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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 | import OAI.LinearAlgebra.MatrixMultiplication.Duality.FiniteRealization
import OAI.LinearAlgebra.MatrixMultiplication.Duality.FiniteRates
import OAI.LinearAlgebra.MatrixMultiplication.Duality.WitnessProgram
import OAI.LinearAlgebra.MatrixMultiplication.Completion.HierarchyPrograms
import OAI.LinearAlgebra.MatrixMultiplication.Completion.Termination
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualWitnessRealization
open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels
open CompletionLabels.TopologicalFlatten CompletionHierarchyWords
open scoped BigOperators Classical
theorem program_context (a : DualWitness.Support) : DualWitness.program.context a = PUnit.unit :=
rfl
theorem program_joint :
Function.Injective (fun a : DualWitness.Support =>
(DualWitness.program.view .B a, DualWitness.program.view .C a,
DualWitness.program.view .A a)) := by
intro a b h
apply Subtype.ext
apply Subtype.ext
exact h
def hierarchy :
ReadableHierarchy DualWitness.Support DualWitness.Word DualWitness.Word DualWitness.Word :=
ReadableHierarchy.ofProgram DualWitness.program PUnit.unit program_context program_joint
theorem hierarchy_complete :
Function.Injective (labelRecordOf hierarchy.labels hierarchy.depth) :=
ReadableHierarchy.ofProgram_complete DualWitness.program PUnit.unit program_context
program_joint DualWitness.program_complete
@[simp] theorem hierarchy_x (a : DualWitness.Support) : hierarchy.x a = a.val.val.1 := rfl
@[simp] theorem hierarchy_y (a : DualWitness.Support) : hierarchy.y a = a.val.val.2.1 := rfl
@[simp] theorem hierarchy_z (a : DualWitness.Support) : hierarchy.z a = a.val.val.2.2 := rfl
def tensor : Tensor ℂ DualWitness.Word DualWitness.Word DualWitness.Word :=
CompletionTermination.tensor DualWitness.source .A
def retainedLeafEquiv : CompletionTermination.Leaf DualWitness.source .A ≃ DualWitness.Support :=
CompletionTermination.retainedLeafEquiv DualWitness.source .A .B .C
(by intro c; cases c <;> simp)
def approximation :
Tensor.PolynomialApproximation tensor (2 ^ 288)
(Script.dual.order 1) (Script.dual.degree 3) :=
CompletionTermination.approximation DualWitness.source .A DualFiniteRealization.dualSource
theorem base_unit (x y z : Fin 2) (h : BaseTwo.tensor x y z ≠ 0) :
BaseTwo.tensor x y z = 1 := by
unfold BaseTwo.tensor at h ⊢
split_ifs with hsum
· rfl
· simp [hsum] at h
theorem tensor_unit_coefficients (x y z : DualWitness.Word) (h : tensor x y z ≠ 0) :
tensor x y z = 1 := by
obtain ⟨hs, ha⟩ := (CompletionTermination.support_iff DualWitness.source .A x y z).mp h
change (if ownerFlag DualWitness.source .A (coordinate (x, y, z) .A) then 0
else DualWitness.source.tensor x y z) = 1
rw [ite_eq_right ha]
exact CompletionFiniteRealization.script_coefficient_one BaseTwo.flagged base_unit
Script.dual x y z hs
theorem source_supported (x y z : DualWitness.Word) (h : tensor x y z ≠ 0) :
∃ a : DualWitness.Support, hierarchy.x a = x ∧ hierarchy.y a = y ∧ hierarchy.z a = z :=
⟨retainedLeafEquiv ⟨(x, y, z), h⟩, rfl, rfl, rfl⟩
theorem source_unit (a : DualWitness.Support) :
tensor (hierarchy.x a) (hierarchy.y a) (hierarchy.z a) = 1 :=
tensor_unit_coefficients _ _ _ (retainedLeafEquiv.symm a).property
def witness (counts : DualWitness.Support → ℕ) (t : ℕ) :
ComplexWitness.FiniteRectangularWitness :=
DualFiniteRealization.witness hierarchy counts t hierarchy_complete approximation
(pow_pos (by decide : 0 < (2 : ℕ)) 288) source_supported source_unit
theorem witness_rank (counts : DualWitness.Support → ℕ) (t : ℕ) :
(witness counts t).rank =
(2 ^ (288 * CompletionFiniteRealization.blocks counts t) *
StageHierarchyResources.stageAuxiliaryBudget counts hierarchy.labels hierarchy.depth t) ^ 2 := by
rw [witness, DualFiniteRealization.witness_rank_stageBudget, ← pow_mul]
theorem source_log : Real.log ((2 ^ 288 : ℕ) : ℝ) = 288 * Real.log 2 := by
rw [Nat.cast_pow, Real.log_pow]
norm_num
theorem source_log_positive : 0 < Real.log ((2 ^ 288 : ℕ) : ℝ) := by
rw [source_log]
exact mul_pos (by norm_num) (Real.log_pos (by norm_num))
end MatrixMultiplication.DualWitnessRealization
end
end OAI
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