openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /CoarseReorder.lean
| 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 | |