Research proof index

Given a finite set of charges c the TenQuanta which do not have exotics, duplicate charges or zero fluxes, which map down to c. This is defined to be as efficient as possible. -/ def liftCharge (c : Finset 𝓩) : Multiset (TenQuanta 𝓩) := /- The {(1, 0), (1, 0), (1, 0)} case. -/ /- The multisets of cardinality 3 containing 3 elements of c. -/ let...

A version of fundamental_theorem_of_variational_calculus' for Continuous f. The proof uses assumption that source of f is finite-dimensional inner-product space, so that a bump function with compact support exists via ContDiffBump.hasCompactSupport from Analysis.Calculus.BumpFunction.Basic. The proof is by contradiction, assume that there is `x₀...

A version of fundamental_theorem_of_variational_calculus' for Continuous f. The proof uses assumption that source of f is finite-dimensional inner-product space, so that a bump function with compact support exists via ContDiffBump.hasCompactSupport from Analysis.Calculus.BumpFunction.Basic. The proof is by contradiction, assume that there is `x₀...

The product of two Levi-Civita-type symbols is a generalized Kronecker delta: δ^{μ}_{·} · δ^{ν}_{·} = δ^{μ}_{ν}, where each single factor is a Kronecker matrix against the identity. This is the Lean form of ε^{μ₁…μₙ} ε_{ν₁…νₙ} = δ^{μ₁…μₙ}_{ν₁…νₙ}.

Generalized Kronecker delta contraction. Summing a generalizedKroneckerDelta over one shared index appended at the end lowers the rank by one and pulls out a factor of card α - n. This is the reusable combinatorial fact behind all epsilon-epsilon identities.

The determinant of an SO(3) matrix minus the identity is equal to zero.