Dappend eq add Cases
Fin.dappend_eq_addCases
Plain-language statement
dappend is equal to addCases. Marked as csimp to allow for switching to the addCases implementation during execution.
Exact Lean statement
@[csimp] theorem dappend_eq_addCases : @dappend = @addCases
Formal artifact
Lean source
@[csimp]theorem dappend_eq_addCases : @dappend = @addCases := by ext m n motive u v i induction n with | zero => simp [dappend, addCases, castLT] | succ n ih => simp only [dappend, dconcat_eq_snoc] have ih' : ∀ (motive : Fin (m + n) → Sort _) (u : (i : Fin m) → motive (castAdd n i)) (v : (i : Fin n) → motive (natAdd m i)), dappend (motive := motive) u v = addCases (motive := motive) u v := by intro motive_1 u_1 v_1 ext x : 1 apply ih rw [ih' (fun i => motive i.castSucc) u (fun i => v (castSucc i))] simp [snoc, addCases, last, castLT, subNat] by_cases h : i.val < m · have : i.val < m + n := by omega simp [h, this] · by_cases h' : i.val < m + n · grind only · have : i.val = m + n := by omega grind only [cases Or]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Fin/Tuple/Lemmas.lean:340-362
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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.
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.
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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.