File size: 3,793 Bytes
e63eb7f | 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 | 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
end OAI
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