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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 | import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CoarseHashing
import Mathlib.Tactic.FinCases
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointCoarseReorder
open JointCoarseHashing
open scoped BigOperators
attribute [local instance] Classical.propDecidable
def tripleSide {P : Type*} (s : Fin 3) (t : Triple P) : Word P :=
![t.1, t.2.1, t.2.2] s
def reorder {P : Type*} (sigma : Equiv.Perm (Fin 3)) (t : Triple P) : Triple P :=
(tripleSide (sigma 0) t, tripleSide (sigma 1) t, tripleSide (sigma 2) t)
@[simp] theorem tripleSide_reorder {P : Type*} (sigma : Equiv.Perm (Fin 3))
(t : Triple P) (s : Fin 3) :
tripleSide s (reorder sigma t) = tripleSide (sigma s) t := by
fin_cases s <;> rfl
theorem reorder_injective {P : Type*} (sigma : Equiv.Perm (Fin 3)) :
Function.Injective (reorder (P := P) sigma) := by
intro t u h
have hs (s : Fin 3) : tripleSide s t = tripleSide s u := by
have hh := congrArg (tripleSide (sigma.symm s)) h
simpa only [tripleSide_reorder, sigma.apply_symm_apply] using hh
exact Prod.ext (hs 0) (Prod.ext (hs 1) (hs 2))
theorem reorder_support {P : Type*} (sigma : Equiv.Perm (Fin 3)) (S : P → ℕ)
(t : Triple P) (ht : HasSupportSum S t) : HasSupportSum S (reorder sigma t) := by
intro p
have hsum : (∑ i : Fin 3, (tripleSide (sigma i) t p).val) =
∑ i : Fin 3, (tripleSide i t p).val :=
Equiv.sum_comp sigma (fun i : Fin 3 => (tripleSide i t p).val)
simpa [reorder, tripleSide, Fin.sum_univ_succ, Nat.add_assoc] using hsum.trans (by
simpa [tripleSide, Fin.sum_univ_succ, Nat.add_assoc] using ht p)
end MatrixMultiplication.JointCoarseReorder
end
end OAI
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