| import OAI.Computability.UniqueGames.Soundness.ZeroInformationLemmas |
|
|
| namespace OAI |
|
|
| /-! |
| # Complete local private inputs after fixing H |
|
|
| The constructors here consume only the appropriate zero-set question tuple: |
| occurrences for the first player, names for the second. A single reference sample |
| from the positive observation fibre is fixed once as part of H-dependent advice. |
| It supplies arbitrary padding at hidden positions and never depends on the actual |
| remaining sample. The reconstruction theorems prove equality of the complete |
| question-plus-row input, and the row maps restrict to the actual equation spaces |
| as shown in `ZeroInformation.supplied_map_pullback`. |
| -/ |
|
|
| namespace UniqueGamesTheorem.Soundness.ConditionalIncidences |
|
|
| noncomputable section |
| |
| instance zeroBits (R : Type) : Zero (ZeroInformation.Bits R) := |
| ⟨ZeroInformation.zero⟩ |
| |
| def membershipBool {P : Type} [DecidableEq P] (J : Finset P) : P → Bool := |
| fun j => decide (j ∈ J) |
| |
| def fullCoefficient {P R : Type} [DecidableEq P] (J : Finset P) |
| (gamma : RawCoefficients J (ZeroInformation.Bits R)) (i : Fin 2) : |
| P → ZeroInformation.Bits R := |
| fun j => if hj : j ∉ J then gamma (some (.inl (⟨j, hj⟩, i))) else ZeroInformation.zero |
| |
| def singleCoefficient {P R : Type} [DecidableEq P] (J : Finset P) |
| (gamma : RawCoefficients J (ZeroInformation.Bits R)) : P → ZeroInformation.Bits R := |
| fun j => if hj : j ∈ J then gamma (some (.inr ⟨j, hj⟩)) else ZeroInformation.zero |
| |
| def fillHidden {P X : Type} [DecidableEq P] (K : Finset P) |
| (fallback : P → X) (inside : PositionInside K → X) : P → X := |
| fun j => if hj : j ∈ K then inside ⟨j, hj⟩ else fallback j |
| |
| theorem singleCoefficient_zero {P R : Type} [DecidableEq P] |
| [Fintype R] [DecidableEq R] (J : Finset P) |
| (gamma : RawCoefficients J (ZeroInformation.Bits R)) (j : P) |
| (hj : j ∈ zeroSet J gamma) : singleCoefficient J gamma j = ZeroInformation.zero := by |
| obtain ⟨hjJ, hz⟩ := (mem_zeroSet J gamma j).mp hj |
| change gamma (some (.inr ⟨j, hjJ⟩)) = ZeroInformation.zero at hz |
| simpa only [singleCoefficient, hjJ, ↓reduceDIte] using hz |
| |
| theorem zeroSet_subset {P D : Type} [DecidableEq P] [Zero D] [DecidableEq D] |
| (J : Finset P) (gamma : RawCoefficients J D) : zeroSet J gamma ⊆ J := by |
| intro j hj |
| exact ((mem_zeroSet J gamma j).mp hj).choose |
| |
| variable {P R O N : Type} [Fintype P] [DecidableEq P] [Fintype R] [DecidableEq R] |
| (J : Finset P) (name : O → Fin 3 → N) |
| (gamma : RawCoefficients J (ZeroInformation.Bits R)) |
| (observed : PositionOutside (zeroSet J gamma) → O × N) |
| |
| def actualFirst (sample : Draw P O) : ZeroInformation.FirstInput P R O := |
| ZeroInformation.firstInput Finset.univ.toList (membershipBool J) (gamma none) |
| (fullCoefficient J gamma 0) (fullCoefficient J gamma 1) (singleCoefficient J gamma) |
| (fun j => (sample j).1) (fun j => (sample j).2) |
| |
| def actualSecond (sample : Draw P O) : ZeroInformation.SecondInput P R O N := |
| ZeroInformation.secondInput Finset.univ.toList (membershipBool J) (gamma none) |
| (fullCoefficient J gamma 0) (fullCoefficient J gamma 1) (singleCoefficient J gamma) |
| (fun j => (sample j).1) (fun j => name (sample j).1 (sample j).2) |
| |
| def localFirstFromH (reference : RawObservationFibre J name gamma observed) |
| (insideOccurrences : PositionInside (zeroSet J gamma) → O) : |
| ZeroInformation.FirstInput P R O := |
| ZeroInformation.reconstructFirst Finset.univ.toList |
| (membershipBool J) (membershipBool (zeroSet J gamma)) (gamma none) |
| (fullCoefficient J gamma 0) (fullCoefficient J gamma 1) (singleCoefficient J gamma) |
| name |
| (fillHidden (zeroSet J gamma) (fun j => (reference.val.2 j).1) insideOccurrences) |
| (fun j => (reference.val.2 j).1) |
| (fun j => name (reference.val.2 j).1 (reference.val.2 j).2) |
| |
| def localSecondFromH (reference : RawObservationFibre J name gamma observed) |
| (insideNames : PositionInside (zeroSet J gamma) → N) : |
| ZeroInformation.SecondInput P R O N := |
| ZeroInformation.reconstructSecond Finset.univ.toList |
| (membershipBool J) (membershipBool (zeroSet J gamma)) (gamma none) |
| (fullCoefficient J gamma 0) (fullCoefficient J gamma 1) (singleCoefficient J gamma) |
| (fun j => (reference.val.2 j).1) |
| (fillHidden (zeroSet J gamma) |
| (fun j => name (reference.val.2 j).1 (reference.val.2 j).2) insideNames) |
| (fun j => name (reference.val.2 j).1 (reference.val.2 j).2) |
| |
| omit [DecidableEq R] in |
| theorem localFirstFromH_question_inside |
| (reference : RawObservationFibre J name gamma observed) |
| (insideOccurrences : PositionInside (zeroSet J gamma) → O) |
| (j : PositionInside (zeroSet J gamma)) : |
| (localFirstFromH J name gamma observed reference insideOccurrences).question j.val = |
| insideOccurrences j := by |
| simp [localFirstFromH, ZeroInformation.reconstructFirst, ZeroInformation.firstInput, |
| ZeroInformation.reconstructQuestion, fillHidden, membershipBool, j.property] |
| |
| omit [DecidableEq R] in |
| theorem localSecondFromH_question_inside |
| (reference : RawObservationFibre J name gamma observed) |
| (insideNames : PositionInside (zeroSet J gamma) → N) |
| (j : PositionInside (zeroSet J gamma)) : |
| (localSecondFromH J name gamma observed reference insideNames).question j.val = |
| Sum.inr (insideNames j) := by |
| have hjJ : j.val ∈ J := zeroSet_subset J gamma j.property |
| simp [localSecondFromH, ZeroInformation.reconstructSecond, ZeroInformation.secondInput, |
| ZeroInformation.partnerQuestion, ZeroInformation.reconstructQuestion, |
| fillHidden, membershipBool, j.property, hjJ] |
| |
| omit [Fintype P] [DecidableEq R] in |
| theorem outside_pair_agree (reference sample : RawObservationFibre J name gamma observed) |
| (j : P) (hj : j ∉ zeroSet J gamma) : |
| incidence name (reference.val.2 j) = incidence name (sample.val.2 j) := by |
| exact congrFun (reference.property.2.trans sample.property.2.symm) ⟨j, hj⟩ |
| |
| omit [Fintype P] [DecidableEq R] in |
| theorem fillHidden_occurrences (reference sample : RawObservationFibre J name gamma observed) : |
| fillHidden (zeroSet J gamma) (fun j => (reference.val.2 j).1) |
| (fun j => (sample.val.2 j.val).1) = |
| (fun j => (sample.val.2 j).1) := by |
| funext j |
| by_cases hj : j ∈ zeroSet J gamma |
| · simp [fillHidden, hj] |
| · simp only [fillHidden, hj, ↓reduceDIte] |
| exact congrArg (fun p : O × N => p.1) |
| (outside_pair_agree J name gamma observed reference sample j hj) |
| |
| omit [Fintype P] [DecidableEq R] in |
| theorem fillHidden_names (reference sample : RawObservationFibre J name gamma observed) : |
| fillHidden (zeroSet J gamma) |
| (fun j => name (reference.val.2 j).1 (reference.val.2 j).2) |
| (fun j => name (sample.val.2 j.val).1 (sample.val.2 j.val).2) = |
| (fun j => name (sample.val.2 j).1 (sample.val.2 j).2) := by |
| funext j |
| by_cases hj : j ∈ zeroSet J gamma |
| · simp [fillHidden, hj] |
| · simp only [fillHidden, hj, ↓reduceDIte] |
| exact congrArg Prod.snd (outside_pair_agree J name gamma observed reference sample j hj) |
| |
| / |
| theorem localFirstFromH_correct |
| (distinct : ∀ o i i', name o i = name o i' → i = i') |
| (reference sample : RawObservationFibre J name gamma observed) : |
| localFirstFromH J name gamma observed reference (fun j => (sample.val.2 j.val).1) = |
| actualFirst J gamma sample.val.2 := by |
| unfold localFirstFromH actualFirst |
| rw [fillHidden_occurrences J name gamma observed reference sample] |
| apply ZeroInformation.first_input_reconstruction |
| · intro j hj |
| exact singleCoefficient_zero J gamma j (of_decide_eq_true hj) |
| · exact distinct |
| · intro j hj |
| have hj' : j ∉ zeroSet J gamma := of_decide_eq_false hj |
| exact congrArg (fun p : O × N => p.1) |
| (outside_pair_agree J name gamma observed reference sample j hj') |
| · intro j hj |
| have hj' : j ∉ zeroSet J gamma := of_decide_eq_false hj |
| exact congrArg Prod.snd (outside_pair_agree J name gamma observed reference sample j hj') |
| |
| omit [DecidableEq R] in |
| / |
| occurrence, equation rhs, or slot from any position in K. -/ |
| theorem localSecondFromH_correct (reference sample : RawObservationFibre J name gamma observed) : |
| localSecondFromH J name gamma observed reference |
| (fun j => name (sample.val.2 j.val).1 (sample.val.2 j.val).2) = |
| actualSecond J name gamma sample.val.2 := by |
| unfold localSecondFromH actualSecond |
| rw [fillHidden_names J name gamma observed reference sample] |
| apply ZeroInformation.second_input_reconstruction |
| · intro j hj |
| exact decide_eq_true (zeroSet_subset J gamma (of_decide_eq_true hj)) |
| · intro j hj |
| have hj' : j ∉ zeroSet J gamma := of_decide_eq_false hj |
| exact congrArg (fun p : O × N => p.1) |
| (outside_pair_agree J name gamma observed reference sample j hj') |
| · intro j hj |
| have hj' : j ∉ zeroSet J gamma := of_decide_eq_false hj |
| exact congrArg Prod.snd (outside_pair_agree J name gamma observed reference sample j hj') |
| |
| end |
| |
| end UniqueGamesTheorem.Soundness.ConditionalIncidences |
| |
| end OAI |
| |
