File size: 3,296 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.Counting.CompanionComputedInitial
import OAI.Combinatorics.ContingencyTables.Machines.PairedCatalogProgram
namespace OAI
/-! # The companion proposal with computed metadata
The executable proposal receives only a binary profile and fresh bits.
It computes the consecutive-pair catalog itself and scans coordinate
exchanges. Its semantic law is the same ordered proposal law.
-/
namespace ContingencyTables.CompanionComputedProposal
open MatchingFPRAS.TreeTyped CompanionFactorialApplication CompanionComputedInitial
open CompanionArithmeticProposal CompanionOrderedProposal CompanionNeighborProgram
open CompanionBooleanPhase
open scoped Classical
def proposalProgram (z : List ℕ×List Bool) : Option (List ℕ) :=
codeProposal ((PairedCatalogProgram.catalog z.1,z.1),z.2)
def catalogInputRealizer : Realizer (fun z : List ℕ×List Bool => PairedCatalogProgram.catalog z.1) :=
composition (f := fun z : List ℕ×List Bool => z.1) (g := PairedCatalogProgram.catalog)
first PairedCatalogProgram.catalogRealizer
def proposalProgramRealizer : Realizer proposalProgram :=
composition (f := fun z : List ℕ×List Bool => ((PairedCatalogProgram.catalog z.1,z.1),z.2))
(g := codeProposal) (pair (pair catalogInputRealizer first) second) codeProposalRealizer
theorem polynomial_proposalProgram : PolynomialTime proposalProgramRealizer :=
polynomial_composition (polynomial_pair (polynomial_pair
(polynomial_composition polynomial_first PairedCatalogProgram.polynomial_catalog)
polynomial_first) polynomial_second) polynomial_codeProposal
noncomputable section
variable {m n : ℕ} (A : Instance (Fin m) (Fin n))
def pairLabels : Fin (CanonicalInitialEncoding.copyOrder (capacities A)).length ≃ Labels A :=
(PairedCatalogProgram.labelOrder (capacities A)).trans (copyOrder A).symm
lemma pair_coordinate (i : Fin (CanonicalInitialEncoding.copyOrder (capacities A)).length)
(side : Bool) :
coordinateOrder A (pairLabels A i,side)=
CanonicalInitialEncoding.coordinateEquiv (capacities A)
(PairedCatalogProgram.labelOrder (capacities A) i,side) := by
simp [coordinateOrder,pairOrder,pairLabels]
lemma generated_catalog (S : Set (Labels A×Bool)) :
PairedCatalogProgram.catalog (encodeSet (coordinateOrder A) S)=
orderedCatalog A (pairLabels A) (coordinateOrder A) := by
have he : encodeSet (coordinateOrder A) S=
encodeSet (CanonicalInitialEncoding.coordinateEquiv (capacities A))
((pairOrder A).symm ⁻¹' S) := rfl
rw [he,PairedCatalogProgram.catalog_encode]
unfold orderedCatalog
apply congrArg List.ofFn
funext i
rw [pair_coordinate,pair_coordinate]
/-- Every proposed code is the encoding of the actual ordered proposal
state, and unused binary indices give precisely the same `none` result. -/
theorem proposalProgram_eq (S : State A) (bits : Fin (proposalBits A) → Bool) :
proposalProgram (encodeSet (coordinateOrder A) S.val,List.ofFn bits)=
(orderedProposal A (coordinateOrder A) S bits).map
(fun T => encodeSet (coordinateOrder A) T.val) := by
unfold proposalProgram
rw [generated_catalog]
exact codeProposal_ordered A (pairLabels A) (coordinateOrder A) S bits
end
end ContingencyTables.CompanionComputedProposal
end OAI
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