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dd14c26 | 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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Probability.MatroidProphet.Main
namespace OAI
/-! The literal conditional cost inequality `eq:expected-cost`. The cost is
zero on an unlisted true group; listing is never conditioned upon. -/
namespace MatroidProphet.AccountingContracts
open Set Finset MainAlgorithm
variable {n : ℕ}
/-- The single path-expansion budget of a listed group, with its real-valued
density contribution. -/
noncomputable def listedCost (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (h : ℕ) : ℝ :=
(groupMask M d w d.C h).ncard +
∑ j ∈ range ((groups M d w).length - (h + 1)),
((groupMask M d w d.C (h + 1 + j)).ncard +
(groupMask M d w d.D (h + 1 + j)).ncard / densityThreshold)
/-- The source's `c_i`, extended by zero to unlisted true groups. -/
noncomputable def trueCost (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) : ℝ := by
classical
exact if i ∈ groups M d w then listedCost M d w ((groups M d w).idxOf i) else 0
theorem listedCost_le_true_counts (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) {h : ℕ} {i : ℤ}
(hi : (groups M d w)[h]? = some i) :
listedCost M d w h ≤
(d.C ∩ trueGroup M d w i).card +
(d.C ∩ higherTrueGroups M d w i).card +
(higherTrueGroups M d w i).card / densityThreshold := by
have hc := listed_cost_le_true_counts M d w hi (2^100)
rw [← trueGroup_eq_candidate_filter M d w i] at hc
have hr : densityThreshold * (groupMask M d w d.C h).ncard +
∑ j ∈ range ((groups M d w).length - (h + 1)),
(densityThreshold * (groupMask M d w d.C (h + 1 + j)).ncard +
(groupMask M d w d.D (h + 1 + j)).ncard) ≤
densityThreshold * (d.C ∩ trueGroup M d w i).card +
densityThreshold * (d.C ∩ higherTrueGroups M d w i).card +
(higherTrueGroups M d w i).card := by
simpa only [densityThreshold, higherTrueGroups, Nat.cast_add, Nat.cast_mul,
Nat.cast_sum, Nat.cast_pow, Nat.cast_ofNat] using
(Nat.cast_le.mpr hc : ( _ : ℝ) ≤ _)
have hk : 0 < densityThreshold := constants_positive.2.1
apply (mul_le_mul_iff_right₀ hk).mp
have hs : densityThreshold * listedCost M d w h =
densityThreshold * (groupMask M d w d.C h).ncard +
∑ j ∈ range ((groups M d w).length - (h + 1)),
(densityThreshold * (groupMask M d w d.C (h + 1 + j)).ncard +
(groupMask M d w d.D (h + 1 + j)).ncard) := by
unfold listedCost
rw [mul_add, mul_sum]
congr 1
apply sum_congr rfl
intro j hj
rw [mul_add]
congr 1
field_simp
rw [hs, mul_add, mul_add]
have he : densityThreshold *
((higherTrueGroups M d w i).card / densityThreshold) =
(higherTrueGroups M d w i).card := by field_simp
rw [he]
exact hr
theorem trueCost_le_true_counts (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) :
trueCost M d w i ≤
(d.C ∩ trueGroup M d w i).card +
(d.C ∩ higherTrueGroups M d w i).card +
(higherTrueGroups M d w i).card / densityThreshold := by
classical
unfold trueCost
split_ifs with hi
· exact listedCost_le_true_counts M d w (groups_get?_idxOf M d w i hi)
· have hk : 0 < densityThreshold := constants_positive.2.1
positivity
/-- Conditional on `H`, the actual randomly listed cost is dominated before
averaging. The higher-group cardinality is the sum over all higher true groups,
including those that the density mask did not list. -/
theorem expected_cost (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) :
mainMean d (fun d' => trueCost M d' w i) ≤
thinningRate * (trueGroup M d w i).card +
(thinningRate + 1 / densityThreshold) * (higherTrueGroups M d w i).card := by
have hm : mainMean d (fun d' => trueCost M d' w i) ≤
mainMean d (fun d' => (d'.C ∩ trueGroup M d w i).card +
(d'.C ∩ higherTrueGroups M d w i).card +
(higherTrueGroups M d w i).card / densityThreshold) := by
apply mainMean_mono
intro d' hH
have hc := trueCost_le_true_counts M d' w i
rwa [trueGroup_eq_of_H_eq M d d' w i hH,
higherTrueGroups_eq_of_H_eq M d d' w i hH] at hc
simp only [mainMean_add, mainMean_guard_count, mainMean_const] at hm
convert hm using 1
ring
/-- Exact finite true-level sum presentation of `eq:expected-cost`. -/
theorem expected_cost_sum (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) :
mainMean d (fun d' => trueCost M d' w i) ≤
thinningRate * (trueGroup M d w i).card +
(thinningRate + 1 / densityThreshold) *
∑ j ∈ trueLevels M w d.H with i < j, ((trueGroup M d w j).card : ℝ) := by
rw [← higherTrueGroups_card_eq_sum M d w i]
exact expected_cost M d w i
end MatroidProphet.AccountingContracts
end OAI
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