Research proof index

Computability of a predicate on a Primcodable type is equivalent to the computability of the corresponding predicate on ℕ obtained by decoding.

A chain rule for conditional multidistance. Let $\pi:G\to H$ be a homomorphism, and suppose the pairs $(X_i,Y_i)$ are independent across the finite index set. Then $D[X\mid Y]=D[X\mid(\pi X,Y)]+D[\pi X\mid Y]+I\!\left[\sum_iX_i:(\pi X_i)_i\,\middle|\,\left(\pi\!\left(\sum_iX_i\right),(Y_i)_i\right)\right].$ The first term measures the remaining fiberwise multidistance after adjoining each image $\pi(X_i)$ to its conditioning data.

The energy eigenstates of the tight binding chain are orthogonal. This is a fundamental quantum mechanical result: eigenstates of a Hermitian operator (the Hamiltonian) with distinct eigenvalues are orthogonal. Here we prove it directly using the periodic boundary conditions which quantize the wavenumbers. The key physical insight is that different wavenu...

The energy eigenstates satisfy the time-independent Schrodinger equation.

The hamiltonian of the tight binding chain is hermitian.

The quantized wavenumbers form a subset of the BrillouinZone. -/ lemma quantaWaveNumber_subset_brillouinZone : T.QuantaWaveNumber ⊆ T.BrillouinZone := by rintro _ ⟨n, rfl⟩ have hT := T.a_pos have hNpos : 0 < T.N := lt_of_le_of_lt (Nat.zero_le _) n.isLt simp only [BrillouinZone, Set.mem_Ico] generalize T.N = x at * generalize T.a = a at * have hx : (0 :...