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cdec422 | 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 | import Mathlib
import OAI.LinearAlgebra.MatrixFields.Entropy.StageCLaw
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
namespace MatrixMultiplication.AllFieldParameters
def statisticLaw (d : ℚ) (k : Fin 5) (i : Fin 6) : ℚ :=
if k.val = 2 then
if i = 2 then 1 - d else if i = 3 then d else 0
else if i = singletonSlot k.val then 1 else 0
def statisticComplement (k : Fin 5) : Fin 5 := ⟨4 - k.val, by omega⟩
theorem statisticLaw_complement (d : ℚ) (k : Fin 5) (i : Fin 6) :
statisticLaw d k (kappa i) = statisticLaw d (statisticComplement k) i := by
fin_cases k <;> fin_cases i <;> rfl
theorem statisticLaw_outside_support (d : ℚ) (k : Fin 5) (i : Fin 6)
(h : statisticWeight i ≠ k.val) : statisticLaw d k i = 0 := by
fin_cases k <;> fin_cases i
all_goals first | rfl | exact False.elim (h (by decide))
theorem littleLaw_eq_statisticLaw (t u : Shape) (w : Fin 3) (hw : u w < 5) (i : Fin 6) :
littleLaw t u w i = statisticLaw (binaryParameter t u) ⟨u w, hw⟩ i := rfl
theorem littleLaw_outside_support (t u : Shape) (w : Fin 3) (hw : u w < 5) (i : Fin 6)
(h : statisticWeight i ≠ u w) : littleLaw t u w i = 0 :=
statisticLaw_outside_support _ ⟨u w, hw⟩ i h
theorem littleLaw_complement (t u : Shape) (w v : Fin 3)
(hw : u w < 5) (hv : u v < 5) (hsum : u w + u v = 4) (i : Fin 6) :
littleLaw t u w (kappa i) = littleLaw t u v i := by
rw [littleLaw_eq_statisticLaw t u w hw, littleLaw_eq_statisticLaw t u v hv]
have hk : (⟨u v, hv⟩ : Fin 5) = statisticComplement ⟨u w, hw⟩ := by
apply Fin.ext
simp only [statisticComplement]
omega
rw [hk]
exact statisticLaw_complement _ _ _
theorem child_statistic_weight_bounded : ∀ t ∈ positiveSecond, ∀ u ∈ below t,
∀ w : Fin 3, u w < 5 := by decide +kernel
theorem pair_complement_involution (i j : Fin 6) :
(kappa (kappa i), kappa (kappa j)) = (i, j) := by simp [kappa_involution]
end MatrixMultiplication.AllFieldParameters
end MatrixAllFields
end OAI
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