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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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.Algebra.DepthFive.Basic
namespace OAI
noncomputable section
universe u
namespace Problem335
/-- The scalars actually occurring at a leaf. -/
def leafCoefficientSet {K : Type u} {n : ℕ} (x : D5Leaf K n) : Finset K := by
classical
exact match x with
| .scalar a => {a}
| .variable _ => ∅
/-- All coefficients occurring in a circuit, including weighted wires. -/
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
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]⟩
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⟩)⟩
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⟩)⟩
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⟩)
/-- The subfield generated by the finitely many coefficients of a circuit. -/
def circuitCoefficientField {K : Type u} [Field K] {n : ℕ}
(c : Depth5Circuit K n) : Subfield K :=
Subfield.closure (↑(circuitCoefficientSet c) : Set K)
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)
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)
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)
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)
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)
end Problem335
end
end OAI
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