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f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexTensor
namespace OAI
/-! Finite coefficient tensors and their algebraic transformations. -/
open scoped BigOperators
namespace MatrixMultiplication.Foundation
namespace Tensor
section Pairing
variable {K U : Type*} [CommSemiring K] [DecidableEq U]
def dotPairing (U : Type*) [DecidableEq U] : Tensor K U U Unit :=
fun i j _ => if i = j then 1 else 0
@[simp] theorem dotPairing_matching (i : U) :
dotPairing (K := K) U i i () = 1 := by
simp [dotPairing]
theorem contract_dotPairing [Fintype U] (a b : U → K) :
contract (dotPairing (K := K) U) a b () = ∑ i, a i * b i := by
classical
simp [contract, dotPairing, ite_mul]
end Pairing
end Tensor
end MatrixMultiplication.Foundation
end OAI
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