File size: 4,180 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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseBinaryGrid
namespace OAI
namespace EditApproximation
namespace BinaryFraction
/-- Child arrays are reused exactly as in the shared rational evaluator.
This composition lemma isolates the inductive recursive-call hypothesis. -/
theorem coarseBinaryChildren_rep {B d : ℕ} {α : Type*} [DecidableEq α]
(k q L : ℕ) (frame : List (Option α)) (two allowance : BinaryFraction)
(children : List (CoarseSourceTree B α d × Fin B × CoarseThresholdDraws B d))
(htwo : two.value = 2) (ha : allowance.representation.den = two.representation.den)
(ht : ∀ draw : Fin B,
(coarseChildThresholdWithWork B q allowance draw).1.value =
coarseChildThreshold B q allowance.value draw)
(hs : (coarseChildAllowanceWithWork q allowance).1.value = allowance.value / q)
(he : ∀ child draw below, (child, draw, below) ∈ children → ∀ start,
CoarseArrayRep (coarseBinaryEval k q L frame two child below
(coarseChildAllowanceWithWork L (coarseChildThresholdWithWork B q allowance draw).1).1 start).1
(coarseArrayEval k q (1 / L) frame child below
((1 / L) * coarseChildThreshold B q allowance.value draw) start).1)
(start : ℕ) :
CoarseArrayRep (coarseBinaryChildren k q L two
(fun child => coarseBinaryEval k q L frame two child) allowance children start).1
(coarseArrayEvalChildren k q (1 / L) (fun child => coarseArrayEval k q (1 / L) frame child)
allowance.value children start).1 := by
induction children generalizing start with
| nil => rfl
| cons data rest ih =>
obtain ⟨child, draw, below⟩ := data
let threshold := coarseChildThresholdWithWork B q allowance draw
let childAllowance := coarseChildAllowanceWithWork L threshold.1
let evaluated := coarseBinaryEval k q L frame two child below childAllowance.1 start
let ratEvaluated := coarseArrayEval k q (1 / L) frame child below
((1 / L) * coarseChildThreshold B q allowance.value draw) start
have hthreshold : threshold.1.representation.den = two.representation.den :=
(coarseChildThresholdWithWork_den B q allowance draw).trans ha
have hchild : childAllowance.1.representation.den = two.representation.den :=
(coarseChildAllowanceWithWork_den L threshold.1).trans hthreshold
have hev : CoarseArrayRep evaluated.1 ratEvaluated.1 := he child draw below List.mem_cons_self start
have hg : CoarseArrayOnGrid two.representation.den evaluated.1 :=
coarseBinaryEval_grid k q L frame two child below childAllowance.1 start hchild
have hp := coarseBinaryPenalty_rep two evaluated.1 ratEvaluated.1 htwo hev hg
have hpg := coarseBinaryPenalty_grid two evaluated.1 hg
have hmap := coarseBinaryMap_rep
(coarseContributionWithWork threshold.1 (coarseChildAllowanceWithWork q allowance).1)
(coarseThresholdContribution (coarseChildThreshold B q allowance.value draw) (allowance.value / q))
(by
intro a
rw [coarseContributionWithWork_value, hs]
exact congrArg (fun t => coarseThresholdContribution t (allowance.value / q) a.value) (ht draw))
(coarseBinaryPenaltyWithWork two evaluated.1).1 (coarseStoredPenalty ratEvaluated.1).1 hp
have hmg := coarseBinaryMap_grid
(coarseContributionWithWork threshold.1 (coarseChildAllowanceWithWork q allowance).1)
(coarseBinaryPenaltyWithWork two evaluated.1).1 two.representation.den hpg (by
intro a hden
exact (coarseContributionWithWork_den _ _ a
(((coarseChildAllowanceWithWork_den q allowance).trans ha).trans hden.symm)).trans hden)
have hr := ih (fun child draw below hmem => he child draw below (List.mem_cons_of_mem _ hmem))
(start + child.word.length)
have hrg := coarseBinaryChildren_grid k q L two
(fun child => coarseBinaryEval k q L frame two child) allowance rest
(start + child.word.length) ha (by
intro child draw below _ a s had
exact coarseBinaryEval_grid k q L frame two child below a s had)
exact coarseBinaryAdd_rep _ _ _ _ two.representation.den hmap hr hmg hrg
end BinaryFraction
end EditApproximation
end OAI
|