openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /ExternalWordInverseEvaluation.lean
| import OAI.SetTheory.PartitionPrinciple.Models.GroundExternalWordQuotient | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.ExternalWordRewrite | |
| open Compression.RecursiveMaps.CosetGeometry | |
| universe u | |
| variable (J S : Type u) [Group J] | |
| theorem symbolValue_inverse (a : Alphabet J S) : | |
| symbolValue J S (externalSymbolInverse a) = (symbolValue J S a)⁻¹ := by | |
| cases a with | |
| | inl g => exact map_inv (old J S) g | |
| | inr p => | |
| change free J S (FreeGroup.mk [(p.1, !p.2)]) = (free J S (FreeGroup.mk [p]))⁻¹ | |
| rw [← map_inv, FreeGroup.inv_mk] | |
| rfl | |
| theorem evalWord_inverse (xs : List (Alphabet J S)) : | |
| evalWord J S ((xs.map externalSymbolInverse).reverse) = (evalWord J S xs)⁻¹ := by | |
| induction xs with | |
| | nil => exact (inv_one : (1 : External J S)⁻¹ = 1).symm | |
| | cons a xs ih => | |
| rw [List.map_cons, List.reverse_cons] | |
| have append (v w : List (Alphabet J S)) : | |
| evalWord J S (v ++ w) = evalWord J S v * evalWord J S w := | |
| (evalWord J S).map_mul v w | |
| rw [append, ih] | |
| have singleton : evalWord J S [externalSymbolInverse a] = (symbolValue J S a)⁻¹ := by | |
| exact symbolValue_inverse J S a | |
| have cons : evalWord J S (a :: xs) = symbolValue J S a * evalWord J S xs := by | |
| exact (evalWord J S).map_mul (FreeMonoid.of a) (FreeMonoid.ofList xs) | |
| rw [singleton, cons, mul_inv_rev] | |
| end PartitionPilot.Forcing.ExternalWordRewrite | |
| end | |
| end OAI | |