Research proof index

A point in the span of the second part of the basis. -/ def P!' (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).LinSols := ∑ i, f i • basis! i lemma P!'_val (f : Fin n → ℚ) : (P!' f).val = P! f := by simp only [P!', P!] funext i rw [sum_of_anomaly_free_linear, sum_of_charges] rfl /-! ### C.11. The basis vectors are linearly independent -/ theorem basis!_linear_ind...

A point in the span of the basis as a charge. -/ def Pa (f : Fin n → ℚ) (g : Fin n → ℚ) : (PureU1 (2 * n + 1)).Charges := P f + P! g /-! ### E.3. Components of the inclusion -/ lemma Pa_oddShiftShiftZero (f g : Fin n.succ → ℚ) : Pa f g oddShiftShiftZero = f 0 := by rw [Pa] simp only [ACCSystemCharges.chargesAddCommMonoid_add] nth_rewrite 1 [oddShiftShiftZ...

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 -/ theorem basisa_linear_indepen...

The basis formed out of our basisa vectors. -/ noncomputable def basisaAsBasis : Basis (Fin n.succ ⊕ Fin n.succ) ℚ (PureU1 (2 * n.succ + 1)).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 n.succ ⊕ Fin n.succ) ℚ (PureU1 (2 * n.succ + 1)).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 + 1)).L...

Swapping the elements oddShiftFst j and oddShiftSnd j is equivalent to adding a vector basis!AsCharges j.