File size: 3,954 Bytes
d5b7741 | 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 78 79 80 | import OAI.Combinatorics.ContingencyTables.Sampling.AllSamplingProgram
import OAI.Combinatorics.ContingencyTables.Sampling.UnitTrialSemantics
import OAI.Combinatorics.ContingencyTables.Sampling.ProfilePublicSemantics
namespace OAI
/-! # One executable program chooses the correct rectangle branch -/
namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms DensePrograms CompletionCounts FirstPaperPhysicalMarginal
open FirstSuccess FairBits SmallContextCoordinates
open scoped BigOperators Classical
noncomputable section
attribute [local irreducible] unitProfileRetry unitCodeOuterDraw
variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
local notation "d" => dimensionAllowance (I := Fin m) (J := Fin n)
local notation "b" => smallBinaryLength r c
local notation "s" => smallProposalBits (I := Fin m) (J := Fin n)
lemma hasLargeRectangle_spec (k : ℕ) :
hasLargeRectangle ((List.ofFn r,List.ofFn c),k)=true ↔
0<(largeIndices (d^20,List.ofFn r)).length ∧
0<(largeIndices (d^20,List.ofFn c)).length := by
simp only [hasLargeRectangle,parameterLargeIndices,marginThreshold_ofFn,
ite_true,Bool.false_eq_true,ite_false,Bool.and_eq_true,decide_eq_true_eq]
lemma allSmall_of_no_large (k : ℕ)
(h : hasLargeRectangle ((List.ofFn r,List.ofFn c),k)≠true) :
∀ i j,paperSmall r c i j=true := by
intro i j
apply Bool.eq_true_of_ne_false
intro hij
obtain ⟨hi,hj⟩ := (firstPaperSmall_false_iff r c (d^20) i j).mp hij
have a := (largeIndexEquiv r (d^20)).symm ⟨i,hi⟩
have b' := (largeIndexEquiv c (d^20)).symm ⟨j,hj⟩
apply h
apply (hasLargeRectangle_spec r c k).mpr
exact ⟨Nat.zero_lt_of_lt a.isLt,Nat.zero_lt_of_lt b'.isLt⟩
lemma parameterUnitWords_ofFn (k : ℕ) :
parameterUnitWords ((List.ofFn r,List.ofFn c),k)=outerTrials d k*(outerSteps d b k*s) := by
rw [parameterUnitWords,parameterTrials_ofFn,parameterSteps_ofFn,parameterProposalWidth_ofFn]
lemma unitCodeBits_le (k : ℕ) : outerTrials d k*(outerSteps d b k*s)≤finiteSamplerBits r c k := by
apply Nat.mul_le_mul_left
exact (Nat.mul_le_mul_left (outerSteps d b k)
(Nat.le_add_right s (25*(d+b+outerAccuracy d b k+1)^62))).trans (Nat.le_add_right _ _)
lemma scheduled_steps_pos (k : ℕ) : 0<outerSteps d b k := by
have hd : 0<d := by unfold dimensionAllowance; omega
have hb := (smallBinaryLength_parameters r c).2.1
unfold outerSteps
exact Nat.mul_pos (pow_pos hd _) (pow_pos (by omega) _)
/-- The all-small branch uses a prefix of the same public fixed-length word. -/
lemma unitSamplingMatrix_code (hsmall : ∀ i j,paperSmall r c i j=true)
(htotal : (∑ i,r i)=∑ j,c j) (k : ℕ) (bits : Fin (finiteSamplerBits r c k) → Bool) :
unitSamplingMatrix (((List.ofFn r,List.ofFn c),k),List.ofFn bits)=
matrixCode (unitCodeOuterDraw r c (computedCellEquiv r c) (allSmallFibre r c hsmall)
(tableState r c (GreedyFeasibleTable.table r c htotal)) (outerSteps d b k) (outerTrials d k)
(GreedyFeasibleTable.table r c htotal) (prefixWord (unitCodeBits_le r c k) bits)).val := by
have hdraw := unitProfileRetry_code r c (computedCellEquiv r c) hsmall htotal
(outerAccuracy d b k) (tableState r c (GreedyFeasibleTable.table r c htotal))
(outerSteps d b k) (outerTrials d k) (25*(d+b+outerAccuracy d b k+1)^62)
(scheduled_steps_pos r c k) (GreedyFeasibleTable.table r c htotal)
(prefixWord (unitCodeBits_le r c k) bits)
unfold unitSamplingMatrix unitSamplingInput unitSamplingBits samplingRetryParameters
rw [parameterUnitWords_ofFn,take_ofFn_prefix (unitCodeBits_le r c k),
initialConfiguration_code r c htotal,samplingSchedule_ofFn,GreedyFeasibleTable.listMatrix_ofFn]
have hcode : matrixCode (GreedyFeasibleTable.table r c htotal).val=
List.ofFn (fun i => List.ofFn (GreedyFeasibleTable.matrix r c i)) := rfl
rw [hcode] at hdraw
simpa only [Table] using hdraw
end
end ContingencyTables.SmallGraphProfiles
end OAI
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