File size: 2,215 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialEntropy
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
namespace MatrixMultiplication.AllFieldInitialOrbit
open AllFieldParameters AllFieldHistory AllFieldInitialEntropy
open scoped BigOperators
theorem source_total (p : PlacedInitial) :
(initialShapeCode p).1.val + (initialShapeCode p).2.1.val +
(initialShapeCode p).2.2.val = 16 := by
unfold initialShapeCode rootShape
rw [encodePhysicalShape_total]
exact (root_shape_spec _ (initialShape_mem _)).2
theorem decodeShape_injective : Function.Injective decodeShape := by
intro u v h
apply Prod.ext
· exact Fin.ext (congrFun h 0)
· apply Prod.ext
· exact Fin.ext (congrFun h 1)
· exact Fin.ext (congrFun h 2)
theorem decodeShape_mem_shapes (u : JointPopulation.Shape)
(hu : u.1.val + u.2.1.val + u.2.2.val = 16) :
decodeShape u ∈ shapes 16 := by
simp only [shapes, List.mem_flatMap, List.mem_map, List.mem_range]
refine ⟨u.1.val, by omega, u.2.1.val, by omega, ?_⟩
funext i
fin_cases i
· rfl
· rfl
· change 16 - u.1.val - u.2.1.val = u.2.2.val
omega
theorem initialShapeCode_eq_iff (p : PlacedInitial) (u : JointPopulation.Shape) :
initialShapeCode p = u ↔
physicalShape p.2 (sortedInitial.get p.1) = decodeShape u := by
have hdecode : decodeShape (initialShapeCode p) =
physicalShape p.2 (sortedInitial.get p.1) := by
funext i
exact initialShapeCode_side p i
rw [← hdecode]
exact decodeShape_injective.eq_iff.symm
theorem weighted_shape_fiber_count : ∀ u ∈ shapes 16,
(∑ p : PlacedInitial,
if physicalShape p.2 (sortedInitial.get p.1) = u
then initialMultiplicity (sortedInitial.get p.1) else 0) = 6 := by
decide +kernel
theorem weighted_fiber_count (u : JointPopulation.Shape)
(hu : u.1.val + u.2.1.val + u.2.2.val = 16) :
(∑ p : PlacedInitial, if initialShapeCode p = u
then initialMultiplicity (sortedInitial.get p.1) else 0) = 6 := by
simp only [initialShapeCode_eq_iff]
exact weighted_shape_fiber_count _ (decodeShape_mem_shapes u hu)
end MatrixMultiplication.AllFieldInitialOrbit
end OAI
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