Research proof index

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 metric tensor is non-degenerate. -/ lemma nondegenerate : (∀ (x : ContrMod d), ⟪x, y⟫ₘ = 0) ↔ y = 0 := by refine Iff.intro (fun h => ?) (fun h => ?) · exact (self_parity_eq_zero_iff _).mp ((symm _ _).trans $ h _) · simp [h] set_option backward.isDefEq.respectTransparency false in lemma matrix_apply_eq_iff_sub : ⟪x, Λ *ᵥ y⟫ₘ = ⟪x, Λ' *ᵥ y⟫ₘ ↔ ⟪x, (Λ...

The metric tensor is non-degenerate. -/ lemma nondegenerate : (∀ (x : ContrMod d), ⟪x, y⟫ₘ = 0) ↔ y = 0 := by refine Iff.intro (fun h => ?) (fun h => ?) · exact (self_parity_eq_zero_iff _).mp ((symm _ _).trans $ h _) · simp [h] set_option backward.isDefEq.respectTransparency false in lemma matrix_apply_eq_iff_sub : ⟪x, Λ *ᵥ y⟫ₘ = ⟪x, Λ' *ᵥ y⟫ₘ ↔ ⟪x, (Λ...

Single contraction. Contracting the last k of k+1 index pairs leaves one free pair σ, τ, with the factorial factor (4-1)(4-2)….

Double contraction. Contracting the last k of k+2 index pairs leaves two free pairs, with value a 2×2 generalized Kronecker delta times the factorial factor.