File size: 2,790 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldParameters.Traversal
import Mathlib.Data.List.Nodup
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldParameters
theorem list_map_involution_perm {α : Type*} (xs : List α) (f : α → α)
(hn : xs.Nodup) (hm : ∀ x ∈ xs, f x ∈ xs)
(hi : ∀ x ∈ xs, f (f x) = x) : (xs.map f).Perm xs := by
have hinj : ∀ x ∈ xs, ∀ y ∈ xs, f x = f y → x = y := by
intro x hx y hy hxy
simpa only [hi x hx, hi y hy] using congrArg f hxy
apply (List.perm_ext_iff_of_nodup (List.Nodup.map_on hinj hn) hn).2
intro y
constructor
· intro hy
obtain ⟨x, hx, rfl⟩ := List.mem_map.mp hy
exact hm x hx
· intro hy
exact List.mem_map.mpr ⟨f y, hm y hy, hi y hy⟩
theorem list_weighted_involution_sum {α : Type*} (xs : List α) (f : α → α)
(hn : xs.Nodup) (hm : ∀ x ∈ xs, f x ∈ xs)
(hi : ∀ x ∈ xs, f (f x) = x) (w h : α → ℝ)
(hw : ∀ x ∈ xs, w (f x) = w x) :
(xs.map (fun x => w x * h (f x))).sum =
(xs.map (fun x => w x * h x)).sum := by
calc
_ = ((xs.map f).map (fun x => w x * h x)).sum := by
congr 1
rw [List.map_map]
apply List.map_congr_left
intro x hx
simp only [Function.comp_apply, hw x hx]
_ = _ := ((list_map_involution_perm xs f hn hm hi).map _).sum_eq
theorem complement_involution_of_le (s u : Shape) (hu : ∀ i, u i ≤ s i) :
complement s (complement s u) = u := by
funext i
exact Nat.sub_sub_self (hu i)
theorem stageA_complement_sum (s : Shape) (hs : s ∈ positiveInitial)
(h : Shape → ℝ) :
((below s).map (fun u => (stageALaw s u : ℝ) * h (complement s u))).sum =
((below s).map (fun u => (stageALaw s u : ℝ) * h u)).sum := by
apply list_weighted_involution_sum (below s) (complement s)
(stageA_children_nodup s hs)
· intro u hu
exact (stageA_support_complement s hs u hu).1
· intro u hu
exact complement_involution_of_le s u (stageA_support_complement s hs u hu).2
· intro u hu
rw [stageALaw_complement s u hs hu]
theorem stageB_complement_sum (s : Shape) (hs : s ∈ positiveSecond)
(h : Shape → ℝ) :
((below s).map (fun u => (stageBLaw s u : ℝ) * h (complement s u))).sum =
((below s).map (fun u => (stageBLaw s u : ℝ) * h u)).sum := by
apply list_weighted_involution_sum (below s) (complement s)
(stageB_children_nodup s hs)
· intro u hu
exact (stageB_support_complement s hs u hu).1
· intro u hu
exact complement_involution_of_le s u (stageB_support_complement s hs u hu).2
· intro u hu
rw [stageBLaw_complement s u hs hu]
end MatrixMultiplication.AllFieldParameters
end
end OAI
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