| import OAI.Algebra.DepthFive.Basic |
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| namespace OAI |
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| noncomputable section |
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| universe u |
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| namespace Problem335 |
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| / |
| def leafCoefficientSet {K : Type u} {n : ℕ} (x : D5Leaf K n) : Finset K := by |
| classical |
| exact match x with |
| | .scalar a => {a} |
| | .variable _ => ∅ |
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| / |
| def circuitCoefficientSet {K : Type u} [CommSemiring K] {n : ℕ} |
| (c : Depth5Circuit K n) : Finset K := by |
| classical |
| exact (Finset.univ.biUnion fun i => leafCoefficientSet (c.leaves i)) ∪ |
| (Finset.univ.biUnion fun i => ((c.bottomInputs i).map Prod.fst).toFinset) ∪ |
| (Finset.univ.biUnion fun i => ((c.middleInputs i).map Prod.fst).toFinset) ∪ |
| (c.outputInputs.map Prod.fst).toFinset |
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| lemma leaf_scalar_mem_coefficientSet {K : Type u} [CommSemiring K] {n : ℕ} |
| (c : Depth5Circuit K n) (i : Fin c.leafCount) (a : K) |
| (h : c.leaves i = .scalar a) : a ∈ circuitCoefficientSet c := by |
| classical |
| simp only [circuitCoefficientSet, Finset.mem_union] |
| left; left; left |
| exact Finset.mem_biUnion.mpr ⟨i, Finset.mem_univ _, by simp [h, leafCoefficientSet]⟩ |
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| lemma bottom_coeff_mem_coefficientSet {K : Type u} [CommSemiring K] {n : ℕ} |
| (c : Depth5Circuit K n) (i : Fin c.bottomCount) (entry : K × Fin c.leafCount) |
| (h : entry ∈ c.bottomInputs i) : entry.1 ∈ circuitCoefficientSet c := by |
| classical |
| simp only [circuitCoefficientSet, Finset.mem_union] |
| left; left; right |
| exact Finset.mem_biUnion.mpr ⟨i, Finset.mem_univ _, List.mem_toFinset.mpr |
| (List.mem_map.mpr ⟨entry, h, rfl⟩)⟩ |
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| lemma middle_coeff_mem_coefficientSet {K : Type u} [CommSemiring K] {n : ℕ} |
| (c : Depth5Circuit K n) (i : Fin c.middleCount) (entry : K × Fin c.lowerCount) |
| (h : entry ∈ c.middleInputs i) : entry.1 ∈ circuitCoefficientSet c := by |
| classical |
| simp only [circuitCoefficientSet, Finset.mem_union] |
| left; right |
| exact Finset.mem_biUnion.mpr ⟨i, Finset.mem_univ _, List.mem_toFinset.mpr |
| (List.mem_map.mpr ⟨entry, h, rfl⟩)⟩ |
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| lemma output_coeff_mem_coefficientSet {K : Type u} [CommSemiring K] {n : ℕ} |
| (c : Depth5Circuit K n) (entry : K × Fin c.upperCount) |
| (h : entry ∈ c.outputInputs) : entry.1 ∈ circuitCoefficientSet c := by |
| classical |
| simp only [circuitCoefficientSet, Finset.mem_union] |
| right |
| exact List.mem_toFinset.mpr (List.mem_map.mpr ⟨entry, h, rfl⟩) |
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| / |
| def circuitCoefficientField {K : Type u} [Field K] {n : ℕ} |
| (c : Depth5Circuit K n) : Subfield K := |
| Subfield.closure (↑(circuitCoefficientSet c) : Set K) |
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| lemma leaf_scalar_mem {K : Type u} [Field K] {n : ℕ} |
| (c : Depth5Circuit K n) (i : Fin c.leafCount) (a : K) |
| (h : c.leaves i = .scalar a) : a ∈ circuitCoefficientField c := |
| Subfield.subset_closure (leaf_scalar_mem_coefficientSet c i a h) |
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| lemma bottom_coeff_mem {K : Type u} [Field K] {n : ℕ} |
| (c : Depth5Circuit K n) (i : Fin c.bottomCount) (entry : K × Fin c.leafCount) |
| (h : entry ∈ c.bottomInputs i) : entry.1 ∈ circuitCoefficientField c := |
| Subfield.subset_closure (bottom_coeff_mem_coefficientSet c i entry h) |
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| lemma middle_coeff_mem {K : Type u} [Field K] {n : ℕ} |
| (c : Depth5Circuit K n) (i : Fin c.middleCount) (entry : K × Fin c.lowerCount) |
| (h : entry ∈ c.middleInputs i) : entry.1 ∈ circuitCoefficientField c := |
| Subfield.subset_closure (middle_coeff_mem_coefficientSet c i entry h) |
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| lemma output_coeff_mem {K : Type u} [Field K] {n : ℕ} |
| (c : Depth5Circuit K n) (entry : K × Fin c.upperCount) |
| (h : entry ∈ c.outputInputs) : entry.1 ∈ circuitCoefficientField c := |
| Subfield.subset_closure (output_coeff_mem_coefficientSet c entry h) |
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| instance circuitCoefficientField_countable {K : Type u} [Field K] {n : ℕ} |
| (c : Depth5Circuit K n) : Countable (circuitCoefficientField c) := by |
| apply Cardinal.mk_le_aleph0_iff.mp |
| exact (Subfield.cardinalMk_closure_le_max (↑(circuitCoefficientSet c) : Set K)).trans |
| (max_le (Cardinal.mk_le_aleph0) le_rfl) |
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| end Problem335 |
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| end |
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| end OAI |
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