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2f887db | 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 | import Mathlib
import OAI.Computability.QuantumFactoring.BitEncoding
namespace OAI
section
open scoped BigOperators
namespace ExactQuantumFactoring
open scoped BigOperators
lemma bitsValue_toNat {b : ℕ} (x : Fin b → Bool) :
(bitsValue x).toNat = ∑ i : Fin b, 2^i.val*(x i).toNat := by
induction b with
| zero => simp [bitsValue]
| succ b ih =>
rw [bitsValue, BitVec.toNat_cons', Nat.shiftLeft_eq, ih, Fin.sum_univ_castSucc]
simp only [Fin.val_last, Fin.val_castSucc]
ring
lemma bitsValue_ofFn {b : ℕ} (v : Fin (2^b)) :
(bitsValue (fun i : Fin b => v.val.testBit i.val)).toNat = v.val := by
have he : (fun i : Fin b => v.val.testBit i.val) =
(fun i : Fin b => (BitVec.ofNat b v.val).getLsbD i.val) := by simp
rw [he, bitsValue_bits, BitVec.toNat_ofNat, Nat.mod_eq_of_lt v.isLt]
end ExactQuantumFactoring
end
end OAI
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