File size: 3,543 Bytes
2c7d975 | 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 74 75 76 77 78 79 80 | 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
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