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3277efa | 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 | 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
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