| import OAI.Combinatorics.MatchingPSD.GoodCells |
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| namespace OAI |
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| universe u_1 |
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| namespace PerfectMatchingPSD |
| open Matrix |
| open scoped BigOperators MatrixOrder |
| attribute [local instance] Classical.propDecidable |
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| lemma domination_decoder {R : Type u_1} [Fintype R] [DecidableEq R] {q : ℕ} |
| (Q : (Fin q → Bool) → Matrix R R ℝ) |
| (T : Fin q → Bool → Matrix R R ℝ) (P E : Fin q → Matrix R R ℝ) |
| (S : Matrix R R ℝ) |
| (hQ : ∀ y, (Q y).PosSemidef) (hT : ∀ i b, (T i b).PosSemidef) |
| (hP : ∀ i, (P i).PosSemidef) (hE : ∀ i, (E i).PosSemidef) (hS : S.PosSemidef) |
| (hdom : ∀ i, T i false + T i true ≤ S+(2:ℝ) • P i+E i) |
| (hzero : ∀ y i, (Q y*P i).trace=0) (hstar : ∀ y, (Q y*S).trace=2) |
| (hbit : ∀ y i, (Q y*T i (y i)).trace=2) |
| {m e δ : ℝ} (hδ : 0 < δ) (hm : ∀ y, (Q y).trace ≤ m) |
| (he : ∀ y i, (Q y*E i).trace ≤ e) |
| (hε : 0 ≤ (q:ℝ)*e+δ*m) |
| (hsmall : (q:ℝ)*Real.sqrt ((q:ℝ)*e+δ*m) ≤ 1/4) : |
| (2:ℝ)^q/2 ≤ Fintype.card R := by |
| let F := fun i => (2:ℝ) • P i+E i |
| have hF (i : Fin q) : (F i).PosSemidef := (hP i).smul (by norm_num) |>.add (hE i) |
| let D := S+(∑ i, F i)+δ • (1 : Matrix R R ℝ) |
| have hsum : (∑ i, F i).PosSemidef := matrix_sum_psd _ _ (fun i _ => hF i) |
| have hdelt : (δ • (1 : Matrix R R ℝ)).PosDef := Matrix.PosDef.one.smul hδ |
| have hD : D.PosDef := by |
| simpa only [D, add_comm] using hdelt.add_posSemidef (hS.add hsum) |
| have hleS : S ≤ D := by |
| have he := add_nonneg hsum.nonneg hdelt.posSemidef.nonneg |
| dsimp [D] |
| simpa only [add_assoc] using le_add_of_nonneg_right he |
| have hdomD (i : Fin q) : T i false+T i true ≤ D := by |
| apply (hdom i).trans |
| have hi : F i ≤ ∑ j, F j := Finset.single_le_sum (fun j _ => (hF j).nonneg) (Finset.mem_univ i) |
| calc |
| S+(2:ℝ) • P i+E i = S+F i := by dsimp [F]; abel |
| _ ≤ S+(∑ j, F j) := by simpa only [add_comm] using add_le_add_left hi S |
| _ ≤ D := le_add_of_nonneg_right hdelt.posSemidef.nonneg |
| have htr (y : Fin q → Bool) : 2 ≤ (Q y*D).trace ∧ (Q y*D).trace ≤ 2+((q:ℝ)*e+δ*m) := by |
| constructor |
| · rw [← hstar y] |
| exact trace_mul_le_of_le (hQ y) hleS |
| · dsimp [D] |
| rw [mul_add, mul_add, Matrix.trace_add, Matrix.trace_add, hstar, |
| Finset.mul_sum, Matrix.trace_sum, mul_smul_comm, Matrix.trace_smul, mul_one] |
| have hsumtr : (∑ i, (Q y*F i).trace) ≤ (q:ℝ)*e := by |
| have hh := Finset.sum_le_sum (s := Finset.univ) (fun i _ => he y i) |
| have hFi (i : Fin q) : (Q y*F i).trace = (Q y*E i).trace := by |
| simp only [F, mul_add, Matrix.trace_add, mul_smul_comm, Matrix.trace_smul, |
| hzero, smul_zero, zero_add] |
| simpa only [hFi, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] using hh |
| have hδm := mul_le_mul_of_nonneg_left (hm y) hδ.le |
| change _ + δ*(Q y).trace ≤ _ |
| linarith |
| obtain ⟨ρ,H,hρ,htρ,hH,hpair,hbit'⟩ := states_of_domination Q T D hD hQ hT hdomD |
| ((q:ℝ)*e+δ*m) hε htr hbit |
| exact nearly_perfect_bit_tests ρ H hρ htρ hH hpair _ hε hsmall hbit' |
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| end PerfectMatchingPSD |
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| end OAI |
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