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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 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 | import OAI.Combinatorics.MatrixRemoval.HostSampling
import OAI.Combinatorics.MatrixRemoval.HostSampledPath
import OAI.Combinatorics.MatrixRemoval.GuardContainment
namespace OAI
/-!
# Compatibility of the path sampler and protected-cell sampler
The quotient/remainder formula and `finProdFinEquiv` select the same
coordinate, identifying path samples with the uniform-fiber sampler.
-/
namespace Problem348.TreeSampling
/-- The equivalence-based and arithmetic mixers select identical coordinates. -/
theorem atLevel_eq_mixFin (h i : ℕ) (hi : i ≤ h) (z u : Fin (2 ^ h)) :
atLevel h i hi z u = SampledPath.mixFin h i hi z u := by
apply Fin.ext
change u.val % (2 ^ (h - i)) + (2 ^ (h - i)) * (z.val / (2 ^ (h - i))) =
z.val / (2 ^ (h - i)) * (2 ^ (h - i)) + u.val % (2 ^ (h - i))
ac_rfl
end Problem348.TreeSampling
namespace Problem348.Construction.HostSampling
/-- An actual sampled variable path point is selected by the marginal sampler. -/
theorem coordinate_path_position (h i : ℕ) (hi : i ≤ h) (sign : Bool)
(z u : Fin (2 ^ h)) :
coordinate z u (HostSampledPath.position h i hi sign z u) =
offset (HostSampledPath.position h i hi sign z u) := by
change TreeSampling.atLevel h (depth (SampledPath.sample h i hi sign z u))
(depth_le _) z u =
(SampledPath.sample h i hi sign z u).2
have hd : depth (SampledPath.sample h i hi sign z u) = i :=
(HostSampledPath.sample_atLevel h i hi sign z u).1
simp only [hd]
exact TreeSampling.atLevel_eq_mixFin h i hi z u
/-- Anchor representatives use the corresponding independent axis seed. -/
@[simp] theorem coordinate_anchor (h : ℕ) (t : Mode h) (v : Fin 64)
(z u : Fin (2 ^ h)) :
coordinate z u (Sum.inl (t, v, u)) = offset (Sum.inl (t, v, u)) := rfl
/-- The selected dummy position uses the same independent axis seed. -/
@[simp] theorem coordinate_path_dummy (h : ℕ) (z u : Fin (2 ^ h)) :
coordinate z u (HostSampledPath.dummy h u) =
offset (HostSampledPath.dummy h u) := rfl
/-- Every pair of path samples, not just the protected pairs, is selected.
The two axes use different offset seeds but the same leaf seed. -/
theorem selected_path_positions (h i j : ℕ) (hi : i ≤ h) (hj : j ≤ h)
(s t : Bool) (z u v : Fin (2 ^ h)) :
Selected (z, u, v) (HostSampledPath.position h i hi s z u)
(HostSampledPath.position h j hj t z v) := by
exact Prod.ext (coordinate_path_position h i hi s z u)
(coordinate_path_position h j hj t z v)
/-- A path-row / selected-anchor-column pair is a selected protected cell. -/
theorem protected_selected_path_anchor (h i : ℕ) (hi : i ≤ h) (sign : Bool)
(t : Mode h) (a : Fin 64) (z u v : Fin (2 ^ h)) :
Protected (HostSampledPath.position h i hi sign z u) (Sum.inl (t, a, v)) ∧
Selected (z, u, v) (HostSampledPath.position h i hi sign z u)
(Sum.inl (t, a, v)) := by
exact ⟨protected_anchor_right _ _,
Prod.ext (coordinate_path_position h i hi sign z u) rfl⟩
/-- A selected-anchor-row / path-column pair is a selected protected cell. -/
theorem protected_selected_anchor_path (h i : ℕ) (hi : i ≤ h) (sign : Bool)
(t : Mode h) (a : Fin 64) (z u v : Fin (2 ^ h)) :
Protected (Sum.inl (t, a, u)) (HostSampledPath.position h i hi sign z v) ∧
Selected (z, u, v) (Sum.inl (t, a, u))
(HostSampledPath.position h i hi sign z v) := by
exact ⟨protected_anchor_left _ _,
Prod.ext rfl (coordinate_path_position h i hi sign z v)⟩
/-- Direct row-field adapter for the geometric guarded-path certificate. -/
theorem rowChosen_path {h n : ℕ} (er : Position h ≃ Fin n)
(i : ℕ) (hi : i ≤ h) (sign : Bool) (s : TreeSampling.Seeds (2 ^ h)) :
RowChosen er s (er (HostSampledPath.position h i hi sign s.1 s.2.1)) := by
simp only [RowChosen, Equiv.symm_apply_apply]
exact coordinate_path_position h i hi sign s.1 s.2.1
/-- Direct column-field adapter for the geometric guarded-path certificate. -/
theorem colChosen_path {h n : ℕ} (ec : Position h ≃ Fin n)
(i : ℕ) (hi : i ≤ h) (sign : Bool) (s : TreeSampling.Seeds (2 ^ h)) :
ColChosen ec s (ec (HostSampledPath.position h i hi sign s.1 s.2.2)) := by
simp only [ColChosen, Equiv.symm_apply_apply]
exact coordinate_path_position h i hi sign s.1 s.2.2
@[simp] theorem rowChosen_path_dummy {h n : ℕ} (er : Position h ≃ Fin n)
(s : TreeSampling.Seeds (2 ^ h)) :
RowChosen er s (er (HostSampledPath.dummy h s.2.1)) := by
simp only [RowChosen, Equiv.symm_apply_apply]
rfl
@[simp] theorem colChosen_path_dummy {h n : ℕ} (ec : Position h ≃ Fin n)
(s : TreeSampling.Seeds (2 ^ h)) :
ColChosen ec s (ec (HostSampledPath.dummy h s.2.2)) := by
simp only [ColChosen, Equiv.symm_apply_apply]
rfl
@[simp] theorem rowChosen_path_anchor {h n : ℕ} (er : Position h ≃ Fin n)
(s : TreeSampling.Seeds (2 ^ h)) (t : Mode h) (a : Fin 64) :
RowChosen er s (er (Sum.inl (t, a, s.2.1))) := by
simp only [RowChosen, Equiv.symm_apply_apply]
rfl
@[simp] theorem colChosen_path_anchor {h n : ℕ} (ec : Position h ≃ Fin n)
(s : TreeSampling.Seeds (2 ^ h)) (t : Mode h) (a : Fin 64) :
ColChosen ec s (ec (Sum.inl (t, a, s.2.2))) := by
simp only [ColChosen, Equiv.symm_apply_apply]
rfl
/-- Both sampled roots belong to protected classes, regardless of sign. -/
theorem protected_path_roots (h : ℕ) (s t : Bool) (z u v : Fin (2 ^ h)) :
Protected (HostSampledPath.position h 0 (Nat.zero_le h) s z u)
(HostSampledPath.position h 0 (Nat.zero_le h) t z v) := by
apply protected_roots
· exact (HostSampledPath.sample_atLevel h 0 (Nat.zero_le h) s z u).1
· exact (HostSampledPath.sample_atLevel h 0 (Nat.zero_le h) t z v).1
end Problem348.Construction.HostSampling
end OAI
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