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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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Populations
import OAI.LinearAlgebra.MatrixMultiplication.FieldParameters.HalfLaws
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldTerminalStatisticLaws
open AllFieldParameters AllFieldHistory
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
abbrev AZero (K : ℕ) :=
{h : AfterA K // aShape h ∉ positiveSecond} × Placement
abbrev BZero (K : ℕ) :=
{h : AfterB K // positive (bShape h) ≠ true} × Placement
abbrev TerminalZero (K : ℕ) := AZero K ⊕ BZero K
variable {K : ℕ}
theorem aShape_mem_shapes (h : AZero K) : aShape h.1.val ∈ shapes 8 :=
aShape_size h.1.val
theorem aShape_not_positive (h : AZero K) : positive (aShape h.1.val) ≠ true := by
intro hp
exact h.1.property (List.mem_filter.mpr ⟨aShape_mem_shapes h, hp⟩)
theorem aShape_mem_zeroSecond (h : AZero K) : aShape h.1.val ∈ zeroSecond := by
simp [zeroSecond, aShape_mem_shapes h, aShape_not_positive h]
def bParent (h : BZero K) : Shape := aShape h.1.val.1.val
theorem bParent_mem_positiveSecond (h : BZero K) : bParent h ∈ positiveSecond :=
h.1.val.1.property
theorem bShape_mem_shapes (h : BZero K) : bShape h.1.val ∈ shapes 4 :=
bShape_size h.1.val
theorem bShape_mem_below (h : BZero K) : bShape h.1.val ∈ below (bParent h) :=
bShape_mem h.1.val
theorem bShape_max_lt_five (h : BZero K) : shapeMax (bShape h.1.val) < 5 := by
have hb := child_statistic_weight_bounded (bParent h)
(bParent_mem_positiveSecond h) (bShape h.1.val) (bShape_mem_below h)
exact max_lt_iff.mpr ⟨hb 0, max_lt_iff.mpr ⟨hb 1, hb 2⟩⟩
def bWeight (h : BZero K) : Fin 5 :=
⟨shapeMax (bShape h.1.val), bShape_max_lt_five h⟩
@[simp] theorem bWeight_val (h : BZero K) : (bWeight h).val = shapeMax (bShape h.1.val) :=
rfl
def Slot : TerminalZero K → Type
| .inl _ => PairSlot
| .inr _ => Fin 6
instance (h : TerminalZero K) : Fintype (Slot h) := by
cases h <;> dsimp [Slot] <;> infer_instance
def history : TerminalZero K → History K
| .inl h => (.afterA h.1.val, h.2)
| .inr h => (.afterB h.1.val, h.2)
@[simp] theorem history_inl (h : AZero K) :
history (.inl h) = (.afterA h.1.val, h.2) := rfl
@[simp] theorem history_inr (h : BZero K) :
history (.inr h) = (.afterB h.1.val, h.2) := rfl
def law : (h : TerminalZero K) → Slot h → ℚ
| .inl h, s => zeroPairLaw (aShape h.1.val) s.1 s.2
| .inr h, s => statisticLaw (binaryParameter (bParent h) (bShape h.1.val)) (bWeight h) s
@[simp] theorem law_inl (h : AZero K) (s : PairSlot) :
law (.inl h) s = zeroPairLaw (aShape h.1.val) s.1 s.2 := rfl
@[simp] theorem law_inr (h : BZero K) (s : Fin 6) :
law (.inr h) s =
statisticLaw (binaryParameter (bParent h) (bShape h.1.val)) (bWeight h) s := rfl
theorem statisticLaw_nonnegative (t u : Shape) (k : Fin 5) (s : Fin 6) :
0 ≤ statisticLaw (binaryParameter t u) k s := by
obtain ⟨hl, hh⟩ := binaryParameter_bounds t u
unfold statisticLaw
split_ifs <;> linarith
theorem statisticLaw_total (d : ℚ) (k : Fin 5) :
∑ s : Fin 6, statisticLaw d k s = 1 := by
fin_cases k <;> simp [statisticLaw, Fin.sum_univ_succ, singletonSlot]
theorem law_nonneg (h : TerminalZero K) (s : Slot h) : 0 ≤ law h s := by
cases h with
| inl h => exact zeroPairLaw_nonnegative (aShape h.1.val) s.1 s.2
| inr h => exact statisticLaw_nonnegative (bParent h) (bShape h.1.val) (bWeight h) s
theorem law_total (h : TerminalZero K) : ∑ s : Slot h, law h s = 1 := by
cases h with
| inl h => exact zeroPairLaw_full_normalized (aShape h.1.val) (aShape_mem_zeroSecond h)
| inr h => exact statisticLaw_total _ (bWeight h)
theorem history_amount_pos (allocation : Allocation) (h : TerminalZero K) :
0 < amount allocation (history h) := by
cases h with
| inl h =>
change 0 < aAmount h.1.val / 6
exact div_pos (aAmount_pos h.1.val) (by norm_num)
| inr h =>
change 0 < bAmount h.1.val / 6
exact div_pos (bAmount_pos h.1.val) (by norm_num)
end MatrixMultiplication.AllFieldTerminalStatisticLaws
end
end OAI
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