| import OAI.NumberTheory.Ostmann.Construction.FullAtomRangeSupport |
| import OAI.NumberTheory.Ostmann.Construction.ConstituentCopiedSupport |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace Ostmann |
|
|
| open scoped Classical |
|
|
| noncomputable def atomInterval {I : Type*} (role : I → CopyScheduleRole) |
| (lo hi : I → ℕ) (n : ℕ) (v : CopyScheduleAtoms role n) : ScheduleAtomRange role n where |
| atoms := [v] |
| lower := lo (copyScheduleOrigin n v.val) |
| upper := hi (copyScheduleOrigin n v.val) |
|
|
| noncomputable def atomIntervalRanges {I : Type*} [Fintype I] (role : I → CopyScheduleRole) |
| (lo hi : I → ℕ) (n : ℕ) : List (ScheduleAtomRange role n) := |
| Finset.univ.toList.map (atomInterval role lo hi n) |
|
|
| theorem atomInterval_holds_iff {I : Type*} (role : I → CopyScheduleRole) |
| (lo hi : I → ℕ) (n : ℕ) (v : CopyScheduleAtoms role n) (x : CopyScheduleAtoms role n → ℕ) : |
| (atomInterval role lo hi n v).Holds x ↔ |
| lo (copyScheduleOrigin n v.val) ≤ x v ∧ x v ≤ hi (copyScheduleOrigin n v.val) := by |
| simp only [atomInterval, ScheduleAtomRange.Holds, List.map_singleton, List.prod_singleton, |
| Set.mem_Icc, Nat.cast_le] |
|
|
| theorem atomIntervalRanges_holds_iff {I : Type*} [Fintype I] (role : I → CopyScheduleRole) |
| (lo hi : I → ℕ) (n : ℕ) (x : CopyScheduleAtoms role n → ℕ) : |
| (∀ r ∈ atomIntervalRanges role lo hi n, r.Holds x) ↔ |
| ∀ v, lo (copyScheduleOrigin n v.val) ≤ x v ∧ x v ≤ hi (copyScheduleOrigin n v.val) := by |
| constructor |
| · intro h v |
| apply (atomInterval_holds_iff role lo hi n v x).mp |
| apply h |
| exact List.mem_map.mpr ⟨v, Finset.mem_toList.mpr (Finset.mem_univ v), rfl⟩ |
| · intro h r hr |
| obtain ⟨v, _, rfl⟩ := List.mem_map.mp hr |
| exact (atomInterval_holds_iff role lo hi n v x).mpr (h v) |
|
|
| /-- Copying preserves the original atom origin and hence its exact interval. -/ |
| theorem atomIntervalRanges_copied {I : Type*} [Fintype I] |
| (role : I → CopyScheduleRole) (lo hi : I → ℕ) (n M : ℕ) |
| (u : CopyScheduleY role n → ℕ) (l r : CopyScheduleH role n → ℕ) |
| (hL : ∀ a ∈ atomIntervalRanges role lo hi n, |
| a.Holds (scheduledInsertedAtoms role n M l u)) |
| (hR : ∀ a ∈ atomIntervalRanges role lo hi n, |
| a.Holds (scheduledInsertedAtoms role n M r u)) : |
| ∀ a ∈ atomIntervalRanges role lo hi (n + 1), |
| a.Holds (scheduledCopiedAssignment role n u l r) := by |
| apply (atomIntervalRanges_holds_iff role lo hi (n + 1) _).mpr |
| have hL' := (atomIntervalRanges_holds_iff role lo hi n _).mp hL |
| have hR' := (atomIntervalRanges_holds_iff role lo hi n _).mp hR |
| intro v |
| obtain ⟨j, rfl⟩ := (copyScheduleSurvivorEquiv role n).symm.surjective v |
| rcases j with ⟨b, h⟩ | y |
| · cases b |
| · have hh := hR' ⟨h.val, h.property.1⟩ |
| change lo (copyScheduleOrigin n h.val) ≤ r h ∧ r h ≤ hi (copyScheduleOrigin n h.val) |
| simpa only [scheduledInsertedAtoms_H] using hh |
| · have hh := hL' ⟨h.val, h.property.1⟩ |
| change lo (copyScheduleOrigin n h.val) ≤ l h ∧ l h ≤ hi (copyScheduleOrigin n h.val) |
| simpa only [scheduledInsertedAtoms_H] using hh |
| · have hh := hL' ⟨y.val, y.property.1⟩ |
| change lo (copyScheduleOrigin n y.val) ≤ u y ∧ u y ≤ hi (copyScheduleOrigin n y.val) |
| simpa only [scheduledInsertedAtoms_Y] using hh |
|
|
| /-- Both old full coefficients enforce every new interval. This discharges |
| the interval clause in the actual guarded arithmetic step. -/ |
| theorem fullAtomTransferWeight_copied_intervals {I : Type*} [Fintype I] |
| (role : I → CopyScheduleRole) (lo hi : I → ℕ) |
| (childBound pivotBound : ℕ → ℕ) (leaf : ScheduleAtomState role → ℤ → ℂ) |
| (n M : ℕ) (u : CopyScheduleY role n → ℕ) (l r : CopyScheduleH role n → ℕ) |
| (t t' : FrequencyTree ℤ n) |
| (hL : fullAtomTransferWeight role childBound pivotBound (atomIntervalRanges role lo hi) leaf n |
| (scheduledInsertedAtoms role n M l u) t ≠ 0) |
| (hR : fullAtomTransferWeight role childBound pivotBound (atomIntervalRanges role lo hi) leaf n |
| (scheduledInsertedAtoms role n M r u) t' ≠ 0) : |
| ∀ a ∈ atomIntervalRanges role lo hi (n + 1), a.Holds (scheduledCopiedAssignment role n u l r) := by |
| exact atomIntervalRanges_copied role lo hi n M u l r |
| (fun a ha => fullAtomTransferWeight_top_range role childBound pivotBound _ leaf n _ t hL a ha) |
| (fun a ha => fullAtomTransferWeight_top_range role childBound pivotBound _ leaf n _ t' hR a ha) |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
