openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /CachedDecisionSemantics.lean
| 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 | |