| import Mathlib |
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| namespace OAI |
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| namespace UniqueGames.Foundations.PCP.AmplificationIteration |
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| 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 |
|
|
| 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) |
|
|
| 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)) |
|
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| 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 UniqueGames.Foundations.PCP.AmplificationIteration |
| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end OAI |
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|
