File size: 1,977 Bytes
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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
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