theorem SymmetricBilinearForms.sum_diagonal_ji (a i j : ℕ) : (∑ x : Fin a, if ↑x = j then if ↑x = i then 1 else 0 else 0) = if i < a ∧ i = j then 1 else 0

Evaluates the (j,i)-entry of the diagonal block in the matrix of the standard F₂ form: summing the indicator [x = j ∧ x = i] over x : Fin a gives 1 if i < a and i = j, otherwise 0. Used in standardFormF2_eq_toMatrix' to compute the diagonal part of the Gram matrix.

theorem SymmetricBilinearForms.sum_hyperbolic_ji (a b i j : ℕ) (_hi : i < a + 2 * b) (hj : j < a + 2 * b) : ∑ x : Fin b, ((if a + 2 * ↑x + 1 = j then if a + 2 * ↑x + 0 = i then 1 else 0 else 0) + if a + 2 * ↑x + 0 = j then if a + 2 * ↑x + 1 = i then 1 else 0 else 0) = if i ≥ a ∧ j ≥ a ∧ (i - a) / 2 = (j - a) / 2 ∧ (i - a) % 2 ≠ (j - a) % 2 then 1 else 0

Evaluates the (j,i)-entry of the hyperbolic-plane block in the matrix of the standard F₂ form: each pair (a+2x, a+2x+1) contributes the antidiagonal [[0,1],[1,0]], so the sum is 1 iff i, j ≥ a and (i-a)/2 = (j-a)/2 while (i-a) % 2 ≠ (j-a) % 2. Used in standardFormF2_eq_toMatrix'.

def SymmetricBilinearForms.prodToFinEquiv (a b : ℕ) : ((Fin a → ZMod 2) × (Fin b → Fin 2 → ZMod 2)) ≃ₗ[ZMod 2] Fin (a + 2 * b) → ZMod 2

Linear isomorphism (Fin a → F₂) × (Fin b → Fin 2 → F₂) ≃ₗ Fin (a + 2 b) → F₂ that interleaves the two factors as in standardFormF2: indices < a come from the first factor, and indices a + 2k + r (with r ∈ {0,1}) come from the r-th coordinate of the k-th block in the second factor. Bridges the abstract product presentation of Proposition 30.6 with the concrete matrix presentation standardFormF2 a b.

theorem SymmetricBilinearForms.standardFormF2_eq_toMatrix' (a b : ℕ) : LinearMap.BilinForm.toMatrix' ((LinearMap.BilinForm.congr (prodToFinEquiv a b)) (BilinFormZMod2.standardF2Form a b)) = standardFormF2 a b

Bridge from the abstract Proposition 30.6 to the concrete standardFormF2 a b. Transporting BilinFormZMod2.standardF2Form a b along the isomorphism prodToFinEquiv a b and then taking its matrix representation reproduces exactly the explicit Gram matrix standardFormF2 a b. This identifies the abstract block-diagonal form of the classification theorem with the concrete (a + 2b) × (a + 2b) matrix used downstream.