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e320e82 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 | import OAI.Combinatorics.GraphThreshold.ProbabilityTree
import OAI.Combinatorics.GraphThreshold.ResamplingEvents
namespace OAI
/-! Fixed-input local masses and the deterministic good-node predicate. -/
noncomputable section
open scoped BigOperators
open Classical
namespace LeanBlast.SecondKahnKalai
section LocalMasses
variable {X B : Type*} [DecidableEq X] [Fintype B]
def successfulTargetMass (μ : FiniteLaw B) (A : B → Finset X) (ρ : ℝ) (m : ℕ)
(W : Finset X) (b : B) : ℝ :=
∑ c, if LocalSuccess μ A ρ m W c b then resamplingWeight μ A ρ W (c, b) else 0
def localFailureMass (μ : FiniteLaw B) (A : B → Finset X) (ρ : ℝ) (m : ℕ)
(W : Finset X) : ℝ :=
∑ b, ∑ c, if LocalFailure μ A ρ m W c b then resamplingWeight μ A ρ W (c, b) else 0
theorem successfulTargetMass_nonneg (μ : FiniteLaw B) (A : B → Finset X)
{ρ : ℝ} (hρ : 0 ≤ ρ) (m : ℕ) (W : Finset X) (b : B) :
0 ≤ successfulTargetMass μ A ρ m W b := by
apply Finset.sum_nonneg
intro c _
split_ifs
· exact resamplingWeight_nonneg μ A hρ W (c, b)
· exact le_rfl
theorem localFailureMass_nonneg (μ : FiniteLaw B) (A : B → Finset X)
{ρ : ℝ} (hρ : 0 ≤ ρ) (m : ℕ) (W : Finset X) :
0 ≤ localFailureMass μ A ρ m W := by
apply Finset.sum_nonneg
intro b _
apply Finset.sum_nonneg
intro c _
split_ifs
· exact resamplingWeight_nonneg μ A hρ W (c, b)
· exact le_rfl
theorem localFailureMass_add_successfulTargetMass (μ : FiniteLaw B)
(A : B → Finset X) {ρ : ℝ} (hμ : μ.IsPositive) (hρ : 0 < ρ)
(m : ℕ) (W : Finset X) :
localFailureMass μ A ρ m W + ∑ b, successfulTargetMass μ A ρ m W b = 1 := by
classical
calc
_ = ∑ b, ∑ c, resamplingWeight μ A ρ W (c, b) := by
unfold localFailureMass successfulTargetMass LocalFailure
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro b _
rw [← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro c _
by_cases h : LocalSuccess μ A ρ m W c b <;> simp [h]
_ = ∑ c, ∑ b, resamplingWeight μ A ρ W (c, b) := Finset.sum_comm
_ = 1 := by
simp_rw [resampling_source_marginal μ A hμ hρ W]
exact μ.total
theorem localFailureMass_eq_prob (μ : FiniteLaw B) (A : B → Finset X)
{ρ : ℝ} (hμ : μ.IsPositive) (hρ : 0 < ρ) (m : ℕ) (W : Finset X) :
localFailureMass μ A ρ m W =
(resamplingLaw μ A ρ W hμ hρ).prob (fun cb => LocalFailure μ A ρ m W cb.1 cb.2) := by
classical
unfold localFailureMass FiniteLaw.prob
rw [Fintype.sum_prod_type]
rw [Finset.sum_comm]
rfl
theorem successfulTargetMass_pos_capacity (μ : FiniteLaw B) (A : B → Finset X)
(ρ : ℝ) (m : ℕ) (W : Finset X) (b : B)
(hb : 0 < successfulTargetMass μ A ρ m W b) : (A b \ W).card ≤ m / 2 := by
by_contra hcap
have hfail (c : B) : ¬LocalSuccess μ A ρ m W c b := fun h => hcap h.2
simp [successfulTargetMass, hfail] at hb
theorem resamplingZ_congr_fragments (μ : FiniteLaw B) (A : B → Finset X)
(ρ : ℝ) {W V : Finset X}
(hWV : ∀ b, resamplingFragment A W b = resamplingFragment A V b) (c : B) :
resamplingZ μ A ρ (W ∪ A c) = resamplingZ μ A ρ (V ∪ A c) := by
unfold resamplingZ
apply Finset.sum_congr rfl
intro b _
have hsub : A b ⊆ resamplingY A W c ↔ A b ⊆ resamplingY A V c := by
rw [resampling_target_subset_iff_fragment, resampling_target_subset_iff_fragment, hWV]
unfold resamplingNumerator
exact if_congr hsub rfl rfl
theorem localSuccess_congr_fragments (μ : FiniteLaw B) (A : B → Finset X)
(ρ : ℝ) (m : ℕ) {W V : Finset X}
(hWV : ∀ b, resamplingFragment A W b = resamplingFragment A V b) (c b : B) :
LocalSuccess μ A ρ m W c b ↔ LocalSuccess μ A ρ m V c b := by
unfold LocalSuccess
rw [resamplingZ_congr_fragments μ A ρ hWV c]
change (_ ∧ (resamplingFragment A W b).card ≤ m / 2) ↔
(_ ∧ (resamplingFragment A V b).card ≤ m / 2)
rw [hWV b]
theorem successfulTargetMass_congr_fragments (μ : FiniteLaw B) (A : B → Finset X)
(ρ : ℝ) (m : ℕ) {W V : Finset X}
(hWV : ∀ b, resamplingFragment A W b = resamplingFragment A V b) (b : B) :
successfulTargetMass μ A ρ m W b = successfulTargetMass μ A ρ m V b := by
unfold successfulTargetMass
apply Finset.sum_congr rfl
intro c _
by_cases h : LocalSuccess μ A ρ m W c b
· have hv := (localSuccess_congr_fragments μ A ρ m hWV c b).mp h
simp [h, hv, resamplingWeight_congr_fragments μ A ρ hWV]
· have hv : ¬LocalSuccess μ A ρ m V c b :=
fun hv => h ((localSuccess_congr_fragments μ A ρ m hWV c b).mpr hv)
simp [h, hv]
theorem localFailureMass_congr_fragments (μ : FiniteLaw B) (A : B → Finset X)
(ρ : ℝ) (m : ℕ) {W V : Finset X}
(hWV : ∀ b, resamplingFragment A W b = resamplingFragment A V b) :
localFailureMass μ A ρ m W = localFailureMass μ A ρ m V := by
unfold localFailureMass LocalFailure
apply Finset.sum_congr rfl
intro b _
apply Finset.sum_congr rfl
intro c _
by_cases h : LocalSuccess μ A ρ m W c b
· have hv := (localSuccess_congr_fragments μ A ρ m hWV c b).mp h
simp [h, hv]
· have hv : ¬LocalSuccess μ A ρ m V c b :=
fun hv => h ((localSuccess_congr_fragments μ A ρ m hWV c b).mpr hv)
simp [h, hv, resamplingWeight_congr_fragments μ A ρ hWV]
theorem fragments_eq_of_inter_label_union_eq (A : B → Finset X) {W V : Finset X}
(hWV : W ∩ Finset.univ.biUnion A = V ∩ Finset.univ.biUnion A) (b : B) :
resamplingFragment A W b = resamplingFragment A V b := by
apply Finset.ext
intro x
by_cases hx : x ∈ A b
· have hxU : x ∈ Finset.univ.biUnion A :=
Finset.mem_biUnion.mpr ⟨b, Finset.mem_univ b, hx⟩
have hmem : x ∈ W ↔ x ∈ V := by
simpa only [Finset.mem_inter, hxU, and_true] using Finset.ext_iff.mp hWV x
simp [resamplingFragment, hx, hmem]
· simp [resamplingFragment, hx]
theorem successfulTargetMass_dependsOnlyOn (μ : FiniteLaw B) (A : B → Finset X)
(ρ : ℝ) (m : ℕ) (b : B) :
DependsOnlyOn (Finset.univ.biUnion A) (fun W => successfulTargetMass μ A ρ m W b) := by
intro W V hWV
exact successfulTargetMass_congr_fragments μ A ρ m
(fragments_eq_of_inter_label_union_eq A hWV) b
theorem localFailureMass_dependsOnlyOn (μ : FiniteLaw B) (A : B → Finset X)
(ρ : ℝ) (m : ℕ) :
DependsOnlyOn (Finset.univ.biUnion A) (fun W => localFailureMass μ A ρ m W) := by
intro W V hWV
exact localFailureMass_congr_fragments μ A ρ m
(fragments_eq_of_inter_label_union_eq A hWV)
end LocalMasses
variable {X : Type} [DecidableEq X]
/-- Goodness is evaluated on the fixed input tree, using local nonfailure
target masses before any pruning or reweighting. -/
def Good (ρ : ℝ) (m : ℕ → ℕ) (W : Finset X) {d : ℕ} (T : ProbabilityTree X d) : Prop :=
match T with
| .leaf => True
| @ProbabilityTree.node _ _ B inst μ _hμ A child =>
1 - Real.exp (-(m 0 : ℝ)) ≤
∑ b, if Good ρ (fun j => m (j + 1)) W (child b) then
successfulTargetMass μ A ρ (m 0) W b else 0
termination_by d
end LeanBlast.SecondKahnKalai
end
end OAI
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