Plain-language statement
drop commutes with update for indices at or after the drop point.
Exact Lean statement
@[simp]
theorem drop_update_of_ge (m : ℕ) (h : m ≤ n) (v : (i : Fin n) → α i) (i : Fin n) (hi : i ≥ m)
(x : α i) : drop m h (update v i x) =
update (drop m h v) ⟨i - m, by omega⟩
(dcast (by
simp only [addNat_mk, cast_mk]
ext
simp only
rw [Nat.sub_add_cancel hi]) x)Formal artifact
Lean source
@[simp]theorem drop_update_of_ge (m : ℕ) (h : m ≤ n) (v : (i : Fin n) → α i) (i : Fin n) (hi : i ≥ m) (x : α i) : drop m h (update v i x) = update (drop m h v) ⟨i - m, by omega⟩ (dcast (by simp only [addNat_mk, cast_mk] ext simp only rw [Nat.sub_add_cancel hi]) x) := by ext j simp only [Fin.cast, val_addNat, drop_apply, update, addNat_mk, cast_mk] split next h_1 => subst h_1 simp_all only [add_tsub_cancel_right, Fin.eta, ↓reduceDIte] simp only [dcast, eqRec_eq_cast] rw [_root_.cast_cast] exact (cast_eq _ x).symm next h_1 => simp_all only [right_eq_dite_iff] intro h_2 subst h_2 simp_all only [Nat.sub_add_cancel, Fin.eta, not_true_eq_false]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Fin/Tuple/TakeDrop.lean:220-242
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Project documentation
This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose coeff
ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
Plain-language statement
The k-th coefficient (k < deg φ) of a gadget-decomposition block is exactly the corresponding digit of the corresponding input coefficient.
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose lawful
ArkLib.Lattices.Ajtai.gadgetDecompose_lawful
Plain-language statement
The base-b gadget decomposition is a lawful gadget decomposition.
Source project: ArkLib
Person-level attribution pending.