Tensor G1 coord diff
ArkLib.Lattices.Hachi.tensorG1_coord_diff
Project documentation
Coordinate isolation at k = 1 (Hachi Lemma 8, case (C), the c4 subtract-and-divide crux): if c ≡ⱼ c', then tensorG1 (c − c') ŵ = (cⱼ − c'ⱼ) · wⱼ where w := G_blocks *ᵥ ŵ is the recomposed carrier.
Exact Lean statement
theorem tensorG1_coord_diff (base : R) (digits : Nat)
{c c' : PolyVec (Rq Φ) blocks} {j : Fin blocks} (h : CoordinateWise.CoordEq j c c')
(x : PolyVec (Rq Φ) (blocks * digits)) :
tensorG1 Φ base digits (c - c') x
= (c j - c' j) * (gadgetMatrix Φ base blocks digits *ᵥ x) jFormal artifact
Lean source
theorem tensorG1_coord_diff (base : R) (digits : Nat) {c c' : PolyVec (Rq Φ) blocks} {j : Fin blocks} (h : CoordinateWise.CoordEq j c c') (x : PolyVec (Rq Φ) (blocks * digits)) : tensorG1 Φ base digits (c - c') x = (c j - c' j) * (gadgetMatrix Φ base blocks digits *ᵥ x) j := by simp only [tensorG1, dot_eq_sum]; rw [Finset.sum_eq_single j] · simp only [Pi.sub_apply] · intro i _ hij have hzero : (c - c') i = 0 := by simp only [Pi.sub_apply]; rw [h.2 i hij, sub_self] rw [hzero, zero_mul] · intro hj; exact absurd (Finset.mem_univ j) hj- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/Hachi/QuadEval/Gadgets.lean:203-213
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