File size: 3,179 Bytes
e63eb7f | 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 75 76 77 78 79 80 81 | import OAI.Combinatorics.EditApproximation.Binary.BinaryAffineGuard
namespace OAI
/-! A structural polynomial bound for finite Boolean arithmetic loops.
The expression language includes only operations whose magnitude is bounded
by a fixed power of a common input bound. -/
namespace EditApproximation
inductive FiniteBitPolynomial (ι : Type*) where
| input (i : ι)
| constant (n : ℕ)
| add (a b : FiniteBitPolynomial ι)
| mul (a b : FiniteBitPolynomial ι)
| pow (a : FiniteBitPolynomial ι) (n : ℕ)
| maximum (a b : FiniteBitPolynomial ι)
| minimum (a b : FiniteBitPolynomial ι)
| bitLength (a : FiniteBitPolynomial ι)
namespace FiniteBitPolynomial
instance {ι : Type*} (n : ℕ) : OfNat (FiniteBitPolynomial ι) n := ⟨.constant n⟩
instance {ι : Type*} : Add (FiniteBitPolynomial ι) := ⟨.add⟩
instance {ι : Type*} : Mul (FiniteBitPolynomial ι) := ⟨.mul⟩
instance {ι : Type*} : Pow (FiniteBitPolynomial ι) ℕ := ⟨.pow⟩
def eval {ι : Type*} (env : ι → ℕ) : FiniteBitPolynomial ι → ℕ
| .input i => env i
| .constant n => n
| .add a b => a.eval env + b.eval env
| .mul a b => a.eval env * b.eval env
| .pow a n => a.eval env ^ n
| .maximum a b => max (a.eval env) (b.eval env)
| .minimum a b => min (a.eval env) (b.eval env)
| .bitLength a => Nat.size (a.eval env)
def degree {ι : Type*} : FiniteBitPolynomial ι → ℕ
| .input _ => 1
| .constant n => Nat.size n
| .add a b => max a.degree b.degree + 1
| .mul a b => a.degree + b.degree
| .pow a n => a.degree * n
| .maximum a b => max a.degree b.degree
| .minimum a b => max a.degree b.degree
| .bitLength a => a.degree
theorem eval_le_pow {ι : Type*} (p : FiniteBitPolynomial ι) (env : ι → ℕ)
(S : ℕ) (hS : 2 ≤ S) (henv : ∀ i, env i ≤ S) :
p.eval env ≤ S ^ p.degree := by
induction p with
| input i => simpa only [eval, degree, pow_one] using henv i
| constant n =>
exact (Nat.lt_size_self n).le.trans (Nat.pow_le_pow_left hS _)
| add a b ha hb =>
have ha' := ha.trans (Nat.pow_le_pow_right (by omega : 0 < S) (Nat.le_max_left a.degree b.degree))
have hb' := hb.trans (Nat.pow_le_pow_right (by omega : 0 < S) (Nat.le_max_right a.degree b.degree))
change a.eval env + b.eval env ≤ S ^ (max a.degree b.degree + 1)
calc
_ ≤ 2 * S ^ max a.degree b.degree := by omega
_ ≤ S * S ^ max a.degree b.degree := Nat.mul_le_mul_right _ hS
_ = _ := (pow_succ' S _).symm
| mul a b ha hb =>
exact (Nat.mul_le_mul ha hb).trans_eq (pow_add S a.degree b.degree).symm
| pow a n ha =>
exact (Nat.pow_le_pow_left ha n).trans_eq (pow_mul S a.degree n).symm
| maximum a b ha hb =>
exact max_le
(ha.trans (Nat.pow_le_pow_right (by omega) (Nat.le_max_left a.degree b.degree)))
(hb.trans (Nat.pow_le_pow_right (by omega) (Nat.le_max_right a.degree b.degree)))
| minimum a b ha _ =>
exact (Nat.min_le_left _ _).trans
(ha.trans (Nat.pow_le_pow_right (by omega) (Nat.le_max_left a.degree b.degree)))
| bitLength a ha =>
exact (Nat.size_le.mpr Nat.lt_two_pow_self).trans ha
end FiniteBitPolynomial
end EditApproximation
end OAI
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