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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 | import Mathlib
import OAI.GroupTheory.SimpleOvergroups.DoubleEffective
import OAI.GroupTheory.SimpleOvergroups.SemanticActionMonoid
import OAI.GroupTheory.SimpleOvergroups.SemanticActions
import OAI.GroupTheory.SimpleOvergroups.SemanticDouble
namespace OAI
noncomputable section
open scoped Classical
namespace SourceActionMonoid
variable {V Y : Type*} [Group V] [MulAction V Y]
instance monoidWithZero : MonoidWithZero (M V Y) where
__ := (inferInstance : Monoid (M V Y))
zero := omega
zero_mul := omega_mul
mul_zero := mul_omega
end SourceActionMonoid
namespace SimpleFInftyOvergroups.SemanticEncoding
open BoundedEncoding EffectiveNames NamedAction StageConstruction
namespace SemanticTests
local instance : Primcodable (ZMod 2) := FiniteAlgebraicEnvelopes.Effect.finitePrimcodable _
local instance : Primcodable (ZMod 3) := FiniteAlgebraicEnvelopes.Effect.finitePrimcodable _
variable {A R : Type} [Primcodable A] [Ring R] [Algebra (ZMod 2) R]
(π : SemanticMatrix.Raw (ZMod 2) A →ₐ[ZMod 2] R)
(σ : R → SemanticMatrix.Raw (ZMod 2) A) (hσ : Function.Injective σ)
{m : ℕ} (C : WordCalculus (FreeMonoid A) m)
abbrev N := MatrixWords.nameOperations (k:=ZMod 2) (n:=7) (MatrixWords.grade C)
abbrev baseC := SemanticMatrix.calculus σ hσ m
abbrev baseI := SemanticMatrix.interpretation π σ hσ (MatrixWords.grade C) m
abbrev doubledC := SemanticDouble.calculus (baseC σ hσ (m:=m))
abbrev I := SemanticDouble.interpretation (N C) (baseC σ hσ (m:=m)) (baseI π σ hσ C)
abbrev N₂ := SemanticDouble.N₂ (N C)
abbrev Symbols := Letter (SemanticDouble.GroupName (G:=MatrixWords.Word (ZMod 2) A 7))
(SemanticDouble.PointName (G:=MatrixWords.Word (ZMod 2) A 7) (X:=MatrixWords.Point (ZMod 2) A 7))
abbrev evaluation : FreeMonoid (Symbols (A:=A)) →* SourceActionMonoid.M
(SourceEnlargement.Double.W (Q R)) (SourceEnlargement.Double.Y (Q:=Q R) (X:=X R)) :=
SemanticActionMonoid.wordMap (I π σ hσ C)
theorem evaluation_surjective (hπ : Function.Surjective π) :
Function.Surjective (evaluation π σ hσ C) :=
SemanticActionMonoid.wordMap_surjective (I π σ hσ C)
(SemanticDouble.groupValue_surjective (N C) (baseC σ hσ (m:=m)) (baseI π σ hσ C)
(SemanticMatrix.groupValue_surjective π hπ))
(SemanticDouble.pointValue_surjective (N C) (baseC σ hσ (m:=m)) (baseI π σ hσ C)
(SemanticMatrix.groupValue_surjective π hπ) (SemanticMatrix.pointValue_surjective π hπ))
theorem full (hs : ∀ a, σ (π a) = C.collect (ZMod 2) a)
(f : List A → Part (List A))
(hf : ∀ u, (C.normal (FreeMonoid.ofList u)).toList ∈ f u)
(w : List (Symbols (A:=A))) :
(N₂ C).totalTest (I π σ hσ C).gkey (I π σ hσ C).xkey w ∈
EffectiveNames.Double.test (IdempotentNames.name (k:=ZMod 2))
(IdempotentNames.name (k:=ZMod 3)) (MatrixWords.grade C) C.omega.toList f w := by
apply Operations.test_full _ _ _ _ _ (d:=(N₂ C).wordCost w) ?_ ?_ w le_rfl
· intro g hg
apply SemanticDouble.partialGroupKey_full (N C) (baseC σ hσ (m:=m)) (baseI π σ hσ C) _ g hg
intro v hv
exact SemanticMatrix.partialGroupKey_full π σ hσ C hs f v hv (fun u _ => hf u)
· intro x hx
apply SemanticDouble.partialPointKey_full (N C) (baseC σ hσ (m:=m)) (baseI π σ hσ C) _ x hx
intro y hy
exact SemanticMatrix.partialPointKey_full π σ hσ C hs f y hy (fun u _ => hf u)
theorem keys_eq_iff (u v : List (Symbols (A:=A))) :
(N₂ C).totalTest (I π σ hσ C).gkey (I π σ hσ C).xkey u =
(N₂ C).totalTest (I π σ hσ C).gkey (I π σ hσ C).xkey v ↔
evaluation π σ hσ C (FreeMonoid.ofList u) = evaluation π σ hσ C (FreeMonoid.ofList v) :=
SemanticActionMonoid.totalTest_eq_iff (I π σ hσ C)
(SemanticDouble.groupKey_injective (baseC σ hσ (m:=m))
(SemanticMatrix.groupKey_injective σ hσ))
(SemanticDouble.pointKey_injective (baseC σ hσ (m:=m))
(SemanticMatrix.pointKey_injective σ hσ)) u v
theorem keys_eq_none (w : List (Symbols (A:=A))) :
(N₂ C).totalTest (I π σ hσ C).gkey (I π σ hσ C).xkey w = none ↔
evaluation π σ hσ C (FreeMonoid.ofList w) = 0 := by
exact keys_eq_iff π σ hσ C w [.inr (.inr none)]
end SemanticTests
end SimpleFInftyOvergroups.SemanticEncoding
end
end OAI
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