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Project-declaredLean 4.33.0-rc1 · mathlib@169c26b52a38

Reconstruct eq constant Coeff of eval eq

Cslib.Crypto.Protocols.SecretSharing.Shamir.Polynomial.reconstruct_eq_constantCoeff_of_eval_eq

computer sciencecomputabilityprogram semanticspolynomial

Plain-language statement

Reconstruction recovers the constant coefficient of any low-degree polynomial from its values at distinct points.

Exact Lean statement

theorem reconstruct_eq_constantCoeff_of_eval_eq
    {x : ι → F} {p : _root_.Polynomial F}
    (hx : Function.Injective x)
    (hdeg : p.degree < Fintype.card ι) :
    reconstruct x (fun i => p.eval (x i)) = p.constantCoeff

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem reconstruct_eq_constantCoeff_of_eval_eq    {x : ι → F} {p : _root_.Polynomial F}    (hx : Function.Injective x)    (hdeg : p.degree < Fintype.card ι) :    reconstruct x (fun i => p.eval (x i)) = p.constantCoeff := by  classical  have hp :      p = _root_.Lagrange.interpolate Finset.univ x (fun i => p.eval (x i)) :=    _root_.Lagrange.eq_interpolate      (s := Finset.univ)      (v := x)      hx.injOn      (by simpa using hdeg)  simpa [reconstruct] using congrArg _root_.Polynomial.constantCoeff hp.symm
Project
Lean Computer Science Library
License
Apache-2.0
Commit
f36649cff2c9
Source
Cslib/Crypto/Protocols/SecretSharing/Shamir/Polynomial.lean:117-130

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Related declarations

Project-declaredLean 4.33.0-rc1

Unique minimal

Cslib.Automata.DA.FinAcc.unique_minimal

Plain-language statement

The minimal DFA M accepting the language l is unique up to unique isomorphism.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

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Project-declaredLean 4.33.0-rc1

Buchi Family cover

Cslib.Automata.NA.Buchi.buchiFamily_cover

Project documentation

na.buchiFamily is a cover if na has only finitely many states. This theorem uses the Ramsey theorem for infinite graphs and does not depend on any details of na.BuchiCongruence other than that it is of finite index.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

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