File size: 2,345 Bytes
cdec422 | 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 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import Mathlib
import OAI.LinearAlgebra.MatrixFields.Arithmetic.ComplexArithmeticLowerBound
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
noncomputable section
open scoped BigOperators
namespace MatrixMultiplication.Foundation.Arithmetic.Expression
variable {Input Output : Type*}
structure CompiledFamily [Fintype Output] (expressions : Output → Expression Input) where
registers : ℕ
program : Program Input registers
output : Output → Fin registers
correct : ∀ inputs i, program.eval inputs (output i) = (expressions i).eval inputs
cost_eq : program.cost = ∑ i, (expressions i).cost
def compileFinFamily : (n : ℕ) → (expressions : Fin n → Expression Input) →
CompiledFamily expressions
| 0, _ =>
{ registers := 0
program := Program.nil
output := Fin.elim0
correct := by
intro inputs i
exact Fin.elim0 i
cost_eq := by simp [Program.cost] }
| n + 1, expressions =>
let p := compile (expressions 0)
let q := compileFinFamily n (fun i => expressions i.succ)
{ registers := p.registers + q.registers
program := p.program.append q.program
output := Fin.cases (Program.oldIndex _ _ p.output)
(fun i => Program.newIndex _ _ (q.output i))
correct := by
intro inputs i
refine Fin.cases ?_ (fun j => ?_) i
· exact (Program.eval_append_old p.program q.program inputs p.output).trans
(p.correct inputs)
· exact (Program.eval_append_new p.program q.program inputs (q.output j)).trans
(q.correct inputs j)
cost_eq := by
rw [Program.cost_append, p.cost_eq, q.cost_eq, Fin.sum_univ_succ] }
def compileFamily [Fintype Output] (expressions : Output → Expression Input) :
CompiledFamily expressions :=
let e := Fintype.equivFin Output
let p := compileFinFamily (Fintype.card Output) (fun i => expressions (e.symm i))
{ registers := p.registers
program := p.program
output := fun i => p.output (e i)
correct := by
intro inputs i
simpa only [Equiv.symm_apply_apply] using p.correct inputs (e i)
cost_eq := by
rw [p.cost_eq]
exact e.symm.sum_comp (fun i => (expressions i).cost) }
end MatrixMultiplication.Foundation.Arithmetic.Expression
end
end MatrixAllFields
end OAI
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