Take drop add Cases
Fin.take_drop_addCases'
Plain-language statement
The concatenation of the first m elements and the last n - m elements of a tuple is the same as the original tuple.
Exact Lean statement
theorem take_drop_addCases' (m : ℕ) (h : m ≤ n) (v : (i : Fin n) → α i) :
Fin.addCases' (take m h v) (drop m h v) =
fun i =>
cast (by
simp only [Fin.cast, append, addCases, castLE, val_castLT, subNat_mk, natAdd_mk,
addNat_mk, eq_rec_constant, left_eq_dite_iff, not_lt]
intro hi; rw! [Nat.sub_add_cancel hi]; rfl)
(v (i.cast (by omega)))Formal artifact
Lean source
theorem take_drop_addCases' (m : ℕ) (h : m ≤ n) (v : (i : Fin n) → α i) : Fin.addCases' (take m h v) (drop m h v) = fun i => cast (by simp only [Fin.cast, append, addCases, castLE, val_castLT, subNat_mk, natAdd_mk, addNat_mk, eq_rec_constant, left_eq_dite_iff, not_lt] intro hi; rw! [Nat.sub_add_cancel hi]; rfl) (v (i.cast (by omega))) := by ext i simp only [castLE, Fin.cast, val_addNat, addCases', addCases, castAdd_castLT, val_castLT, take_apply, eq_mpr_eq_cast, cast, subNat_mk, natAdd_mk, drop_apply, addNat_mk, cast_mk] split · simp · have : i.val - m + m = i.val := by omega rw! [this] simp only [eqRec_eq_cast] rw [_root_.cast_cast]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Fin/Tuple/TakeDrop.lean:346-362
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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.
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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.
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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.