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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialOrbit
import Mathlib.Data.Rat.BigOperators
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldInitialGibbs
open AllFieldParameters AllFieldHistory AllFieldInitialEntropy AllFieldInitialOrbit
open MatrixMultiplication.Foundation
open scoped BigOperators
def coordinateWeight (k : Fin 17) : ℝ :=
(weight (initialInteger k.val) : ℝ)
def orderedWeight (u : JointPopulation.Shape) : ℝ :=
∏ s : Fin 3, coordinateWeight (JointPopulation.shapeSide s u)
def initialNormalizer : ℝ := ((sortedInitial.map initialWeight).sum : ℚ)
def initialPotential (k : Fin 17) : ℝ := Real.log (coordinateWeight k)
def initialLogNormalizer : ℝ := Real.log initialNormalizer
theorem coordinateWeight_pos (k : Fin 17) : 0 < coordinateWeight k := by
apply Rat.cast_pos.mpr
apply weight_positive_of_argument_bound
exact assigned_argument_bound _ tablea initial_table_arguments_bounded _
theorem orderedWeight_pos (u : JointPopulation.Shape) : 0 < orderedWeight u :=
Finset.prod_pos (fun _ _ => coordinateWeight_pos _)
theorem initialNormalizer_pos : 0 < initialNormalizer := by
apply Rat.cast_pos.mpr
exact list_sum_map_positive _ _ (by decide +kernel)
(fun s _ => initialWeight_positive s)
theorem orderedWeight_source (p : PlacedInitial) :
orderedWeight (initialShapeCode p) =
∏ s : Fin 3, (weight (initialInteger (sortedInitial.get p.1 s)) : ℝ) := by
simp only [orderedWeight, coordinateWeight, initialShapeCode_side]
exact p.2.symm.prod_comp
(fun s : Fin 3 => (weight (initialInteger (sortedInitial.get p.1 s)) : ℝ))
theorem initialWeight_source (p : PlacedInitial) :
(initialWeight (sortedInitial.get p.1) : ℝ) =
(initialMultiplicity (sortedInitial.get p.1) : ℝ) *
orderedWeight (initialShapeCode p) := by
rw [orderedWeight_source]
simp only [initialWeight]
push_cast
rfl
theorem placedInitialLaw_mass (p : PlacedInitial) :
placedInitialLaw.mass p =
(initialMultiplicity (sortedInitial.get p.1) : ℝ) *
orderedWeight (initialShapeCode p) / initialNormalizer / 6 := by
change (initialLaw (sortedInitial.get p.1) : ℝ) / 6 = _
rw [initialLaw, ite_eq_left (List.get_mem _ _), Rat.cast_div, initialWeight_source p]
rfl
theorem orderedInitialLaw_mass (u : JointPopulation.Shape)
(hu : u.1.val + u.2.1.val + u.2.2.val = 16) :
orderedInitialLaw.mass u = orderedWeight u / initialNormalizer := by
classical
have hcount : (∑ p : PlacedInitial,
if initialShapeCode p = u then
(initialMultiplicity (sortedInitial.get p.1) : ℝ) else 0) = 6 := by
exact_mod_cast weighted_fiber_count u hu
simp only [orderedInitialLaw, FiniteLaw.map_mass, placedInitialLaw_mass]
calc
_ = (∑ p : PlacedInitial, if initialShapeCode p = u then
(initialMultiplicity (sortedInitial.get p.1) : ℝ) else 0) *
(orderedWeight u / initialNormalizer / 6) := by
rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro p _
by_cases hp : initialShapeCode p = u
· simp only [hp, ite_true]
ring
· simp only [hp, ite_false, zero_mul]
_ = _ := by rw [hcount]; ring
theorem orderedInitialLaw_zero (u : JointPopulation.Shape)
(hu : u.1.val + u.2.1.val + u.2.2.val ≠ 16) :
orderedInitialLaw.mass u = 0 := by
classical
simp only [orderedInitialLaw, FiniteLaw.map_mass]
apply Finset.sum_eq_zero
intro p _
apply ite_eq_right
intro hp
have ht := source_total p
rw [hp] at ht
exact hu ht
theorem orderedInitialLaw_mass_eq (u : JointPopulation.Shape) :
orderedInitialLaw.mass u =
if u.1.val + u.2.1.val + u.2.2.val = 16
then orderedWeight u / initialNormalizer else 0 := by
split_ifs with hu
· exact orderedInitialLaw_mass u hu
· exact orderedInitialLaw_zero u hu
theorem orderedInitialLaw_positive (u : JointPopulation.Shape)
(hu : u.1.val + u.2.1.val + u.2.2.val = 16) :
0 < orderedInitialLaw.mass u := by
rw [orderedInitialLaw_mass u hu]
exact div_pos (orderedWeight_pos u) initialNormalizer_pos
theorem orderedInitialLaw_support (u : JointPopulation.Shape)
(hu : 0 < orderedInitialLaw.mass u) :
u.1.val + u.2.1.val + u.2.2.val = 16 := by
by_contra hn
rw [orderedInitialLaw_zero u hn] at hu
exact lt_irrefl _ hu
theorem log_orderedWeight (u : JointPopulation.Shape) :
Real.log (orderedWeight u) =
initialPotential (JointPopulation.shapeSide 0 u) +
initialPotential (JointPopulation.shapeSide 1 u) +
initialPotential (JointPopulation.shapeSide 2 u) := by
rw [orderedWeight, Real.log_prod
(fun s _ => ne_of_gt (coordinateWeight_pos (JointPopulation.shapeSide s u)))]
simp only [Fin.sum_univ_three, initialPotential]
theorem orderedInitialLaw_log (u : JointPopulation.Shape)
(hu : 0 < orderedInitialLaw.mass u) :
Real.log (orderedInitialLaw.mass u) =
initialPotential (JointPopulation.shapeSide 0 u) +
initialPotential (JointPopulation.shapeSide 1 u) +
initialPotential (JointPopulation.shapeSide 2 u) - initialLogNormalizer := by
rw [orderedInitialLaw_mass u (orderedInitialLaw_support u hu),
Real.log_div (ne_of_gt (orderedWeight_pos u)) (ne_of_gt initialNormalizer_pos),
log_orderedWeight]
rfl
end MatrixMultiplication.AllFieldInitialGibbs
end
end OAI
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