DeterminantalConjecture.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Determinantal conjecture

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

theorem DeterminantalConjecture.determinantal_conjecture Does the determinant of the sum sorry line 46 source
Source documentation
Does the determinant of the sum $A + B$ of two $n \times n$ normal complex matrices $A$ and $B$ always lie in the convex hull of the $n!$ points $\prod\_i (\lambda(A)\_i + \lambda(B)\_{\sigma(i)})$? Here the numbers $\lambda(A)\_i$ and $\lambda(B)\_i$ are the eigenvalues of $A$ and $B$, and $\sigma$ is an element of the symmetric group $S\_n$.

depends on sorry what this means
Statement uses: Fintype, DecidableEq, Complex, Subtype, Matrix, Membership.mem, Submonoid, unitary, Set, ClosureOperator, convexHull, Real, Set.ofPred, Exists, Equiv.Perm, Eq, Finset.prod, Finset.univ, HAdd.hAdd, Matrix.det and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
103 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem determinantal_conjecture
    (n : Type) [Fintype n] [DecidableEq n]
    (d1 d2 : n → ℂ) (U1 U2 : unitary (Matrix n n ℂ)) :
    (U1 * Matrix.diagonal d1 * star U1 + U2 * Matrix.diagonal d2 * star U2).det
      ∈ convexHull ℝ { ∏ i, (d1 i + d2 (σ i)) | σ : Equiv.Perm n } := by
…
```