| import OAI.Combinatorics.EditApproximation.Binary.BinaryCutReads |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryScoreReads |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryPreparedRepresentative |
| import OAI.Combinatorics.EditApproximation.Definitions |
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| namespace OAI |
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| /-! Representative bands with charged child-table reads in every score |
| and prefix predicate over the materialized wide grid. -/ |
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| namespace EditApproximation |
| namespace BinaryFraction |
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| theorem bandFromActionsReadWithWork_eq_cost {nx ny M : ℕ} (parent : TargetInterval nx) |
| (r : TargetInterval ny) (b : ℕ) (tau theta velocity : BinaryFraction) |
| (read : Fin M → TargetInterval ny → BinaryFraction × ℕ) |
| (actions : List (Fin M → TargetInterval ny)) (hne : actions ≠ []) (R : ℕ) |
| (hr : ∀ action ∈ actions, ∀ i, (read i (action i)).2 ≤ R) : |
| (bandFromActionsReadWithWork parent r b tau theta velocity read actions hne).1 = |
| (bandFromActionsWithWork parent r b tau theta velocity (fun i q => (read i q).1) actions hne).1 ∧ |
| (bandFromActionsReadWithWork parent r b tau theta velocity read actions hne).2 ≤ |
| (bandFromActionsWithWork parent r b tau theta velocity (fun i q => (read i q).1) actions hne).2 + |
| actions.length * (M * ((actions.length + 1) * (M * (R + 4)))) := by |
| have costBound (a a' m g A : ℕ) (ha : a ≤ a' + A) : |
| m + g + a + 2 ≤ m + g + a' + 2 + A := by |
| omega |
| have h := filterWithWork_extra |
| (bandPredicateReadWithWork parent r b tau theta velocity actions hne read) |
| (bandPredicateWithWork parent r b tau theta velocity actions hne (fun i q => (read i q).1)) |
| (fun _ => M * ((actions.length + 1) * (M * (R + 4)))) actions |
| (fun states _ => (bandPredicateReadWithWork_eq_cost parent r b tau theta velocity actions hne read states R hr).1) |
| (fun states _ => (bandPredicateReadWithWork_eq_cost parent r b tau theta velocity actions hne read states R hr).2) |
| simp only [List.map_const', List.sum_replicate, smul_eq_mul] at h |
| unfold bandFromActionsReadWithWork bandFromActionsWithWork |
| dsimp only |
| split_ifs |
| · exact ⟨h.1, costBound _ _ _ _ _ h.2⟩ |
| · exact ⟨rfl, Nat.le_add_right _ _⟩ |
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| def representativeReadOverhead (M vertices R : ℕ) : ℕ := |
| (vertices + 1) * (M * (R + 4)) + vertices * (M * ((vertices + 1) * (M * (R + 4)))) |
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| theorem representativeFromActionsReadWithWork_eq_cost {nx ny M : ℕ} (parent : TargetInterval nx) |
| (r : TargetInterval ny) (sourceLength b : ℕ) (tau theta : BinaryFraction) |
| (read : Fin M → TargetInterval ny → BinaryFraction × ℕ) |
| (actions : List (Fin M → TargetInterval ny)) (R : ℕ) |
| (hr : ∀ action ∈ actions, ∀ i, (read i (action i)).2 ≤ R) : |
| (representativeFromActionsReadWithWork parent r sourceLength b tau theta read actions).1 = |
| (representativeFromActionsWithWork parent r sourceLength b tau theta (fun i q => (read i q).1) actions).1 ∧ |
| (representativeFromActionsReadWithWork parent r sourceLength b tau theta read actions).2 ≤ |
| (representativeFromActionsWithWork parent r sourceLength b tau theta (fun i q => (read i q).1) actions).2 + |
| representativeReadOverhead M actions.length R := by |
| have costBound (a a' c c' x y z A B : ℕ) |
| (ha : a ≤ a' + A) (hc : c ≤ c' + B) : |
| a + x + y + z + c + 3 ≤ a' + x + y + z + c' + 3 + (A + B) := by |
| omega |
| have shortCostBound (a a' x A B : ℕ) (ha : a ≤ a' + A) : |
| a + x + 2 ≤ a' + x + 2 + (A + B) := by |
| omega |
| unfold representativeFromActionsReadWithWork representativeFromActionsWithWork |
| by_cases hne : actions = [] |
| · simp only [dite_eq_left hne] |
| exact ⟨trivial, Nat.le_add_right _ _⟩ |
| · simp only [dite_eq_right hne] |
| have hz := wideMinimumReadWithWork_cost r read actions hne R hr |
| have hmono : (actions.length + 1) * (M * (R + 3)) ≤ (actions.length + 1) * (M * (R + 4)) := by |
| exact Nat.mul_le_mul_left _ (Nat.mul_le_mul_left _ |
| (Nat.add_le_add_left (by decide : 3 ≤ 4) R)) |
| have hz' := hz.trans (Nat.add_le_add_left hmono _) |
| rw [wideMinimumReadWithWork_eq] |
| split_ifs |
| · have hb := bandFromActionsReadWithWork_eq_cost parent r b tau theta |
| (divWithWork (stateGapWithWork sourceLength r).1 |
| (wideMinimumWithWork r (fun i q => (read i q).1) actions hne).1).1 read actions hne R hr |
| refine ⟨hb.1, ?_⟩ |
| exact costBound _ _ _ _ _ _ _ _ _ hz' hb.2 |
| · exact ⟨rfl, shortCostBound _ _ _ _ _ hz'⟩ |
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| theorem preparedRepresentativeReadWithWork_eq_cost {nx ny M : ℕ} (parent : TargetInterval nx) |
| (center : TargetInterval ny) (sourceLength b P : ℕ) (tau theta : BinaryFraction) |
| (read : Fin M → TargetInterval ny → BinaryFraction × ℕ) (R : ℕ) |
| (hr : ∀ i q, (read i q).2 ≤ R) : |
| (preparedRepresentativeReadWithWork parent center sourceLength b P tau theta read).1 = |
| (preparedRepresentativeWithWork parent center sourceLength b P tau theta (fun i q => (read i q).1)).1 ∧ |
| (preparedRepresentativeReadWithWork parent center sourceLength b P tau theta read).2 ≤ |
| (preparedRepresentativeWithWork parent center sourceLength b P tau theta (fun i q => (read i q).1)).2 + |
| representativeReadOverhead M (wideActionGridWithWork M parent center b P).1.length R := by |
| have h := representativeFromActionsReadWithWork_eq_cost parent center sourceLength b tau theta read |
| ((wideActionGridWithWork M parent center b P).1.map Vector.get) R (fun _ _ i => hr i _) |
| simp only [List.length_map] at h |
| unfold preparedRepresentativeReadWithWork preparedRepresentativeWithWork |
| dsimp only |
| exact ⟨h.1, by omega⟩ |
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| end BinaryFraction |
| end EditApproximation |
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| end OAI |
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