| import OAI.Algebra.DepthFive.BlockCircuit |
| import OAI.Algebra.DepthFive.IndexedCircuitValue |
| import OAI.Algebra.DepthFive.BlockMiddleSemantics |
| import OAI.Algebra.DepthFive.ImmBlocks |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace Problem335.BlockData |
|
|
| variable {n : ℕ} |
|
|
| @[simp] theorem indexedCircuit_bottomValue (K : Type*) [CommSemiring K] |
| (blocks : List (List (Fin n))) (z : Fin n) (v : Fin n × Fin n × Fin n) : |
| (indexedCircuit K blocks z).bottomValue v = MvPolynomial.X v := by |
| simp [IndexedDepth5Circuit.bottomValue, indexedCircuit, d5LeafValue] |
|
|
| @[simp] theorem indexedCircuit_lowerValue (K : Type*) [CommSemiring K] |
| (blocks : List (List (Fin n))) (z : Fin n) (p : Lower blocks) : |
| (indexedCircuit K blocks z).lowerValue p = lowerMonomial K blocks p := by |
| simp [IndexedDepth5Circuit.lowerValue, indexedCircuit, lowerMonomial, |
| IndexedDepth5Circuit.bottomValue, d5LeafValue] |
|
|
| @[simp] theorem indexedCircuit_middleValue (K : Type*) [CommSemiring K] |
| (blocks : List (List (Fin n))) (z : Fin n) (m : Middle blocks) : |
| (indexedCircuit K blocks z).middleValue m = |
| immBlock K n (blocks.get m.1) m.2.1 m.2.2 := by |
| change ((middleInputs K blocks m).map (fun e => |
| MvPolynomial.C e.1 * (indexedCircuit K blocks z).lowerValue e.2)).sum = _ |
| simp_rw [indexedCircuit_lowerValue] |
| exact middleInputs_value K blocks m |
|
|
| @[simp] theorem indexedCircuit_upperValue (K : Type*) [CommSemiring K] |
| (blocks : List (List (Fin n))) (z : Fin n) (p : Upper blocks z) : |
| (indexedCircuit K blocks z).upperValue p = |
| ((upperInputs blocks z p).map (fun e => |
| immBlock K n (blocks.get e.1) e.2.1 e.2.2)).prod := by |
| change ((upperInputs blocks z p).map |
| (fun m => (indexedCircuit K blocks z).middleValue m)).prod = _ |
| simp only [indexedCircuit_middleValue] |
|
|
| /-- The block circuit computes the corresponding entry of the product of block matrices. -/ |
| theorem indexedCircuit_circuitValue (K : Type*) [CommSemiring K] |
| (blocks : List (List (Fin n))) (z : Fin n) : |
| (indexedCircuit K blocks z).circuitValue = |
| ((blocks.map (immBlock K n)).prod) z z := by |
| change ((outputInputs K blocks z).map (fun e => |
| MvPolynomial.C e.1 * (indexedCircuit K blocks z).upperValue e.2)).sum = _ |
| simp only [indexedCircuit_upperValue, outputInputs, List.map_ofFn, |
| Function.comp_def, MvPolynomial.C_1, one_mul, List.sum_ofFn] |
| change (∑ k : Fin (matrixEntryPaths (List.finRange blocks.length) z z).length, |
| (((matrixEntryPaths (List.finRange blocks.length) z z).get k).map |
| (fun e => immBlock K n (blocks.get e.1) e.2.1 e.2.2)).prod) = _ |
| rw [matrixEntryPaths_fin_sum_prod (fun j : Fin blocks.length => |
| immBlock K n (blocks.get j))] |
| simp [← List.ofFn_eq_map] |
|
|
| /-- Semantic correctness after converting all gate indices to `Fin`. -/ |
| theorem circuit_value (K : Type*) [CommSemiring K] |
| (blocks : List (List (Fin n))) (z : Fin n) : |
| Problem335.circuitValue (circuit K blocks z) = |
| ((blocks.map (immBlock K n)).prod) z z := by |
| rw [circuit, IndexedDepth5Circuit.circuitValue_toDepth5Circuit, |
| indexedCircuit_circuitValue] |
|
|
| /-- Any partition of the layers produces a circuit computing the qualified IMM polynomial. -/ |
| theorem circuit_value_eq_imm (K : Type*) [CommSemiring K] |
| (hn : 0 < n) (blocks : List (List (Fin n))) |
| (hblocks : blocks.flatten = List.finRange n) : |
| Problem335.circuitValue (circuit K blocks ⟨0, hn⟩) = imm K n := by |
| rw [circuit_value, imm_eq_blocks_prod hn blocks hblocks] |
|
|
| end Problem335.BlockData |
|
|
| end |
|
|
| end OAI |
|
|
