| import OAI.Combinatorics.EditApproximation.Coarse.CoarseBinaryEvaluation |
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseArraySimulation |
|
|
| namespace OAI |
|
|
| namespace EditApproximation |
| namespace BinaryFraction |
|
|
| theorem coarseBinaryChildren_grid {B d : ℕ} {α : Type*} (k q L : ℕ) (two : BinaryFraction) |
| (evaluate : CoarseSourceTree B α d → CoarseThresholdDraws B d → BinaryFraction → ℕ → |
| CoarseBinaryArray × ℕ) (allowance : BinaryFraction) |
| (children : List (CoarseSourceTree B α d × Fin B × CoarseThresholdDraws B d)) (start : ℕ) |
| (ha : allowance.representation.den = two.representation.den) |
| (he : ∀ child draw below, (child, draw, below) ∈ children → ∀ (a : BinaryFraction) (s : ℕ), |
| a.representation.den = two.representation.den → CoarseArrayOnGrid two.representation.den |
| (evaluate child below a s).1) : |
| CoarseArrayOnGrid two.representation.den |
| (coarseBinaryChildren k q L two evaluate allowance children start).1 := by |
| induction children generalizing start with |
| | nil => intro values hv; cases hv |
| | cons data rest ih => |
| obtain ⟨child, draw, below⟩ := data |
| have ht := (coarseChildThresholdWithWork_den B q allowance draw).trans ha |
| have hc := (coarseChildAllowanceWithWork_den L (coarseChildThresholdWithWork B q allowance draw).1).trans ht |
| have hev := he child draw below List.mem_cons_self |
| (coarseChildAllowanceWithWork L (coarseChildThresholdWithWork B q allowance draw).1).1 start hc |
| have hp := coarseBinaryPenalty_grid two _ hev |
| have hs := (coarseChildAllowanceWithWork_den q allowance).trans ha |
| have hm := coarseBinaryMap_grid |
| (coarseContributionWithWork (coarseChildThresholdWithWork B q allowance draw).1 |
| (coarseChildAllowanceWithWork q allowance).1) |
| (coarseBinaryPenaltyWithWork two |
| (evaluate child below (coarseChildAllowanceWithWork L |
| (coarseChildThresholdWithWork B q allowance draw).1).1 start).1).1 |
| two.representation.den hp (by |
| intro z hz |
| exact (coarseContributionWithWork_den _ _ z (hs.trans hz.symm)).trans hz) |
| have hr := ih (start + child.word.length) |
| (fun child draw below hmem => he child draw below (List.mem_cons_of_mem _ hmem)) |
| exact coarseBinaryAdd_grid _ _ _ hm hr |
|
|
| theorem coarseBinaryEval_grid {B d : ℕ} {α : Type*} [DecidableEq α] |
| (k q L : ℕ) (frame : List (Option α)) (two : BinaryFraction) |
| (tree : CoarseSourceTree B α d) (draws : CoarseThresholdDraws B d) |
| (allowance : BinaryFraction) (start : ℕ) |
| (ha : allowance.representation.den = two.representation.den) : |
| CoarseArrayOnGrid two.representation.den |
| (coarseBinaryEval k q L frame two tree draws allowance start).1 := by |
| induction tree generalizing allowance start with |
| | leaf letter => |
| rw [coarseBinaryEval] |
| split_ifs |
| · intro values hv; cases hv |
| · intro values hv a hamem |
| have heq : (arithmeticMapWithWork (fun shift : Fin (2 * k + 1) => |
| coarseGridNatWithWork allowance.denominator allowance.denominator_pos |
| (coarseShiftCost k frame (.leaf (B := B) letter) start shift)) |
| (List.finRange (2 * k + 1))).1 = values := Option.some.inj hv |
| rw [← heq, arithmeticMapWithWork_value] at hamem |
| obtain ⟨shift, _, rfl⟩ := List.mem_map.mp hamem |
| exact (coarseGridNatWithWork_den _ _ _).trans ha |
| | branch children ih => |
| obtain ⟨draws, below⟩ := draws |
| rw [coarseBinaryEval] |
| split_ifs |
| · intro values hv; cases hv |
| · apply coarseBinaryChildren_grid k q L two _ allowance _ start ha |
| intro child draw childBelow hmem a s hag |
| obtain ⟨i, hi⟩ := List.mem_ofFn.mp hmem |
| cases hi |
| exact ih i (below i) a s hag |
|
|
| end BinaryFraction |
| end EditApproximation |
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|
| end OAI |
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