File size: 2,287 Bytes
ea76aac | 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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Rat.BigOperators
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Positivity
namespace OAI
/-! Polynomial-time approximation of weighted matchings with singleton loops. -/
namespace LoopMatching.Accumulator
open scoped BigOperators
def Q (a : ℕ → ℕ) : ℕ → ℕ
| 0 => a 0
| n + 1 => (n + 1) * Q a n + a (n + 1)
@[simp] theorem Q_zero (a : ℕ → ℕ) : Q a 0 = a 0 := rfl
@[simp] theorem Q_succ (a : ℕ → ℕ) (n : ℕ) :
Q a (n + 1) = (n + 1) * Q a n + a (n + 1) := rfl
theorem denominator_pos (n : ℕ) : 0 < n.factorial := Nat.factorial_pos n
theorem quotient_eq_sum (a : ℕ → ℕ) (n : ℕ) :
(Q a n : ℚ) / (n.factorial : ℚ) =
∑ j ∈ Finset.range (n + 1), (a j : ℚ) / (j.factorial : ℚ) := by
induction n with
| zero => simp [Q]
| succ n ih =>
rw [Finset.sum_range_succ, ← ih, Q_succ, Nat.factorial_succ]
push_cast
have hn : (n : ℚ) + 1 ≠ 0 := by positivity
have hf : (n.factorial : ℚ) ≠ 0 := by
exact_mod_cast Nat.factorial_ne_zero n
field_simp [hn, hf]
theorem Q_le_factorial_mul (a : ℕ → ℕ) (n B : ℕ)
(ha : ∀ j ≤ n, a j ≤ B) :
Q a n ≤ (n + 1).factorial * B := by
induction n generalizing B with
| zero => simpa using ha 0 (Nat.le_refl 0)
| succ n ih =>
have hprev : Q a n ≤ (n + 1).factorial * B :=
ih B (fun j hj => ha j (Nat.le_trans hj (Nat.le_succ n)))
have hlast : a (n + 1) ≤ B := ha (n + 1) (Nat.le_refl _)
have hfac : 1 ≤ (n + 1).factorial := Nat.factorial_pos (n + 1)
have hB : B ≤ (n + 1).factorial * B := by
simpa only [one_mul] using Nat.mul_le_mul_right B hfac
calc
Q a (n + 1) = (n + 1) * Q a n + a (n + 1) := rfl
_ ≤ (n + 1) * ((n + 1).factorial * B) + B :=
Nat.add_le_add (Nat.mul_le_mul_left (n + 1) hprev) hlast
_ ≤ (n + 1) * ((n + 1).factorial * B) +
(n + 1).factorial * B := Nat.add_le_add_left hB _
_ = (n + 1 + 1).factorial * B := by
rw [Nat.factorial_succ (n + 1)]
ring
end LoopMatching.Accumulator
end OAI
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