| import OAI.Algebra.DepthFive.IndexedCircuit | |
| import OAI.Algebra.DepthFive.UpperBlockData | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem335.BlockData | |
| variable {n : ℕ} | |
| theorem middleInputs_empty_degree (K : Type*) [One K] | |
| (blocks : List (List (Fin n))) (m : Middle blocks) | |
| (h : middleInputs K blocks m = []) : middleDegree blocks m = 0 := by | |
| by_cases hb : blocks.get m.1 = [] | |
| · change (blocks.get m.1).length = 0 | |
| rw [hb] | |
| rfl | |
| · have hc := congrArg List.length h | |
| simp only [middleInputs, List.length_ofFn, List.length_nil] at hc | |
| have hp := matrixEntryPaths_length (blocks.get m.1) hb m.2.1 m.2.2 | |
| have hn : 0 < n := Nat.zero_lt_of_lt m.2.1.isLt | |
| have hpos : 0 < (paths blocks m).length := by | |
| rw [paths, hp, Fintype.card_fin] | |
| exact Nat.pow_pos hn | |
| omega | |
| theorem outputInputs_empty_degree (K : Type*) [One K] | |
| (blocks : List (List (Fin n))) (z : Fin n) | |
| (h : outputInputs K blocks z = []) : (blocks.map List.length).sum = 0 := by | |
| by_cases hb : blocks = [] | |
| · simp [hb] | |
| · have hc := congrArg List.length h | |
| simp only [outputInputs, List.length_ofFn, List.length_nil] at hc | |
| have hp := matrixEntryPaths_length (List.finRange blocks.length) | |
| (by simpa using hb) z z | |
| have hn : 0 < n := Nat.zero_lt_of_lt z.isLt | |
| have hpos : 0 < (upperPaths blocks z).length := by | |
| rw [upperPaths, hp, Fintype.card_fin] | |
| exact Nat.pow_pos hn | |
| omega | |
| /-- The actual homogeneous five-layer block-expansion circuit, before finite reindexing. -/ | |
| def indexedCircuit (K : Type*) [CommSemiring K] | |
| (blocks : List (List (Fin n))) (z : Fin n) : | |
| IndexedDepth5Circuit K n | |
| (Fin n × Fin n × Fin n) (Fin n × Fin n × Fin n) | |
| (Lower blocks) (Middle blocks) (Upper blocks z) where | |
| leaves := D5Leaf.variable | |
| bottomInputs v := [(1, v)] | |
| bottomDegree _ := 1 | |
| bottomHomogeneous := by | |
| intro i entry h | |
| simp only [List.mem_singleton] at h | |
| subst entry | |
| rfl | |
| bottomEmpty := by simp | |
| lowerInputs := lowerInputs blocks | |
| middleInputs := middleInputs K blocks | |
| middleDegree := middleDegree blocks | |
| middleHomogeneous := by | |
| intro m entry h | |
| simpa using middleInputs_homogeneous K blocks m entry h | |
| middleEmpty := middleInputs_empty_degree K blocks | |
| upperInputs := upperInputs blocks z | |
| outputInputs := outputInputs K blocks z | |
| outputDegree := (blocks.map List.length).sum | |
| outputHomogeneous := by | |
| intro entry _ | |
| exact upperInputs_degree blocks z entry.2 | |
| outputEmpty := outputInputs_empty_degree K blocks z | |
| /-- The qualified `Fin`-indexed circuit produced by block expansion. -/ | |
| def circuit (K : Type*) [CommSemiring K] | |
| (blocks : List (List (Fin n))) (z : Fin n) : Depth5Circuit K n := | |
| (indexedCircuit K blocks z).toDepth5Circuit | |
| @[simp] theorem circuit_outputDegree (K : Type*) [CommSemiring K] | |
| (blocks : List (List (Fin n))) (z : Fin n) : | |
| (circuit K blocks z).outputDegree = (blocks.map List.length).sum := rfl | |
| theorem circuit_size (K : Type*) [CommSemiring K] | |
| (blocks : List (List (Fin n))) (z : Fin n) : | |
| circuitSize (circuit K blocks z) = 2 * n ^ 3 + | |
| Fintype.card (Lower blocks) + Fintype.card (Middle blocks) + | |
| Fintype.card (Upper blocks z) + 1 := by | |
| rw [circuit, IndexedDepth5Circuit.circuitSize_toDepth5Circuit] | |
| simp only [Fintype.card_prod, Fintype.card_fin] | |
| ring | |
| end Problem335.BlockData | |
| end | |
| end OAI | |