openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Coarse /CoarseChildrenSimulation.lean
| 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 | |