File size: 4,964 Bytes
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 | import OAI.LinearAlgebra.MatrixMultiplication.Entropy.BaseConditionalReaders
import OAI.LinearAlgebra.MatrixMultiplication.Arithmetic.Compatibility
import OAI.LinearAlgebra.MatrixMultiplication.Completion.HierarchyPrograms
import OAI.LinearAlgebra.MatrixMultiplication.Completion.SquareRates
import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.SquareCounts
namespace OAI
/-! Explicit complex square and rectangular matrix multiplication bounds. -/
noncomputable section
namespace MatrixMultiplication.SquareBound
open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels
open CompletionLabels.TopologicalFlatten CompletionHierarchyWords Filter
open scoped BigOperators Topology Classical
abbrev Coordinate := Script.Coord (Fin 3) Script.square
abbrev Carrier := RetainedLeaf SquareCounts.completed .B .C
def conditional : ReadableTensor SquareCounts.completed :=
ReadableTensor.script BaseThree.flagged BaseConditionalReaders.baseThreeReadable Script.square
def program :
Program Carrier (CompletionLabels.Coordinate Coordinate Coordinate Coordinate)
PUnit (Color ⊕ conditional.Code) (1 + conditional.depth) :=
terminationProgram SquareCounts.completed .B .C (by decide)
conditional.program conditional.context_eq conditional.view_eq
theorem program_context (a : Carrier) : program.context a = PUnit.unit := rfl
theorem program_view (side : Color) (a : Carrier) :
program.view side a = coordinate a.val.val side := rfl
theorem program_joint :
Function.Injective (fun a : Carrier => (program.view .B a, program.view .C a,
program.view .A a)) := by
intro a b h
apply Subtype.ext
apply Subtype.ext
exact h
def hierarchy : ReadableHierarchy Carrier Coordinate Coordinate Coordinate :=
ReadableHierarchy.ofProgram program PUnit.unit program_context program_joint
theorem hierarchy_complete :
Function.Injective (labelRecordOf hierarchy.labels hierarchy.depth) := by
apply ReadableHierarchy.ofProgram_complete program PUnit.unit program_context program_joint
exact terminationProgram_complete SquareCounts.completed .B .C (by decide)
conditional.program conditional.context_eq conditional.view_eq conditional.complete
@[simp] theorem hierarchy_x (a : Carrier) : hierarchy.x a = a.val.val.1 := rfl
@[simp] theorem hierarchy_y (a : Carrier) : hierarchy.y a = a.val.val.2.1 := rfl
@[simp] theorem hierarchy_z (a : Carrier) : hierarchy.z a = a.val.val.2.2 := rfl
theorem source_supported (x y z : Coordinate) (h : SquareCounts.tensor x y z ≠ 0) :
∃ a : Carrier, hierarchy.x a = x ∧ hierarchy.y a = y ∧ hierarchy.z a = z := by
exact ⟨SquareCounts.retainedLeafEquiv ⟨(x, y, z), h⟩, rfl, rfl, rfl⟩
theorem source_unit (a : Carrier) :
SquareCounts.tensor (hierarchy.x a) (hierarchy.y a) (hierarchy.z a) = 1 := by
exact SquareCounts.tensor_unit_coefficients _ _ _
(SquareCounts.retainedLeafEquiv.symm a).property
def counts (_ : Carrier) : ℕ := 1
theorem card_carrier : Fintype.card Carrier = SquareCounts.leafCount :=
(Fintype.card_congr SquareCounts.retainedLeafEquiv.symm).trans SquareCounts.card_leaf
theorem total_counts : (∑ a : Carrier, counts a) = SquareCounts.leafCount := by
exact (Finset.card_eq_sum_ones Finset.univ).symm.trans card_carrier
theorem total_counts_pos : 0 < ∑ a : Carrier, counts a := by
rw [total_counts]
exact SquareCounts.leafCount_pos
theorem count_entropy :
finiteEntropy (CompletionSquareRates.countMass counts) = Real.log SquareCounts.leafCount := by
have hm : CompletionSquareRates.countMass counts = SquareCounts.retainedUniformLaw.mass := by
funext a
rw [SquareCounts.retainedUniformLaw_mass]
rw [CompletionSquareRates.countMass, total_counts]
simp only [counts, Nat.cast_one, one_div]
rw [hm, SquareCounts.retainedUniformLaw_entropy]
def family (n : ℕ) : FiniteMMRealization :=
CompletionSquareRates.positiveRealization hierarchy counts total_counts_pos
hierarchy_complete SquareCounts.approximation source_supported source_unit (by decide) n
theorem score_limit (β : ℝ) :
Tendsto (fun n => (family n).rates.score β) atTop
(𝓝 (β / 216 * Real.log SquareWitness.leafCount)) := by
have h := CompletionSquareRates.score_limit hierarchy counts total_counts_pos
hierarchy_complete SquareCounts.approximation source_supported source_unit (by decide) β
rw [count_entropy] at h
convert h using 1 <;> norm_num [family, SquareCounts.leafCount, SquareWitness.leafCount]
theorem omega_lt_2267_div_1000 :
MatrixMultiplication.Foundation.Arithmetic.omega < (2267 : ℝ) / 1000 :=
SquareWitness.omega_lt_of_certified_uniform_square_family family
(score_limit ((2267 : ℝ) / 1000))
theorem complex_omega_lt_2267_div_1000 :
MatrixMultiplication.Arithmetic.omega ℂ < (2267 : ℝ) / 1000 := by
rw [MatrixMultiplication.Arithmetic.complex_omega_eq]
exact omega_lt_2267_div_1000
end MatrixMultiplication.SquareBound
end
end OAI
|