚ) 1) (K L : ℕ) |
| (anchor : Fin (g.vertices*2^q)) |
| (i j : Fin ((g.vertices*2^q)*(Unweighted.searchPrime (L*(g.vertices*2^q+1)^2))^6)) : |
| let w := (ofInstance g).gridWeights t ht (vertexEquiv g.vertices q) anchor K |
| let out := Unweighted.deterministicOutput L w (aggregate_symm _) |
| out.graph.adj i j = decide (adjacency q t ht K L (bits g) i.val j.val ≠ 0) := by |
| dsimp only |
| let p := Unweighted.searchPrime (L*(g.vertices*2^q+1)^2) |
| let : Fact p.Prime := ⟨(Unweighted.searchPrime_spec _).1⟩ |
| have hparam : param q L (bits g) = p := by simp [param,p] |
| change decide (_ ∧ _ ∈ _) = _ |
| apply Bool.decide_congr |
| rw [Unweighted.residue_membership] |
| rw [weights_bits] |
| have hf : ⌊(p:ℚ)*((weights q t ht K (bits g) (i.val/p^6) (j.val/p^6):ℕ):ℚ)/ |
| (gridSize q K (bits g):ℚ)⌋₊ = p*weights q t ht K (bits g) (i.val/p^6) (j.val/p^6)/gridSize q K (bits g) := by |
| rw [← Nat.cast_mul,Nat.floor_div_natCast,Nat.floor_natCast] |
| have hd : (Unweighted.dot ((Unweighted.vertexEquiv (g.vertices*2^q)).symm i).2 |
| ((Unweighted.vertexEquiv (g.vertices*2^q)).symm j).2).val = pairing p i.val j.val := by |
| rw [Unweighted.dot_val] |
| simp only [Unweighted.vertexEquiv_symm_second,pairing] |
| rfl |
| simp only [mul_div_assoc, Unweighted.vertexEquiv_symm_first] at * |
| rw [hf,hd] |
| have hne : ((Unweighted.vertexEquiv (g.vertices*2^q)).symm i).1 ≠ |
| ((Unweighted.vertexEquiv (g.vertices*2^q)).symm j).1 ↔ i.val/p^6 ≠ j.val/p^6 := by |
| rw [ne_eq, Fin.ext_iff] |
| rfl |
| rw [hne] |
| unfold adjacency |
| rw [hparam] |
| change (_ ∧ _) ↔ ((if i.val/p^6=j.val/p^6 then 0 else _) ≠ 0) |
| by_cases hij : i.val/p^6=j.val/p^6 |
| · simp [hij] |
| · simp only [ne_eq,hij,not_false_eq_true,true_and,ite_false] |
| simp only [sum_eq_zero_iff,mem_range,ite_eq_right_iff,one_ne_zero,imp_false,not_forall,not_not] |
| constructor |
| · rintro ⟨r,hr,he⟩; exact ⟨r,hr,he⟩ |
| · rintro ⟨r,hr,he⟩; exact ⟨r,hr,he⟩ |
| end OptimalMaxCut.RawGrid |
|
|
| end |
|
|
| end OAI |
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|
