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5d5048e | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationProgram
namespace OAI
/-! # One-sided soundness of the arbitrary-input representation runtime -/
noncomputable section
namespace LowSpaceSubsetSum
namespace RepresentationSetup
variable {n : ℕ} (R : RepresentationSetup n)
theorem rawLeaf_background_subset_outside (z : Layout n) (g : Guesses n)
(f : R.Partitions z) (B : Fin n → Fin 2 × Fin 4) (j : Fin 2) :
RepresentationUniverse.backgroundDomain (R.rawLeaf z g f B j) ⊆
Finset.univ \ R.mixerUnion z := by
change ((R.rawBackground z B j 0 ∪ R.rawBackground z B j 1) ∪
R.rawBackground z B j 2) ∪ R.rawBackground z B j 3 ⊆ _
repeat' apply Finset.union_subset
all_goals exact Finset.filter_subset _ _
theorem rawLeaf_background_color (z : Layout n) (g : Guesses n)
(f : R.Partitions z) (B : Fin n → Fin 2 × Fin 4) (j : Fin 2) (x : Fin n)
(hx : x ∈ RepresentationUniverse.backgroundDomain (R.rawLeaf z g f B j)) :
(B x).1 = j := by
change x ∈ ((R.rawBackground z B j 0 ∪ R.rawBackground z B j 1) ∪
R.rawBackground z B j 2) ∪ R.rawBackground z B j 3 at hx
simp only [Finset.mem_union] at hx
rcases hx with ((hx | hx) | hx) | hx
all_goals exact congrArg Prod.fst (Finset.mem_filter.mp hx).2
theorem rawLeaf_background_sides_disjoint (z : Layout n) (g : Guesses n)
(f : R.Partitions z) (B : Fin n → Fin 2 × Fin 4) :
Disjoint (RepresentationUniverse.backgroundDomain (R.rawLeaf z g f B 0))
(RepresentationUniverse.backgroundDomain (R.rawLeaf z g f B 1)) := by
apply Finset.disjoint_left.mpr
intro x hx hy
have h0 := R.rawLeaf_background_color z g f B 0 x hx
have h1 := R.rawLeaf_background_color z g f B 1 x hy
have h : (0 : Fin 2) = 1 := h0.symm.trans h1
exact (by decide : (0 : Fin 2) ≠ 1) h
/-- Only top-domain indices can be shared by the left and right parents. -/
theorem raw_parentDomain_shared (z : Layout n) (g : Guesses n)
(f : R.Partitions z) (B : Fin n → Fin 2 × Fin 4) :
RepresentationUniverse.parentDomain (R.rawLeaf z g f B 0) ∩
RepresentationUniverse.parentDomain (R.rawLeaf z g f B 1) ⊆
R.orderedMixers z 0 := by
have hbg := R.rawLeaf_background_sides_disjoint z g f B
have hb0 := R.rawLeaf_background_subset_outside z g f B 0
have hb1 := R.rawLeaf_background_subset_outside z g f B 1
have hm1 := R.mixer_disjoint_rawOutside z 1
have hm2 := R.mixer_disjoint_rawOutside z 2
have hm12 := R.orderedMixers_disjoint z (by decide : (1 : Fin 3) ≠ 2)
intro x hx
obtain ⟨hx, hy⟩ := Finset.mem_inter.mp hx
simp only [RepresentationUniverse.parentDomain, R.rawLeaf_topDomain,
R.rawLeaf_mixerDomain, Finset.mem_union] at hx hy
rcases hx with (ht | hm) | hb
· exact ht
· rcases hy with (ht | hm') | hb'
· exact ht
· exact (Finset.disjoint_left.mp hm12 hm hm').elim
· exact (Finset.disjoint_left.mp hm1 hm (hb1 hb')).elim
· rcases hy with (ht | hm') | hb'
· exact ht
· exact (Finset.disjoint_left.mp hm2 hm' (hb0 hb)).elim
· exact (Finset.disjoint_left.mp hbg hb hb').elim
end RepresentationSetup
theorem representation_capacity_pair_le (n : ℕ) (hn : 10 ≤ n) :
compressionDictionaryCapacity n + 1 + (compressionDictionaryCapacity n + 1) ≤ 2 ^ n := by
have hC : 0 < compressionDictionaryCapacity n := by
unfold compressionDictionaryCapacity
positivity
have hk : 199 * n / 1000 + 2 ≤ n := by omega
have hp : 4 * compressionDictionaryCapacity n ≤ 2 ^ n := by
calc
_ = 2 ^ (199 * n / 1000 + 2) := by
rw [pow_add]
unfold compressionDictionaryCapacity
ring
_ ≤ _ := Nat.pow_le_pow_right (by decide : 0 < 2) hk
omega
namespace RepresentationData
variable {n : ℕ} (D : RepresentationData n)
/-- Every accepting random history yields an indexed solution, for an
arbitrary input and without any survival, resource or sampling assumptions. -/
theorem trial_sound (h : D.zeroSetup.PrimeHistory) (r : D.zeroSetup.Residues h.2)
(hn : 10000 ≤ n)
(hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000)
(q : ℕ) (hq : 0 < q) (B : ℕ)
(families : Fin 4 → Array (Finset (Fin (2 * RepresentationSetup.blockHalf n 0))))
(hyes : D.trial h r hn hlog q hq B families = true) : Solvable D.input := by
let R := D.zeroSetup
let z := h.1.1.1
let g := h.1.1.2
let f := h.1.2
let bg := R.allocatedBackground z g hn hlog
apply (D.scheduleData h r hn hlog q hq).run_sound B families
(fun j => (R.rawLeaf_topDomain z g f bg j).trans (R.topCoordinates_domain z f).symm)
(fun j => (R.rawLeaf_domains_disjoint z g f bg j).1)
(fun j => (R.rawLeaf_domains_disjoint z g f bg j).2.1)
(fun j => (R.rawLeaf_domains_disjoint z g f bg j).2.2)
?_ (representation_capacity_pair_le n (by omega)) hyes
change RepresentationUniverse.parentDomain (R.rawLeaf z g f bg 0) ∩
RepresentationUniverse.parentDomain (R.rawLeaf z g f bg 1) ⊆
(R.topCoordinates z f).domain
rw [R.topCoordinates_domain]
exact R.raw_parentDomain_shared z g f bg
end RepresentationData
end LowSpaceSubsetSum
end
end OAI
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