openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Entropy /ComplexAsymptoticRecurrence.lean
| import Mathlib.Basic.Real.Basic | |
| import Mathlib.Tactic.FieldSimp | |
| import Mathlib.Tactic.Linarith | |
| import Mathlib.Tactic.Ring | |
| namespace OAI | |
| /-! Finite entropy, rate estimates and ordered asymptotic limits. -/ | |
| namespace MatrixMultiplication.Foundation | |
| theorem affine_recurrence_shifted (x : ℕ → ℝ) (a c : ℝ) (ha : 0 ≤ a) | |
| (hstep : ∀ t, x (t + 1) ≤ a * x t + (a - 1) * c) : | |
| ∀ t, x t + c ≤ a ^ t * (x 0 + c) := by | |
| intro t | |
| induction t with | |
| | zero => simp | |
| | succ t ih => | |
| calc | |
| x (t + 1) + c ≤ a * x t + (a - 1) * c + c := add_le_add_left (hstep t) c | |
| _ = a * (x t + c) := by ring | |
| _ ≤ a * (a ^ t * (x 0 + c)) := mul_le_mul_of_nonneg_left ih ha | |
| _ = a ^ (t + 1) * (x 0 + c) := by rw [pow_succ]; ring | |
| theorem affine_recurrence_bound (x : ℕ → ℝ) (a b : ℝ) (ha : 1 < a) (hb : 0 ≤ b) | |
| (hstep : ∀ t, x (t + 1) ≤ a * x t + b) : | |
| ∀ t, x t ≤ a ^ t * (x 0 + b / (a - 1)) := by | |
| have hpos : 0 < a - 1 := sub_pos.mpr ha | |
| have hcancel : (a - 1) * (b / (a - 1)) = b := by | |
| field_simp [ne_of_gt hpos] | |
| have hshift := affine_recurrence_shifted x a (b / (a - 1)) (by linarith) | |
| (fun t => by rw [hcancel]; exact hstep t) | |
| intro t | |
| have hc : 0 ≤ b / (a - 1) := div_nonneg hb hpos.le | |
| exact (le_add_of_nonneg_right hc).trans (hshift t) | |
| end MatrixMultiplication.Foundation | |
| end OAI | |