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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 | import Mathlib
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
section
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 (hstep t) (le_refl 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
end MatrixAllFields
end OAI
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