File size: 3,616 Bytes
97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Machines.CachedProfileDecision
namespace OAI
/-! # Cached completions implement the same outer-profile transition -/
namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms DensePrograms PhysicalCompletionFibres FirstPaperPhysicalMarginal
open SmallContextCoordinates CompletionCounts CompletionTranslation
open scoped BigOperators Classical
noncomputable section
variable {m n q : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
variable (e : Fin q ≃ Cells (paperSmall r c))
variable (hr : 0<(largeIndices ((dimensionAllowance (I := Fin m) (J := Fin n))^20,List.ofFn r)).length)
variable (hc : 0<(largeIndices ((dimensionAllowance (I := Fin m) (J := Fin n))^20,List.ofFn c)).length)
local notation "d" => dimensionAllowance (I := Fin m) (J := Fin n)
local notation "R" => listedRows r c (d^20) (d^12) (stateCapacity r c) hr
local notation "P" => listedColumns r c (d^20) (d^12) (stateCapacity r c) hc
local notation "s" => smallProposalBits (I := Fin m) (J := Fin n)
def encodedShortStep (h : ℕ) (x : PaperStates r c) (word : Fin s → Bool) : ShortStep :=
(((stateCodeData r c e x,d^12),(d,h)),List.ofFn word)
/-- This identity holds for every feasible cached completion, including
those not selected by a particular finite random word. -/
theorem cachedDecision_encoded (htotal : ∑ i,r i=∑ j,c j) (h : ℕ)
(x : PaperStates r c) (word : Fin s → Bool) (A : Table (R x) (P x)) :
cachedDecision (encodedShortStep r c e h x word,tableOutput A)=
stateCode r c e (completionDecision R P x (decodedCodeProposal r c e x word) A) := by
have hp := codeProposal_encoded r c e htotal x word
cases hd : decodedCodeProposal r c e x word with
| none =>
rw [hd,Option.map_none] at hp
have hviews : stepCandidateViews (shortStepInput (encodedShortStep r c e h x word))=
stateCode r c e x := by
simp only [stepCandidateViews,stepProposal,shortStepInput,encodedShortStep,hp,
Option.getD_none]
rfl
unfold cachedDecision
rw [hviews]
split_ifs <;> rfl
| some y =>
rw [hd,Option.map_some] at hp
have hparams : stepCandidateParameters (shortStepInput (encodedShortStep r c e h x word))=
((stateCodeData r c e y,d^12),(d,h)) := by
simp only [stepCandidateParameters,stepCandidate,stepCandidateViews,stepProposal,
shortStepInput,encodedShortStep,hp,Option.getD_some]
rfl
have hdata : cachedDecisionInput (encodedShortStep r c e h x word,tableOutput A)=
(((marginCode (R x),marginCode (P x)),(marginCode (R y),marginCode (P y))),
tableOutput A) := by
unfold cachedDecisionInput stepOldMargins stepNewMargins
rw [hparams]
dsimp only [shortStepInput,encodedShortStep]
rw [residualParameters_code,residualParameters_code]
have hviews : stepCandidateViews (shortStepInput (encodedShortStep r c e h x word))=
stateCode r c e y := by
simp only [stepCandidateViews,stepProposal,shortStepInput,encodedShortStep,hp,
Option.getD_some]
have hv := translationValid_code (R x) (R y) (P x) (P y) A
unfold cachedDecision
rw [hdata,hviews]
change (if translationValid _ then stateCode r c e y else stateCode r c e x)=
stateCode r c e (if ∀ i j,0 ≤ (A.val i j:ℤ)+between (R x) (R y) (P x) (P y) i j
then y else x)
by_cases ht : ∀ i j,0 ≤ (A.val i j:ℤ)+between (R x) (R y) (P x) (P y) i j
· rw [ite_eq_left ht,ite_eq_left (hv.mpr ht)]
· rw [ite_eq_right ht,ite_eq_right (fun hv' => ht (hv.mp hv'))]
end
end ContingencyTables.SmallGraphProfiles
end OAI
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