Eq sum eval T of single tensor basis
TensorSpecies.Tensor.eq_sum_evalT_of_single_tensor_basis
Plain-language statement
Basis expansion of a one-index tensor: every t : Tensor S ![c] is the sum over basis indices i of its evaluation coefficient toField (evalT 0 i t) times the corresponding basis tensor.
Exact Lean statement
lemma eq_sum_evalT_of_single_tensor_basis {c : C} (t : Tensor S ![c]) :
t = ∑ i, toField (evalT 0 i t) • basis ![c] (ComponentIdx.single.symm
(basisIdxCongr (by simp) i))Formal artifact
Lean source
lemma eq_sum_evalT_of_single_tensor_basis {c : C} (t : Tensor S ![c]) : t = ∑ i, toField (evalT 0 i t) • basis ![c] (ComponentIdx.single.symm (basisIdxCongr (by simp) i)) := by induction' t using Tensor.induction_on_basis with b a t h t1 t2 h1 h2 · obtain ⟨i, rfl⟩ := ComponentIdx.single.symm.surjective b conv_rhs => enter [2, i]; rw [evalT_basis_single] simp · simp · conv_lhs => rw [h] simp [Finset.smul_sum, smul_smul] · simp [add_smul, Finset.sum_add_distrib] grind- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/Evaluation.lean:346-357
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