Research proof index

A point in the span of the whole basis. -/ def Pa' (f : (Fin n.succ) ⊕ (Fin n) → ℚ) : (PureU1 (2 * n.succ)).LinSols := ∑ i, f i • basisa i lemma Pa'_P'_P!' (f : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr) := by exact Fintype.sum_sum_type _ /-! ### E.6. The combined basis vectors are linearly independent -/ theorem basisa_lin...

The basis formed out of our basisa vectors. -/ noncomputable def basisaAsBasis : Basis (Fin (succ n) ⊕ Fin n) ℚ (PureU1 (2 * succ n)).LinSols := basisOfLinearIndependentOfCardEqFinrank (@basisa_linear_independent n) basisa_card /-! ## F. Every Lienar solution is the sum of a point from each plane

The basis formed out of our basisa vectors. -/ noncomputable def basisaAsBasis : Basis (Fin (succ n) ⊕ Fin n) ℚ (PureU1 (2 * succ n)).LinSols := basisOfLinearIndependentOfCardEqFinrank (@basisa_linear_independent n) basisa_card /-! ## F. Every Lienar solution is the sum of a point from each plane -/ lemma span_basis (S : (PureU1 (2 * n.succ)).LinSols) :...

Swapping the elements evenShiftFst j and evenShiftSnd j is equivalent to adding a vector basis!AsCharges j.

A point in the span of the first part of the basis. -/ def P' (f : Fin n.succ → ℚ) : (PureU1 (2 * n.succ)).LinSols := ∑ i, f i • basis i lemma P'_val (f : Fin n.succ → ℚ) : (P' f).val = P f := by simp only [succ_eq_add_one, P', P] funext i rw [sum_of_anomaly_free_linear, sum_of_charges] rfl /-! ### B.11. The basis vectors are linearly independent -/ theor...

A point in the span of the whole basis. -/ def Pa' (f : (Fin n) ⊕ (Fin n) → ℚ) : (PureU1 (2 * n + 1)).LinSols := ∑ i, f i • basisa i lemma Pa'_P'_P!' (f : (Fin n) ⊕ (Fin n) → ℚ) : Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr) := by exact Fintype.sum_sum_type _ /-! ### E.6. The combined basis vectors are linearly independent