File size: 3,155 Bytes
e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.Combinatorics.MatchingPSD.GoodCells
namespace OAI
universe u_1
namespace PerfectMatchingPSD
open Matrix
open scoped BigOperators MatrixOrder
attribute [local instance] Classical.propDecidable
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'
end PerfectMatchingPSD
end OAI
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