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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 | import Mathlib
import OAI.GroupTheory.SimpleOvergroups.RingAssemblyTools
import OAI.GroupTheory.SimpleOvergroups.SourceModel
namespace OAI
noncomputable section
namespace SimpleFInftyOvergroups.SourceModel
open scoped Classical TensorProduct FiniteAlgebraicEnvelopes.Compiler.HeadCodec
open FiniteAlgebraicEnvelopes Bootstrap Compiler Compiler.Simulator MachineRealization
open ContextualEncoding SemanticEncoding EffectiveNames RingAssembly
variable {G : Type} [Group G] {k : ℕ} (s : Fin k → G)
variable (h : ComputablePred (fun w => FiniteAlgebraicEnvelopes.evalWord s w = 1))
local instance : LinearOrder (Control.Sector Sigma (Bool × family.Γ) (ParameterMachine.State family.Q (index s h))) :=
ParameterCode.finiteOrder (index s h)
theorem realizes_processor : Word.Realizes KAlphabet (KMachine s h) (tableNormal (processor s h)) :=
index_fixed s h ▸ family.realizes (index s h)
def rawCopy : L s h →ₙ* K s h :=
Quotient.rawMap (processor s h) wf (index_good s h) (index_stable s h)
KAlphabet (KMachine s h) family.decode (realizes_processor s h)
theorem rawCopy_injective : Function.Injective (rawCopy s h) :=
Quotient.rawMap_injective (processor s h) wf (index_good s h) (index_stable s h)
KAlphabet (KMachine s h) family.decode (realizes_processor s h)
theorem rawCopy_ne_one (x : L s h) : rawCopy s h x ≠ 1 :=
Quotient.rawMap_ne_one (processor s h) wf (index_good s h) (index_stable s h)
KAlphabet (KMachine s h) family.decode (realizes_processor s h) x
theorem rawCopy_ne_zero (x : L s h) : rawCopy s h x ≠ 0 :=
Quotient.rawMap_ne_zero (processor s h) wf (index_good s h) (index_stable s h)
KAlphabet (KMachine s h) family.decode (realizes_processor s h) x
def i : Astar s h →ₙₐ[ZMod 2] B s h := Contracted.embedding (rawCopy s h)
theorem i_injective : Function.Injective (i s h) :=
Contracted.embedding_injective (rawCopy s h) (rawCopy_injective s h) (rawCopy_ne_zero s h)
theorem one_not_mem_i : (1 : B s h) ∉ Set.range (i s h) :=
Contracted.one_not_mem_range (rawCopy s h) (rawCopy_ne_one s h)
def κ : B s h ⊗[ZMod 2] MonoidAlgebra (ZMod 2) PartialAffine.monoid →ₐ[ZMod 2] Astar s h :=
(Contracted.map (codeProduct₀ s h)).comp (ZeroProduct.tensorMap (ZMod 2) (K s h) PartialAffine.monoid)
theorem κ_injective : Function.Injective (κ s h) :=
(Contracted.map_injective (codeProduct₀ s h) (codeProduct_injective s h)).comp
(ZeroProduct.tensorMap_bijective (ZMod 2) (K s h) PartialAffine.monoid).1
def ξ : B s h ⊗[ZMod 2] MonoidAlgebra (ZMod 2) PartialAffine.monoid →ₙₐ[ZMod 2] B s h :=
(i s h).comp (κ s h).toNonUnitalAlgHom
theorem ξ_injective : Function.Injective (ξ s h) := (i_injective s h).comp (κ_injective s h)
def φ : B s h →ₙ+* B s h := binaryPhi (ξ s h)
theorem φ_injective : Function.Injective (φ s h) := binaryPhi_injective (ξ s h) (ξ_injective s h)
def binary : CompressedSplittings.BinaryData (φ s h) := binaryData (ξ s h)
def e : B s h := i s h 1
theorem e_idempotent : IsIdempotentElem (e s h) := supportIdempotent (i s h : Astar s h →ₙ+* B s h)
theorem e_nonzero : e s h ≠ 0 := by
intro he
exact one_ne_zero (i_injective s h (he.trans (map_zero (i s h)).symm))
theorem e_ne_one : e s h ≠ 1 := fun he => one_not_mem_i s h ⟨1,he⟩
def cornerMap : Astar s h →+* (e_idempotent s h).Corner := toCorner (i s h : Astar s h →ₙ+* B s h)
theorem cornerMap_injective : Function.Injective (cornerMap s h) := toCorner_injective _ (i_injective s h)
abbrev Generators := CommonCompiler.FiniteLetter Sigma family.Γ family.Q (index s h)
def Relations : FreeAlgebra (ZMod 2) (Generators s h) → FreeAlgebra (ZMod 2) (Generators s h) → Prop :=
ContractedPresentation.Rel (ZMod 2) (Generators s h) (Word.Rel KAlphabet (KMachine s h)) [.zero]
def presentation : RingQuot (Relations s h) ≃ₐ[ZMod 2] B s h :=
ContractedPresentation.equivalence (ZMod 2) (Generators s h) (Word.Rel KAlphabet (KMachine s h))
(MulEquiv.refl (K s h)) [.zero] rfl
theorem relations_finite : {p : FreeAlgebra (ZMod 2) (Generators s h) × FreeAlgebra (ZMod 2) (Generators s h) |
Relations s h p.1 p.2}.Finite :=
ContractedPresentation.finite_relations (ZMod 2) (Generators s h) (Word.Rel KAlphabet (KMachine s h))
[.zero] (Word.rules_finite KAlphabet (KMachine s h))
def presentationMap : FreeAlgebra (ZMod 2) (Generators s h) →ₐ[ZMod 2] B s h :=
(presentation s h).toAlgHom.comp (RingQuot.mkAlgHom (ZMod 2) (Relations s h))
theorem presentationMap_surjective : Function.Surjective (presentationMap s h) :=
(presentation s h).surjective.comp (RingQuot.mkAlgHom_surjective (ZMod 2) (Relations s h))
theorem presentationMap_kernel : TwoSidedIdeal.ker (presentationMap s h).toRingHom =
TwoSidedIdeal.span (differences (Relations s h)) := by
rw [←ker_ringQuot]
ext a
simp only [TwoSidedIdeal.mem_ker]
change presentation s h (RingQuot.mkAlgHom (ZMod 2) (Relations s h) a)=0 ↔
RingQuot.mkRingHom (Relations s h) a=0
rw [←RingQuot.mkAlgHom_coe (ZMod 2) (Relations s h)]
exact (presentation s h).map_eq_zero_iff
end SimpleFInftyOvergroups.SourceModel
end
end OAI
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