lean/OAI/LinearAlgebra/MatrixFields/Parameters/ComplementSums.lean

import Mathlib
import OAI.LinearAlgebra.MatrixFields.Entropy.StageCLaw

namespace OAI

namespace MatrixAllFields

open scoped BigOperators Topology Polynomial

namespace MatrixMultiplication.AllFieldParameters

def placements (s : Shape) : List Shape :=
  [shape (s 0) (s 1) (s 2), shape (s 0) (s 2) (s 1),
   shape (s 1) (s 0) (s 2), shape (s 1) (s 2) (s 0),
   shape (s 2) (s 0) (s 1), shape (s 2) (s 1) (s 0)]

theorem initialMultiplicity_counts_placements : ∀ s ∈ sortedInitial,
    (placements s).eraseDups.length = initialMultiplicity s := by decide +kernel

theorem parents_nodup : sortedInitial.Nodup ∧ positiveInitial.Nodup ∧
    positiveSecond.Nodup ∧ zeroSecond.Nodup := by decide +kernel

theorem split_keys_nodup : (splitKeys positiveInitial).Nodup ∧
    (splitKeys positiveSecond).Nodup := by decide +kernel

theorem binary_keys_nodup : binaryKeys.Nodup := by decide +kernel

theorem pair_keys_nodup : pairKeys.Nodup := by decide +kernel

theorem stageA_children_nodup : ∀ s ∈ positiveInitial, (below s).Nodup := by decide +kernel

theorem stageB_children_nodup : ∀ s ∈ positiveSecond, (below s).Nodup := by decide +kernel

theorem ordered_pairs_nodup : ∀ t ∈ zeroSecond, (orderedPairs t).Nodup := by decide +kernel

theorem initial_placements_partition :
    (sortedInitial.flatMap (fun s => (placements s).eraseDups)).Perm (shapes 16) := by
  decide +kernel

end MatrixMultiplication.AllFieldParameters

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 MatrixAllFields

end OAI