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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 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | import OAI.Algebra.DepthFive.Basic
namespace OAI
noncomputable section
universe u v w
namespace Problem335
namespace D5Leaf
def mapCoefficients {K : Type u} {L : Type v} {n : ℕ}
(f : K → L) : D5Leaf K n → D5Leaf L n
| .scalar a => .scalar (f a)
| .variable x => .variable x
@[simp] theorem degree_mapCoefficients {K : Type u} {L : Type v} {n : ℕ}
(f : K → L) (x : D5Leaf K n) :
d5LeafDegree (mapCoefficients f x) = d5LeafDegree x := by
cases x <;> rfl
@[simp] theorem mapCoefficients_id {K : Type u} {n : ℕ} (x : D5Leaf K n) :
mapCoefficients id x = x := by cases x <;> rfl
@[simp] theorem mapCoefficients_comp {K : Type u} {L : Type v} {M : Type w}
{n : ℕ} (f : K → L) (g : L → M) (x : D5Leaf K n) :
mapCoefficients g (mapCoefficients f x) = mapCoefficients (g ∘ f) x := by
cases x <;> rfl
end D5Leaf
/-- Pure coefficient relabeling; no algebraic properties are needed for syntactic homogeneity. -/
def Depth5Circuit.mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K → L) (c : Depth5Circuit K n) : Depth5Circuit L n where
leafCount := c.leafCount
leaves := fun i => D5Leaf.mapCoefficients f (c.leaves i)
bottomCount := c.bottomCount
bottomInputs := fun i => (c.bottomInputs i).map (fun e => (f e.1, e.2))
bottomDegree := c.bottomDegree
bottomHomogeneous := by
intro i entry h
obtain ⟨e, he, rfl⟩ := List.mem_map.mp h
simpa using c.bottomHomogeneous i e he
bottomEmpty := by
intro i h
exact c.bottomEmpty i (List.map_eq_nil_iff.mp h)
lowerCount := c.lowerCount
lowerInputs := c.lowerInputs
middleCount := c.middleCount
middleInputs := fun i => (c.middleInputs i).map (fun e => (f e.1, e.2))
middleDegree := c.middleDegree
middleHomogeneous := by
intro i entry h
obtain ⟨e, he, rfl⟩ := List.mem_map.mp h
exact c.middleHomogeneous i e he
middleEmpty := by
intro i h
exact c.middleEmpty i (List.map_eq_nil_iff.mp h)
upperCount := c.upperCount
upperInputs := c.upperInputs
outputInputs := c.outputInputs.map (fun e => (f e.1, e.2))
outputDegree := c.outputDegree
outputHomogeneous := by
intro entry h
obtain ⟨e, he, rfl⟩ := List.mem_map.mp h
exact c.outputHomogeneous e he
outputEmpty := by
intro h
exact c.outputEmpty (List.map_eq_nil_iff.mp h)
@[simp] theorem Depth5Circuit.lowerInputs_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K → L) (c : Depth5Circuit K n) (i : Fin c.lowerCount) :
(c.mapCoefficients f).lowerInputs i = c.lowerInputs i := rfl
@[simp] theorem Depth5Circuit.middleInputs_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K → L) (c : Depth5Circuit K n) (i : Fin c.middleCount) :
(c.mapCoefficients f).middleInputs i =
(c.middleInputs i).map (fun e => (f e.1, e.2)) := rfl
@[simp] theorem Depth5Circuit.upperInputs_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K → L) (c : Depth5Circuit K n) (i : Fin c.upperCount) :
(c.mapCoefficients f).upperInputs i = c.upperInputs i := rfl
@[simp] theorem Depth5Circuit.outputInputs_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K → L) (c : Depth5Circuit K n) :
(c.mapCoefficients f).outputInputs =
c.outputInputs.map (fun e => (f e.1, e.2)) := rfl
@[simp] theorem circuitSize_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K → L) (c : Depth5Circuit K n) :
circuitSize (c.mapCoefficients f) = circuitSize c := rfl
@[simp] theorem d5LeafValue_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K →+* L) (x : D5Leaf K n) :
d5LeafValue (x.mapCoefficients f) = MvPolynomial.map f (d5LeafValue x) := by
cases x <;> simp [D5Leaf.mapCoefficients, d5LeafValue]
@[simp] theorem bottomValue_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K →+* L) (c : Depth5Circuit K n) (i : Fin c.bottomCount) :
bottomValue (c.mapCoefficients f) i = MvPolynomial.map f (bottomValue c i) := by
simp [bottomValue, Depth5Circuit.mapCoefficients, List.map_map,
Function.comp_def, map_list_sum]
@[simp] theorem lowerValue_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K →+* L) (c : Depth5Circuit K n) (i : Fin c.lowerCount) :
lowerValue (c.mapCoefficients f) i = MvPolynomial.map f (lowerValue c i) := by
simp [lowerValue, bottomValue, Depth5Circuit.mapCoefficients, List.map_map,
Function.comp_def, map_list_sum, map_list_prod]
@[simp] theorem middleValue_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K →+* L) (c : Depth5Circuit K n) (i : Fin c.middleCount) :
middleValue (c.mapCoefficients f) i = MvPolynomial.map f (middleValue c i) := by
simp [middleValue, lowerValue, bottomValue, Depth5Circuit.mapCoefficients, List.map_map,
Function.comp_def, map_list_sum, map_list_prod]
@[simp] theorem upperValue_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K →+* L) (c : Depth5Circuit K n) (i : Fin c.upperCount) :
upperValue (c.mapCoefficients f) i = MvPolynomial.map f (upperValue c i) := by
simp [upperValue, middleValue, lowerValue, bottomValue, Depth5Circuit.mapCoefficients, List.map_map,
Function.comp_def, map_list_sum, map_list_prod]
@[simp] theorem circuitValue_mapCoefficients {K : Type u} {L : Type v}
[CommSemiring K] [CommSemiring L] {n : ℕ}
(f : K →+* L) (c : Depth5Circuit K n) :
circuitValue (c.mapCoefficients f) = MvPolynomial.map f (circuitValue c) := by
simp [circuitValue, upperValue, middleValue, lowerValue, bottomValue, Depth5Circuit.mapCoefficients, List.map_map,
Function.comp_def, map_list_sum, map_list_prod]
@[simp] theorem Depth5Circuit.mapCoefficients_id {K : Type u}
[CommSemiring K] {n : ℕ} (c : Depth5Circuit K n) :
c.mapCoefficients id = c := by
cases c
simp [Depth5Circuit.mapCoefficients]
@[simp] theorem Depth5Circuit.mapCoefficients_comp {K : Type u} {L : Type v}
{M : Type w} [CommSemiring K] [CommSemiring L] [CommSemiring M]
{n : ℕ} (f : K → L) (g : L → M) (c : Depth5Circuit K n) :
(c.mapCoefficients f).mapCoefficients g = c.mapCoefficients (g ∘ f) := by
cases c
simp [Depth5Circuit.mapCoefficients, List.map_map, Function.comp_def]
end Problem335
end
end OAI
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