File size: 2,679 Bytes
e5a97c6 | 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 66 67 68 69 70 71 72 73 74 | import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedRangeInitialization
namespace OAI
/-! # Measured permutation seed generation
Each range call computes its modulus, draws and retains the actual candidate
tuple, and caches its accepted digit. Permutation values are composition
closures; their application or array materialization remains a separate
measured operation, as in `PermutationArray`.
-/
noncomputable section
namespace LowSpaceSubsetSum.WordSeedMeasured
def permutation (w n : ℕ) : (k : ℕ) → (hw : Nat.clog 2 k ≤ w) →
Execution (WordSeedRealization.permutation w n k hw)
| 0, _ => by
exact reprogram (pure (w := w) (Equiv.refl (Fin 0))) (by
unfold WordSeedRealization.permutation
exact mpr_program (Nat.zero_mul (n^3)) _)
| k+1, hw => by
have hk : Nat.clog 2 k ≤ w :=
(Nat.clog_mono_right 2 (by omega : k ≤ k+1)).trans hw
exact reprogram
(map (prod (fin w n (k+1) hw) (permutation w n k hk))
Equiv.Perm.decomposeFin.symm
(fun p => ((Equiv.Perm.decomposeFin.symm p,6),8)) (fun _ => rfl)) (by
conv_lhs => rw [WordSeedRealization.permutation]
exact mpr_program (w := w) (α := Equiv.Perm (Fin (k+1)))
(by simp only [Nat.add_mul,Nat.one_mul,Nat.add_comm] :
(k+1)*n^3 = n^3+k*n^3) _)
theorem permutation_bounded (w n : ℕ) : ∀ (k : ℕ) (hw : Nat.clog 2 k ≤ w),
(permutation w n k hw).Bounded
(k*(13*n^3+5*k+64)+1) ((k+1)*(2*n^3+256)) ((k+1)*(n^3+64)) := by
intro k
induction k with
| zero =>
intro hw stream cursor
change 1 ≤ _ ∧ 4 ≤ _ ∧ 4 ≤ _
constructor
· omega
· constructor <;> omega
| succ k ih =>
intro hw
have hk : Nat.clog 2 k ≤ w :=
(Nat.clog_mono_right 2 (by omega : k ≤ k+1)).trans hw
have ht := ih hk
have hf := fin_bounded w n (k+1) hw
have hp := prod_bounded (fin w n (k+1) hw) (permutation w n k hk)
_ _ _ _ _ _ hf ht
have hm := map_bounded _ Equiv.Perm.decomposeFin.symm
(fun p => ((Equiv.Perm.decomposeFin.symm p,6),8)) (fun _ => rfl)
_ _ _ 6 8 hp (fun _ => le_rfl) (fun _ => le_rfl)
have hclog : Nat.clog 2 (k+1) ≤ k+1 :=
Nat.clog_le_of_le_pow Nat.lt_two_pow_self.le
intro stream cursor
have h := hm stream cursor
change _ ≤ (k+1)*(13*n^3+5*(k+1)+64)+1 ∧
_ ≤ (k+1+1)*(2*n^3+256) ∧ _ ≤ (k+1+1)*(n^3+64)
refine ⟨h.1.trans (by nlinarith),h.2.1.trans ?_,h.2.2.trans (by nlinarith)⟩
apply max_le
· apply max_le
· apply max_le <;> nlinarith
· nlinarith
· nlinarith
end LowSpaceSubsetSum.WordSeedMeasured
end
end OAI
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