Datasets:

roseking
/

openAI-math-dataset

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

	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