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0dd3af4 | 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 | import Mathlib
import OAI.GroupTheory.SimpleOvergroups.BoundedForms
import OAI.GroupTheory.SimpleOvergroups.SemanticEnlargements
namespace OAI
noncomputable section
open scoped Classical
namespace SimpleFInftyOvergroups.SemanticEncoding
open BoundedEncoding EffectiveNames NamedAction SourceEnlargement
namespace SemanticDouble
local instance : Primcodable (ZMod 2) := FiniteAlgebraicEnvelopes.Effect.finitePrimcodable _
local instance : Primcodable (ZMod 3) := FiniteAlgebraicEnvelopes.Effect.finitePrimcodable _
theorem orbit_step {V Y : Type} [Group V] [MulAction V Y]
(w : Double.W V) (v : Double.W₁ V) (y : Y) :
(w * groupEmbedding (ZMod 3) v) • Double.ambientPoint y =
w • pointEmbedding (ZMod 3) (v • pointEmbedding (ZMod 2) y) := by
rw [mul_smul]
exact congrArg (fun x => w • x) (groupEmbedding_pointEmbedding (ZMod 3) v _)
theorem orbit_surjective {V Y U B P : Type} [Group V] [MulAction V Y]
(f : U → Double.W V) (t : B → Y) (p : P → Double.Y (Q:=V) (X:=Y))
(b : U → B → P) (hp : ∀ u x, (p (b u x)).val = f u • Double.ambientPoint (t x))
(hf : Function.Surjective f) (ht : Function.Surjective t) : Function.Surjective p := by
intro y
obtain ⟨w,x,h⟩ := y.property
obtain ⟨u,hu⟩ := hf w
obtain ⟨a,ha⟩ := ht x
refine ⟨b u a,Subtype.ext ?_⟩
exact (hp u a).trans ((congrArg₂ (fun (w : Double.W V) (x : Double.AmbientY Y) => w • x)
hu (congrArg Double.ambientPoint ha)).trans h)
variable {G X V Y K L : Type} [Group V] [MulAction V Y] [Nonempty Y]
[Primcodable K] [Primcodable L] {m : ℕ}
(N : Operations G X) (C : ActionCalculus V Y K L m) (I : Interpretation N C)
abbrev P₁ := IdempotentNames.Index (ZMod 2)
abbrev P₂ := IdempotentNames.Index (ZMod 3)
abbrev N₁ := Enlargement.operations (P:=P₁) N
abbrev N₂ := Enlargement.operations (P:=P₂) (N₁ N)
abbrev C₁ := SemanticEnlargement.calculus (k:=ZMod 2) C
abbrev I₁ := SemanticEnlargement.interpretation (k:=ZMod 2) N C I
abbrev C₂ := SemanticEnlargement.calculus (k:=ZMod 3) (C₁ C)
abbrev I₂ := SemanticEnlargement.interpretation (k:=ZMod 3) (N₁ N) (C₁ C) (I₁ N C I)
abbrev GroupName := Enlargement.Word P₂ (Enlargement.Word P₁ G)
abbrev PointName := Enlargement.Point P₂ (Enlargement.Word P₁ G) (Enlargement.Point P₁ G X)
theorem point_in_orbit (x : PointName (G:=G) (X:=X)) :
(I₂ N C I |>.point x).val ∈ Double.orbitSet (Q:=V) (X:=Y) := by
refine ⟨(I₂ N C I |>.group x.1).val *
groupEmbedding (ZMod 3) (I₁ N C I |>.group x.2.1).val, (I.point x.2.2).val, ?_⟩
exact orbit_step _ _ _
def calculus : ActionCalculus (Double.W V) (Double.Y (Q:=V) (X:=Y)) (ℕ × ℕ) ℕ m where
group := (C₂ C).group
point :=
{ ball := fun b y => (C₂ C).point.ball b y.val
mono := fun _ _ h _ hy => (C₂ C).point.mono h hy
key := fun y => (C₂ C).point.key y.val }
one_mem := (C₂ C).one_mem
inv_mem := (C₂ C).inv_mem
mul_mem := (C₂ C).mul_mem
smul_mem := (C₂ C).smul_mem
inv_congr := (C₂ C).inv_congr
mul_congr := (C₂ C).mul_congr
smul_congr := (C₂ C).smul_congr
def interpretation : Interpretation (N₂ N) (calculus C) where
group := (I₂ N C I).group
point x := ⟨⟨(I₂ N C I |>.point x).val,point_in_orbit N C I x⟩,
(I₂ N C I |>.point x).cost, (I₂ N C I |>.point x).mem⟩
group_cost := (I₂ N C I).group_cost
point_cost := (I₂ N C I).point_cost
one := (I₂ N C I).one
mul := (I₂ N C I).mul
inv := (I₂ N C I).inv
act g x := WithBound.ext'
(Subtype.ext (congrArg WithBound.val ((I₂ N C I).act g x)))
(congrArg (fun t : WithBound (C₂ C).point => t.cost) ((I₂ N C I).act g x))
theorem groupKey_injective (hg : Function.Injective C.group.key) :
Function.Injective (calculus C).group.key :=
SemanticEnlargement.groupKey_injective (C₁ C)
(SemanticEnlargement.groupKey_injective C hg)
theorem pointKey_injective (hx : Function.Injective C.point.key) :
Function.Injective (calculus C).point.key := by
intro a b h
apply Subtype.ext
exact SemanticEnlargement.pointKey_injective (C₁ C)
(SemanticEnlargement.pointKey_injective C hx) h
theorem groupValue_surjective (hg : Function.Surjective (fun g => (I.group g).val)) :
Function.Surjective (fun g => (interpretation N C I |>.group g).val) :=
SemanticEnlargement.groupValue_surjective (N₁ N) (C₁ C) (I₁ N C I)
(SemanticEnlargement.groupValue_surjective N C I hg)
theorem pointValue_base (u : GroupName (G:=G)) (b : X) :
(I₂ N C I |>.point (u,([],b))).val =
(I₂ N C I |>.group u).val • Double.ambientPoint (I.point b).val := by
exact congrArg (fun y => (I₂ N C I |>.group u).val • pointEmbedding (ZMod 3) y)
(SemanticEnlargement.pointValue_basis (k:=ZMod 2) N C I b)
theorem pointValue_surjective (hg : Function.Surjective (fun g => (I.group g).val))
(hx : Function.Surjective (fun x => (I.point x).val)) :
Function.Surjective (fun x => (interpretation N C I |>.point x).val) :=
orbit_surjective (fun g => (I₂ N C I |>.group g).val) (fun x => (I.point x).val)
(fun x => (interpretation N C I |>.point x).val) (fun u b => (u,([],b)))
(pointValue_base N C I) (groupValue_surjective N C I hg) hx
variable [Primcodable G] [Primcodable X]
omit [Primcodable G] [Primcodable X] in
theorem partialGroupKey_full (f : G → Part K) (w : GroupName (G:=G)) {d : ℕ}
(hw : (N₂ N).gcost w ≤ d) (hf : ∀ g, N.gcost g ≤ d → I.gkey g ∈ f g) :
(interpretation N C I).gkey w ∈
Enlargement.partialGroupKey (N₁ N) (IdempotentNames.name (k:=ZMod 3))
(Enlargement.partialGroupKey N (IdempotentNames.name (k:=ZMod 2)) f) w := by
apply SemanticEnlargement.partialGroupKey_full (N₁ N) (C₁ C) (I₁ N C I) _ w hw
intro v hv
exact SemanticEnlargement.partialGroupKey_full N C I f v hv hf
omit [Primcodable G] [Primcodable X] in
theorem partialPointKey_full (f : X → Part L) (x : PointName (G:=G) (X:=X)) {d : ℕ}
(hx : (N₂ N).xcost x ≤ d) (hf : ∀ y, N.xcost y ≤ d → I.xkey y ∈ f y) :
(interpretation N C I).xkey x ∈
Enlargement.partialPointKey (N₁ N) (IdempotentNames.name (k:=ZMod 3))
(Enlargement.partialPointKey N (IdempotentNames.name (k:=ZMod 2)) f) x := by
apply SemanticEnlargement.partialPointKey_full (N₁ N) (C₁ C) (I₁ N C I) _ x hx
intro y hy
exact SemanticEnlargement.partialPointKey_full N C I f y hy hf
end SemanticDouble
end SimpleFInftyOvergroups.SemanticEncoding
end
end OAI
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