Span basis
PureU1.VectorLikeOddPlane.span_basis
Project documentation
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
Exact Lean statement
lemma span_basis (S : (PureU1 (2 * n.succ + 1)).LinSols) :
∃ (g f : Fin n.succ → ℚ), S.val = P g + P! fFormal artifact
Lean source
lemma span_basis (S : (PureU1 (2 * n.succ + 1)).LinSols) : ∃ (g f : Fin n.succ → ℚ), S.val = P g + P! f := by obtain ⟨f, hf⟩ := (Submodule.mem_span_range_iff_exists_fun ℚ).mp (Basis.mem_span basisaAsBasis S) simp only [succ_eq_add_one, basisaAsBasis, coe_basisOfLinearIndependentOfCardEqFinrank, Fintype.sum_sum_type] at hf change P' _ + P!' _ = S at hf refine ⟨f ∘ Sum.inl, f ∘ Sum.inr, ?_⟩ rw [← hf] simp only [succ_eq_add_one, ACCSystemLinear.linSolsAddCommMonoid_add_val, P'_val, P!'_val] rfl- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QFT/QED/AnomalyCancellation/Odd/BasisLinear.lean:864-874
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