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refactor(BitString,AffineDisperser): move bitEquiv_symm_foldl_xor to ZModTwo (#949) #973

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Merged
SamuelSchlesinger merged 1 commit into from
Oct 25, 2026
Merged

refactor(BitString,AffineDisperser): move bitEquiv_symm_foldl_xor to ZModTwo (#949) #973

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6 changes: 0 additions & 6 deletions Complexity/Computability/Complexity/AffineDisperser/Coordinates.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -77,12 +77,6 @@ theorem repr_fieldOfBits (B : Basis (Fin n) (ZMod 2) F) (a : List Bool) (u : Fin
theorem fieldOfBits_nil (B : Basis (Fin n) (ZMod 2) F) : fieldOfBits B [] = 0 := by
simp [fieldOfBits]

theorem bitEquiv_symm_foldl_xor (l : List Bool) :
bitEquiv.symm (l.foldl xor false) = (l.map bitEquiv.symm).sum := by
rw [← map_list_sum]
congr 1
simp [List.sum_eq_foldl, Bool.add_eq_xor, Bool.zero_eq_false]

/-- The coordinates of a sum: `listXor` adds coordinate words. -/
theorem fieldOfBits_listXor (B : Basis (Fin n) (ZMod 2) F) (words : List (List Bool)) :
fieldOfBits B (listXor n words) = (words.map (fieldOfBits B)).sum := by
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8 changes: 8 additions & 0 deletions Complexity/Foundations/Data/BitString/ZModTwo.lean
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Expand Up @@ -30,6 +30,7 @@ identifies `ZMod 2` with `Bool`; it equals Mathlib's `ZMod.ringEquivOfPrime`
* `Cslib.BitString.bitEquiv_eq_ringEquivOfPrime`: `bitEquiv` is `ZMod.ringEquivOfPrime Bool`.
* `Cslib.BitString.bitEquiv_symm_apply`, `Cslib.BitString.bitEquiv_symm_false`,
`Cslib.BitString.bitEquiv_symm_true`: `bitEquiv.symm` reads `false` as `0` and `true` as `1`.
* `Cslib.BitString.bitEquiv_symm_foldl_xor`: `bitEquiv.symm` turns a fold over `xor` into a sum.

## References

Expand Down Expand Up @@ -67,4 +68,11 @@ theorem bitEquiv_symm_false : bitEquiv.symm false = 0 := rfl
@[simp]
theorem bitEquiv_symm_true : bitEquiv.symm true = 1 := rfl

/-- The inverse ring isomorphism maps a fold over `xor` to the sum of the mapped bits. -/
theorem bitEquiv_symm_foldl_xor (l : List Bool) :
bitEquiv.symm (l.foldl xor false) = (l.map bitEquiv.symm).sum := by
rw [← map_list_sum]
congr 1
simp [List.sum_eq_foldl, Bool.add_eq_xor, Bool.zero_eq_false]

end Cslib.BitString
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