openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Tensor /ComplexElementary.lean
| import Std | |
| namespace OAI | |
| /-! Finite coefficient tensors and their algebraic transformations. -/ | |
| namespace MatrixMultiplication.Foundation.Elementary | |
| inductive BoundaryTerm where | |
| | a0 | a1 | b0 | b1 | c | |
| deriving DecidableEq, BEq, Repr | |
| namespace BoundaryTerm | |
| def x : BoundaryTerm → Nat | |
| | .a0 | .a1 | .c => 0 | |
| | .b0 => 2 | |
| | .b1 => 1 | |
| def y : BoundaryTerm → Nat | |
| | .a0 | .b0 => 0 | |
| | .a1 | .b1 => 1 | |
| | .c => 2 | |
| def z : BoundaryTerm → Nat | |
| | .a0 => 2 | |
| | .a1 => 1 | |
| | .b0 | .b1 | .c => 0 | |
| def all : List BoundaryTerm := [.a0, .a1, .b0, .b1, .c] | |
| end BoundaryTerm | |
| def testA (w : List BoundaryTerm) : Bool := w.all (fun s => s.z != 0) | |
| def testB (w : List BoundaryTerm) : Bool := w.any (fun s => s.x != 0) | |
| def testC (w : List BoundaryTerm) : Bool := w.any (fun s => s.y == 2) | |
| theorem local_tests_exhaust (w : List BoundaryTerm) : | |
| testA w = (!testB w && !testC w) := by | |
| induction w with | |
| | nil => rfl | |
| | cons s w ih => | |
| cases s <;> | |
| simp_all [testA, testB, testC, BoundaryTerm.x, BoundaryTerm.y, BoundaryTerm.z] | |
| def localDegree (w : List BoundaryTerm) : Nat := | |
| (testA w).toNat + (testB w).toNat + (testC w).toNat | |
| theorem localDegree_eq (w : List BoundaryTerm) : | |
| localDegree w = if testB w && testC w then 2 else 1 := by | |
| unfold localDegree | |
| rw [local_tests_exhaust] | |
| cases testB w <;> cases testC w <;> rfl | |
| def retained (w : List BoundaryTerm) : Bool := | |
| !(testB w && testC w) && !testA w | |
| def triples : List (List BoundaryTerm) := | |
| BoundaryTerm.all.flatMap fun a => | |
| BoundaryTerm.all.flatMap fun b => | |
| BoundaryTerm.all.map fun third => [a, b, third] | |
| theorem retained_triples_count : (triples.filter retained).length = 75 := by decide | |
| theorem retained_b_triples_count : | |
| (triples.filter (fun w => retained w && testB w)).length = 56 := by decide | |
| theorem retained_c_triples_count : | |
| (triples.filter (fun w => retained w && testC w)).length = 19 := by decide | |
| end MatrixMultiplication.Foundation.Elementary | |
| end OAI | |