Research proof index

Rewriting basis for the product in terms of the tensor product basis. -/ lemma basis_prod_eq {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C} : basis (S := S) (Fin.append c c1) = (((Tensor.basis (S := S) c).tensorProduct (Tensor.basis (S := S) c1)).reindex (ComponentIdx.prod.symm)).map tensorEquivProd := by ext b simp [ComponentIdx.prod, tensorEquivProd] rw [pr...

Rewriting basis for the product in terms of the tensor product basis. -/ lemma basis_prod_eq {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C} : basis (S := S) (Fin.append c c1) = (((Tensor.basis (S := S) c).tensorProduct (Tensor.basis (S := S) c1)).reindex (ComponentIdx.prod.symm)).map tensorEquivProd := by ext b simp [ComponentIdx.prod, tensorEquivProd] rw [pr...

Rewriting basis for the product in terms of the tensor product basis. -/ lemma basis_prod_eq {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C} : basis (S := S) (Fin.append c c1) = (((Tensor.basis (S := S) c).tensorProduct (Tensor.basis (S := S) c1)).reindex (ComponentIdx.prod.symm)).map tensorEquivProd := by ext b simp [ComponentIdx.prod, tensorEquivProd] rw [pr...

Rewriting basis for the product in terms of the tensor product basis. -/ lemma basis_prod_eq {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C} : basis (S := S) (Fin.append c c1) = (((Tensor.basis (S := S) c).tensorProduct (Tensor.basis (S := S) c1)).reindex (ComponentIdx.prod.symm)).map tensorEquivProd := by ext b simp [ComponentIdx.prod, tensorEquivProd] rw [pr...

Rewriting basis for the product in terms of the tensor product basis. -/ lemma basis_prod_eq {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C} : basis (S := S) (Fin.append c c1) = (((Tensor.basis (S := S) c).tensorProduct (Tensor.basis (S := S) c1)).reindex (ComponentIdx.prod.symm)).map tensorEquivProd := by ext b simp [ComponentIdx.prod, tensorEquivProd] rw [pr...

Given two pure tensors p1 : Pure S c and p2 : Pure S c, prodP p p2 is the tensor product of those tensors returning an element in Pure S (Sum.elim c c1 ∘ ⇑finSumFinEquiv.symm). -/ def Pure.prodP {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C} (p1 : Pure S c) (p2 : Pure S c1) : Pure S (Fin.append c c1) := Fin.addCases (fun i => LinearEquiv.cast (R := k)...