lean/OAI/Computability/UniqueGames/PCP/AmplificationIteration.lean

import Mathlib.Data.Nat.Log
import Mathlib.Basic.Real.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring

namespace OAI

section

/-!
The finite iteration and size arithmetic for an amplification transformation.
The transformation itself must be supplied by the actual graph construction;
these theorems do not assert that such a transformation exists.  In particular,
the polynomial size estimate below is not a machine runtime certificate.
-/

namespace UniqueGamesTheorem.Foundations.PCP.AmplificationIteration

def run {X : Type*} (step : X → X) : Nat → X → X
  | 0, x => x
  | n + 1, x => step (run step n x)

private theorem min_double_min (u cap : ℝ) (hc : 0 ≤ cap) :
    min (2 * min u cap) cap = min (2 * u) cap := by
  by_cases hu : u ≤ cap
  · rw [min_eq_left hu]
  · have hcu : cap ≤ u := le_of_not_ge hu
    rw [min_eq_right hcu, min_eq_right (by linarith : cap ≤ 2 * cap),
      min_eq_right (by linarith : cap ≤ 2 * u)]

theorem run_gap {X : Type*} (step : X → X) (gap : X → ℝ)
    (cap : ℝ) (hc : 0 ≤ cap)
    (amplifies : ∀ x, min (2 * gap x) cap ≤ gap (step x))
    (n : Nat) (x : X) :
    min (2 ^ n * gap x) cap ≤ gap (run step n x) := by
  induction n with
  | zero => simp [run]
  | succ n ih =>
    have hm : min (2 * min (2 ^ n * gap x) cap) cap ≤
        min (2 * gap (run step n x)) cap :=
      min_le_min (mul_le_mul_of_nonneg_left ih (by norm_num)) le_rfl
    rw [min_double_min _ _ hc] at hm
    have h := hm.trans (amplifies (run step n x))
    simpa [run, pow_succ, mul_assoc, mul_left_comm, mul_comm] using h

theorem run_preserves {X : Type*} (step : X → X) (P : X → Prop)
    (preserves : ∀ x, P x → P (step x)) (n : Nat) (x : X) (hx : P x) :
    P (run step n x) := by
  induction n with
  | zero => exact hx
  | succ n ih => exact preserves _ ih

theorem run_size {X : Type*} (step : X → X) (size : X → Nat) (C : Nat)
    (blowup : ∀ x, size (step x) ≤ C * size x) (n : Nat) (x : X) :
    size (run step n x) ≤ C ^ n * size x := by
  induction n with
  | zero => simp [run]
  | succ n ih =>
    have h := (blowup (run step n x)).trans (Nat.mul_le_mul_left C ih)
    simpa [run, pow_succ, Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm] using h

/-- A concrete logarithmic iteration count, including the zero input. -/
def rounds (n : Nat) : Nat := Nat.log2 n + 1

theorem rounds_large (n : Nat) : n < 2 ^ rounds n := by
  simpa [rounds, Nat.log2_eq_log_two] using
    (Nat.lt_pow_succ_log_self (by decide : 1 < 2) n)

theorem rounds_power_le {n : Nat} (hn : 0 < n) : 2 ^ rounds n ≤ 2 * n := by
  have h := Nat.pow_log_le_self 2 (Nat.ne_of_gt hn)
  simpa [rounds, Nat.log2_eq_log_two, pow_succ, Nat.mul_comm] using
    (Nat.mul_le_mul_right 2 h)

/-- The constant blowup raised to the actual iteration count is polynomial in `n`. -/
theorem blowup_rounds_le {C degree n : Nat} (hn : 0 < n) (hC : C ≤ 2 ^ degree) :
    C ^ rounds n ≤ (2 * n) ^ degree := by
  calc
    C ^ rounds n ≤ (2 ^ degree) ^ rounds n := Nat.pow_le_pow_left hC _
    _ = (2 ^ rounds n) ^ degree := by
      rw [← pow_mul, ← pow_mul, Nat.mul_comm degree (rounds n)]
    _ ≤ (2 * n) ^ degree := Nat.pow_le_pow_left (rounds_power_le hn) _

theorem run_size_polynomial {X : Type*} (step : X → X) (size : X → Nat)
    {C degree n : Nat} (hn : 0 < n) (hC : C ≤ 2 ^ degree)
    (blowup : ∀ x, size (step x) ≤ C * size x) (x : X) :
    size (run step (rounds n) x) ≤ (2 * n) ^ degree * size x :=
  (run_size step size C blowup (rounds n) x).trans
    (Nat.mul_le_mul_right _ (blowup_rounds_le hn hC))

/-- An inverse-size initial gap reaches the fixed threshold after the concrete loop. -/
theorem run_reaches_cap {X : Type*} (step : X → X) (gap : X → ℝ)
    (cap : ℝ) (hc : 0 ≤ cap) (hc1 : cap ≤ 1)
    (amplifies : ∀ x, min (2 * gap x) cap ≤ gap (step x))
    (n : Nat) (x : X) (hg : 0 ≤ gap x) (hstart : 1 ≤ (n : ℝ) * gap x) :
    cap ≤ gap (run step (rounds n) x) := by
  have hp : (n : ℝ) ≤ (2 : ℝ) ^ rounds n := by
    exact_mod_cast (Nat.le_of_lt (rounds_large n))
  have hcap : cap ≤ (2 : ℝ) ^ rounds n * gap x :=
    hc1.trans (hstart.trans (mul_le_mul_of_nonneg_right hp hg))
  have h := run_gap step gap cap hc amplifies (rounds n) x
  rwa [min_eq_right hcap] at h

end UniqueGamesTheorem.Foundations.PCP.AmplificationIteration

end

end OAI