| import Mathlib |
| import OAI.Computability.QuantumFactoring.BooleanFold |
| import OAI.Computability.QuantumFactoring.StaticPredicates |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
| |
|
|
| namespace ExactQuantumFactoring.BooleanNetwork |
| |
| lemma all_ofFn_eval {n k : ℕ} (cs : Fin k → BooleanNetwork n 1) (x : Basis n) : |
| (all (List.ofFn cs)).eval x 0=true ↔ ∀ i, (cs i).eval x 0=true := by |
| rw [all_eval] |
| simp |
|
|
| lemma any_ofFn_eval {n k : ℕ} (cs : Fin k → BooleanNetwork n 1) (x : Basis n) : |
| (any (List.ofFn cs)).eval x 0=true ↔ ∃ i, (cs i).eval x 0=true := by |
| rw [any_eval] |
| simp |
|
|
| lemma bnot_value {n : ℕ} (c : BooleanNetwork n 1) (x : Basis n) : |
| c.bnot.eval x 0=true ↔ ¬ c.eval x 0=true := by simp |
|
|
| end ExactQuantumFactoring.BooleanNetwork |
| |
| namespace ExactQuantumFactoring.BitArithmetic |
| open BooleanNetwork |
| |
| def orderTests (w s k : ℕ) : BooleanNetwork w 1 := |
| all (List.ofFn (fun i : Fin k => (powerOneNet w s (i.val+1)).bnot)) |
| |
| lemma orderTests_value (w s k : ℕ) (x : Basis w) (hs : 2 ≤ s) |
| (hsw : s < 2^w) (hkw : k < 2^w) : |
| (orderTests w s k).eval x 0=true ↔ |
| ∀ j : ℕ, 1 ≤ j → j ≤ k → (bitsValue x).toNat^j%s ≠ 1 := by |
| rw [orderTests,all_ofFn_eval] |
| constructor |
| · intro h j hj hjk |
| have hh := (bnot_value _ _).mp (h (⟨j-1,by omega⟩ : Fin k)) |
| rw [powerOneNet_value _ _ _ _ hs hsw (by omega)] at hh |
| simpa only [show j-1+1=j by omega] using hh |
| · intro h i |
| rw [bnot_value,powerOneNet_value _ _ _ _ hs hsw (by have := i.isLt |
| exact h (i.val+1) (by omega) (by omega) |
|
|
| lemma orderTests_count (w s k : ℕ) : (orderTests w s k).net.count ≤ |
| k*(100*w+23+w*(3672*w*w+436*w+36))+1 := by |
| have h := all_count (c := 100*w+22+w*(3672*w*w+436*w+36)) |
| (List.ofFn (fun i : Fin k => (powerOneNet w s (i.val+1)).bnot)) (by |
| intro c hc |
| obtain ⟨i,rfl⟩ := List.mem_ofFn.mp hc |
| rw [count_bnot] |
| have := powerOneNet_count w s (i.val+1) |
| omega) |
| convert h using 1 < |
| ring |
|
|
| def properDivisor (w d : ℕ) : BooleanNetwork w 1 := |
| (dividesNet w d).band (wordLt (wordConstant (BitVec.ofNat w d)) (select id)) |
|
|
| lemma properDivisor_value (w d : ℕ) (x : Basis w) (hd : d < 2^w) : |
| (properDivisor w d).eval x 0=true ↔ d ∣ (bitsValue x).toNat ∧ d < (bitsValue x).toNat := by |
| rw [properDivisor,eval_band,Bool.and_eq_true,dividesNet_value _ _ _ hd, |
| wordLt_eval,wordConstant_eval,eval_select,Function.comp_id,BitVec.toNat_ofNat, |
| Nat.mod_eq_of_lt hd,decide_eq_true_eq] |
|
|
| lemma properDivisor_count (w d : ℕ) : |
| (properDivisor w d).net.count ≤ 216*w*w+203*w+36 := by |
| have h := wordLt_count (wordConstant (n := w) (BitVec.ofNat w d)) (select id) |
| rw [wordConstant_count,count_select] at h |
| have hd := dividesNet_count w d |
| rw [properDivisor,count_band] |
| omega |
|
|
| def noSmallDivisor (w B : ℕ) : BooleanNetwork w 1 := |
| all (List.ofFn (fun i : Fin (B-1) => (properDivisor w (i.val+2)).bnot)) |
| |
| lemma noSmallDivisor_value (w B : ℕ) (x : Basis w) (hB : B < 2^w) : |
| (noSmallDivisor w B).eval x 0=true ↔ |
| ∀ d : ℕ, 2 ≤ d → d ≤ B → d < (bitsValue x).toNat → ¬ d ∣ (bitsValue x).toNat := by |
| rw [noSmallDivisor,all_ofFn_eval] |
| constructor |
| · intro h d hd hdB hdm hdiv |
| have hh := (bnot_value _ _).mp (h (⟨d-2,by omega⟩ : Fin (B-1))) |
| rw [properDivisor_value _ _ _ (by omega)] at hh |
| exact hh (by simpa [show d-2+2=d by omega] using And.intro hdiv hdm) |
| · intro h i |
| rw [bnot_value,properDivisor_value _ _ _ (by have := i.isLt |
| rintro ⟨hd,hlt⟩ |
| exact h (i.val+2) (by omega) (by omega) hlt hd |
|
|
| lemma noSmallDivisor_count (w B : ℕ) : (noSmallDivisor w B).net.count ≤ |
| (B-1)*(216*w*w+203*w+38)+1 := by |
| have h := all_count (c := 216*w*w+203*w+37) |
| (List.ofFn (fun i : Fin (B-1) => (properDivisor w (i.val+2)).bnot)) (by |
| intro c hc |
| obtain ⟨i,rfl⟩ := List.mem_ofFn.mp hc |
| rw [count_bnot] |
| have := properDivisor_count w (i.val+2) |
| omega) |
| simpa only [noSmallDivisor,List.length_ofFn] using h |
|
|
| end ExactQuantumFactoring.BitArithmetic |
| |
|
|
| end |
|
|
| end OAI |
|
|
