| import Mathlib |
| import OAI.GroupTheory.SimpleOvergroups.RingAssemblyTools |
| import OAI.GroupTheory.SimpleOvergroups.SourceModel |
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|
| namespace OAI |
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| noncomputable section |
| namespace SimpleFInftyOvergroups.SourceModel |
| open scoped Classical TensorProduct FiniteAlgebraicEnvelopes.Compiler.HeadCodec |
| open FiniteAlgebraicEnvelopes Bootstrap Compiler Compiler.Simulator MachineRealization |
| open ContextualEncoding SemanticEncoding EffectiveNames RingAssembly |
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| 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) |
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| theorem realizes_processor : Word.Realizes KAlphabet (KMachine s h) (tableNormal (processor s h)) := |
| index_fixed s h ▸ family.realizes (index s h) |
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| 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 |
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| 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) |
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| 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 |
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| 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) |
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| 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) |
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| 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⟩ |
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| 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) |
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|
| 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] |
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| 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)) |
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| 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 |
|
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| end OAI |
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