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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 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionFactorialApplication
import OAI.Combinatorics.ContingencyTables.Transport.InitialTransversals
namespace OAI
/-! # Exact initial normalizer for the literal companion completion count -/
namespace ContingencyTables.CompanionInitialization
open scoped BigOperators Classical
open CommonBasesFPRAS CompanionFactorialApplication LabelledFactorial TwoLiftStates
variable {I J : Type*} [Fintype I] [Fintype J]
noncomputable def freeTotal (A : Instance I J) : ℝ :=
∏ a : ForestAssignmentSplit.FreeCell A.small A.T,
((A.b a.val.1 a.val.2+1:ℕ):ℝ)
theorem h_zero (A : Instance I J) (z : View A → ℕ) : h A 0 z=freeTotal A :=
ForestCountIdentification.sourceMarginal_zero A.small A.T A.d A.b A.r A.c z
theorem denominator_eq (A : Instance I J) :
CompanionFactorialApplication.denominator A=
InitialTransversals.denominator (fun s : SmallCell A => A.b s.val.1 s.val.2) := by
unfold CompanionFactorialApplication.denominator TwoLiftStates.denominator InitialTransversals.denominator
simp_rw [label_width]
/-- C(0) is derived for the actual source f, using the finite
paired-transversal enumeration and the literal free-completion count. -/
theorem total_zero (A : Instance I J) :
slotTotal pairs (f A 0) ∅=freeTotal A*∏ s : SmallCell A, ((A.b s.val.1 s.val.2:ℝ)+1) := by
have he : f A 0=firstLift (group (labelCell A)) (fun _ => freeTotal A)
(CompanionFactorialApplication.denominator A) := by
funext S
unfold f firstLift
rw [h_zero]
rw [he,denominator_eq]
convert InitialTransversals.slotTotal_constant
(fun s : SmallCell A => A.b s.val.1 s.val.2) (freeTotal A) using 1
congr 1
end ContingencyTables.CompanionInitialization
end OAI
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