openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /ExternalWordRewriteRealization.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundExternalRewriteSteps | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| open SetRealization SetFormula | |
| universe u | |
| variable {J L : Type u} [Group J] (s : Presentation J) (t : Presentation L) | |
| theorem externalWordAtomicCode_iff (xs ys : List (J ⊕ (L × Bool))) : | |
| ExternalWordAtomicCode s.carrier t.carrier | |
| ((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1) | |
| (finiteWordGraph (externalGeneratorAlphabetPresentation s t) xs) | |
| (finiteWordGraph (externalGeneratorAlphabetPresentation s t) ys) ↔ | |
| ExternalWordRewrite.Atomic xs ys := by | |
| let a := externalGeneratorAlphabetPresentation s t | |
| have oldOne (g : J) : finiteWordGraph a [.inl g] = wordSingleton (externalActorSymbol (s.encode g)) := | |
| finiteWordGraph_singleton a _ | |
| have oldTwo (g h : J) : finiteWordGraph a [.inl g, .inl h] = | |
| wordDouble (externalActorSymbol (s.encode g)) (externalActorSymbol (s.encode h)) := | |
| finiteWordGraph_double a _ _ | |
| have signedTwo (l : L) (b c : Bool) : finiteWordGraph a [.inr (l, b), .inr (l, c)] = | |
| wordDouble (externalLetterSymbol (t.encode l) b) (externalLetterSymbol (t.encode l) c) := | |
| finiteWordGraph_double a _ _ | |
| constructor | |
| · intro atomic | |
| rcases atomic with ⟨_, left, right⟩ | ⟨g, hg, h, hh, k, hk, product, left, right⟩ | | |
| ⟨l, hl, left, right⟩ | |
| · have xsEq := finiteWordGraph_injective a (left.trans (oldOne 1).symm) | |
| have ysEq := finiteWordGraph_injective a (right.trans (finiteWordGraph_nil a).symm) | |
| subst xs ys | |
| exact .old_one | |
| · obtain ⟨g, rfl⟩ := (s.mem_carrier g).1 hg | |
| obtain ⟨h, rfl⟩ := (s.mem_carrier h).1 hh | |
| obtain ⟨k, rfl⟩ := (s.mem_carrier k).1 hk | |
| have productEq := ((s.product s).pair_mem_graph s (fun p => p.1 * p.2) (g, h) k).1 product | |
| have xsEq := finiteWordGraph_injective a (left.trans (oldTwo g h).symm) | |
| have ysEq := finiteWordGraph_injective a (right.trans (oldOne k).symm) | |
| subst xs ys k | |
| exact .old_mul g h | |
| · obtain ⟨l, rfl⟩ := (t.mem_carrier l).1 hl | |
| have ysEq := finiteWordGraph_injective a (right.trans (finiteWordGraph_nil a).symm) | |
| subst ys | |
| rcases left with left | left | |
| · have xsEq := finiteWordGraph_injective a (left.trans (signedTwo l false true).symm) | |
| subst xs | |
| exact .cancel l false | |
| · have xsEq := finiteWordGraph_injective a (left.trans (signedTwo l true false).symm) | |
| subst xs | |
| exact .cancel l true | |
| · intro atomic | |
| cases atomic with | |
| | old_one => exact Or.inl ⟨s.encode_mem 1, oldOne 1, finiteWordGraph_nil a⟩ | |
| | old_mul g h => | |
| exact Or.inr (Or.inl ⟨s.encode g, s.encode_mem g, s.encode h, s.encode_mem h, | |
| s.encode (g * h), s.encode_mem (g * h), | |
| ((s.product s).pair_mem_graph s (fun p => p.1 * p.2) (g, h) (g * h)).2 rfl, | |
| oldTwo g h, oldOne (g * h)⟩) | |
| | cancel l b => | |
| refine Or.inr (Or.inr ⟨t.encode l, t.encode_mem l, ?_, finiteWordGraph_nil a⟩) | |
| cases b | |
| · exact Or.inl (signedTwo l false true) | |
| · exact Or.inr (signedTwo l true false) | |
| namespace TransitiveGround | |
| variable (M : TransitiveGround.{u}) | |
| variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| variable (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| variable (separation : SeparationSchema M.sets) | |
| variable (infinite : Realize M.sets infinity Fin.elim0) | |
| variable (hAlphabet : (externalGeneratorAlphabetPresentation s t).carrier ∈ M.sets) | |
| include pairing unions separation infinite hAlphabet | |
| theorem externalWordStepCode_iff (xs ys : List (J ⊕ (L × Bool))) : | |
| ExternalWordStepCode (M.finiteWords (externalGeneratorAlphabet s.carrier t.carrier)) | |
| (M.finiteWordAppendGraph (externalGeneratorAlphabet s.carrier t.carrier)) | |
| s.carrier t.carrier ((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1) | |
| (finiteWordGraph (externalGeneratorAlphabetPresentation s t) xs) | |
| (finiteWordGraph (externalGeneratorAlphabetPresentation s t) ys) ↔ | |
| ExternalWordRewrite.Step xs ys := by | |
| let a := externalGeneratorAlphabetPresentation s t | |
| have word (v : List (J ⊕ (L × Bool))) := | |
| finiteWordGraph_mem_finiteWords a M pairing unions separation infinite hAlphabet v | |
| have decode (v : ZFSet.{u}) := finiteWords_exact_lists a M pairing unions separation infinite hAlphabet v | |
| have append (v w : List (J ⊕ (L × Bool))) : | |
| ZFSet.pair (ZFSet.pair (finiteWordGraph a v) (finiteWordGraph a w)) | |
| (finiteWordGraph a (v ++ w)) ∈ M.finiteWordAppendGraph a.carrier := | |
| (M.pair_mem_finiteWordAppendGraph _ _ _ _).2 | |
| ⟨word v, word w, word (v ++ w), finiteWordGraph_append_code a v w⟩ | |
| have appendEq (v w z : List (J ⊕ (L × Bool))) | |
| (entry : ZFSet.pair (ZFSet.pair (finiteWordGraph a v) (finiteWordGraph a w)) | |
| (finiteWordGraph a z) ∈ M.finiteWordAppendGraph a.carrier) : z = v ++ w := | |
| finiteWordGraph_injective a ((finiteWordAppendCode_iff a v w _).1 | |
| ((M.pair_mem_finiteWordAppendGraph _ _ _ _).1 entry).2.2.2) | |
| constructor | |
| · rintro ⟨pre, hp, post, hq, l, hl, r, hr, pl, _, pr, _, atomic, left, right, direction⟩ | |
| obtain ⟨pre, rfl⟩ := (decode pre).1 hp | |
| obtain ⟨post, rfl⟩ := (decode post).1 hq | |
| obtain ⟨l, rfl⟩ := (decode l).1 hl | |
| obtain ⟨r, rfl⟩ := (decode r).1 hr | |
| have leftEq := (finiteWordAppendCode_iff a pre l pl).1 | |
| ((M.pair_mem_finiteWordAppendGraph _ _ _ _).1 left).2.2.2 | |
| have rightEq := (finiteWordAppendCode_iff a pre r pr).1 | |
| ((M.pair_mem_finiteWordAppendGraph _ _ _ _).1 right).2.2.2 | |
| subst pl pr | |
| refine ⟨pre, post, l, r, (externalWordAtomicCode_iff s t l r).1 atomic, ?_⟩ | |
| rcases direction with ⟨first, second⟩ | ⟨first, second⟩ | |
| · exact Or.inl ⟨appendEq _ _ _ first, appendEq _ _ _ second⟩ | |
| · exact Or.inr ⟨appendEq _ _ _ first, appendEq _ _ _ second⟩ | |
| · rintro ⟨pre, post, l, r, atomic, direction⟩ | |
| refine ⟨finiteWordGraph a pre, word pre, finiteWordGraph a post, word post, | |
| finiteWordGraph a l, word l, finiteWordGraph a r, word r, | |
| finiteWordGraph a (pre ++ l), word (pre ++ l), | |
| finiteWordGraph a (pre ++ r), word (pre ++ r), | |
| (externalWordAtomicCode_iff s t l r).2 atomic, append pre l, append pre r, ?_⟩ | |
| rcases direction with ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ | |
| · exact Or.inl ⟨append (pre ++ l) post, append (pre ++ r) post⟩ | |
| · exact Or.inr ⟨append (pre ++ l) post, append (pre ++ r) post⟩ | |
| theorem pair_mem_externalWordSteps (xs ys : List (J ⊕ (L × Bool))) : | |
| ZFSet.pair (finiteWordGraph (externalGeneratorAlphabetPresentation s t) xs) | |
| (finiteWordGraph (externalGeneratorAlphabetPresentation s t) ys) ∈ | |
| M.externalWordSteps s.carrier t.carrier ((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1) ↔ | |
| ExternalWordRewrite.Step xs ys := by | |
| rw [externalWordSteps, pair_mem_externalWordStepGraph, | |
| M.externalWordStepCode_iff s t pairing unions separation infinite hAlphabet xs ys] | |
| exact ⟨fun h => h.2.2, fun h => ⟨ | |
| finiteWordGraph_mem_finiteWords _ M pairing unions separation infinite hAlphabet xs, | |
| finiteWordGraph_mem_finiteWords _ M pairing unions separation infinite hAlphabet ys, h⟩⟩ | |
| end TransitiveGround | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |