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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 96 97 98 99 100 101 102 103 | import Mathlib
import OAI.GroupTheory.BooneHigman.MonoidAlgebras
import OAI.GroupTheory.SimpleOvergroups.ContractedNormalForms
import OAI.GroupTheory.SimpleOvergroups.FirstFamilies
import OAI.GroupTheory.SimpleOvergroups.VariableCode
namespace OAI
namespace SimpleFInftyOvergroups.SemanticEncoding
noncomputable section
open scoped Classical
open FiniteAlgebraicEnvelopes Bootstrap
open ContextualEncoding
namespace SemanticProduct
variable {A K : Type*} [DecidableEq A] [MonoidWithZero K]
(P : NormalPresentation (FreeMonoid A) K)
abbrev Mon := FiniteAlgebraicEnvelopes.ZeroProduct.Mon K PartialAffine.monoid
def pairEval : FreeMonoid (ProductFamily.Letter A) →* K × PartialAffine.monoid where
toFun w := (P.q (FreeMonoid.ofList (Projection.left w.toList)),PartialAffine.wordValue (Projection.right w.toList))
map_one' := by simp only [FreeMonoid.toList_one, Projection.left, List.filterMap_nil,
FreeMonoid.ofList_nil, map_one]; rfl
map_mul' u v := by
apply Prod.ext
· simp only [FreeMonoid.toList_mul,Projection.left_append,FreeMonoid.ofList_append,map_mul]
rfl
· apply Subtype.ext
simp [PartialAffine.wordValue,PartialAffine.value]
def evaluation : FreeMonoid (ProductFamily.Letter A) →* Mon (K:=K) :=
(FiniteAlgebraicEnvelopes.ZeroProduct.pair K PartialAffine.monoid).comp (pairEval P)
omit [DecidableEq A] in
theorem pairEval_surjective : Function.Surjective (pairEval P) := by
rintro ⟨x,y⟩
obtain ⟨u,rfl⟩ := P.surjective x
obtain ⟨v,rfl⟩ := PartialAffine.wordValue_surjective y
refine ⟨FreeMonoid.ofList (u.toList.map Sum.inl++v.map Sum.inr),?_⟩
simp [pairEval,Projection.left,Projection.right,List.filterMap_map]
omit [DecidableEq A] in
theorem evaluation_surjective : Function.Surjective (evaluation P) :=
(FiniteAlgebraicEnvelopes.ZeroProduct.pair_surjective _ _).comp (pairEval_surjective P)
def normal (w : List (ProductFamily.Letter A)) : List A :=
(P.normal (FreeMonoid.ofList (Projection.left w))).toList
def value (w : List (ProductFamily.Letter A)) : Option (ProductFamily.Key A) :=
ProductFamily.combine P.omega.toList (normal P w,PartialAffine.testedClass (Projection.right w))
omit [DecidableEq A] in
theorem normal_eq_iff (u v : List (ProductFamily.Letter A)) : normal P u = normal P v ↔
P.q (FreeMonoid.ofList (Projection.left u)) = P.q (FreeMonoid.ofList (Projection.left v)) := by
rw [P.complete]
exact FreeMonoid.toList.injective.eq_iff
omit [DecidableEq A] in
theorem normal_eq_zero (w : List (ProductFamily.Letter A)) : normal P w = P.omega.toList ↔
P.q (FreeMonoid.ofList (Projection.left w)) = 0 := by
change (P.normal (FreeMonoid.ofList (Projection.left w))).toList = P.omega.toList ↔ _
rw [FreeMonoid.toList.injective.eq_iff,P.normal_eq_zero_iff]
theorem value_eq_iff (u v : List (ProductFamily.Letter A)) :
value P u = value P v ↔ evaluation P (FreeMonoid.ofList u) = evaluation P (FreeMonoid.ofList v) := by
change value P u = value P v ↔
FiniteAlgebraicEnvelopes.ZeroProduct.pair _ _ _ = FiniteAlgebraicEnvelopes.ZeroProduct.pair _ _ _
rw [FiniteAlgebraicEnvelopes.ZeroProduct.pair_eq_iff]
change value P u = value P v ↔
(P.q (FreeMonoid.ofList (Projection.left u)),PartialAffine.wordValue (Projection.right u)) =
(P.q (FreeMonoid.ofList (Projection.left v)),PartialAffine.wordValue (Projection.right v)) ∨
P.q (FreeMonoid.ofList (Projection.left u)) = 0 ∧ P.q (FreeMonoid.ofList (Projection.left v)) = 0
rw [Prod.ext_iff,←normal_eq_iff P,←normal_eq_zero P,←normal_eq_zero P]
have hp : PartialAffine.wordValue (Projection.right u) = PartialAffine.wordValue (Projection.right v) ↔
PartialAffine.testedClass (Projection.right u) = PartialAffine.testedClass (Projection.right v) := by
rw [PartialAffine.testedClass_eq_iff]
exact Subtype.ext_iff
rw [hp]
by_cases hu : normal P u = P.omega.toList <;>
by_cases hv : normal P v = P.omega.toList <;>
simp [value,ProductFamily.combine,hu,hv]
aesop
theorem value_eq_none (w : List (ProductFamily.Letter A)) :
value P w = none ↔ evaluation P (FreeMonoid.ofList w) = 0 := by
change value P w = none ↔ FiniteAlgebraicEnvelopes.ZeroProduct.pair _ _ _ = 0
rw [FiniteAlgebraicEnvelopes.ZeroProduct.pair_eq_zero]
change value P w = none ↔ P.q (FreeMonoid.ofList (Projection.left w))=0
rw [←normal_eq_zero P]
simp [value,ProductFamily.combine]
theorem full (f : List A → Part (List A))
(hf : ∀ w, (P.normal (FreeMonoid.ofList w)).toList ∈ f w)
(w : List (ProductFamily.Letter A)) : value P w ∈ ProductFamily.test P.omega.toList f w :=
(Part.mem_map_iff _).mpr ⟨_,hf _,rfl⟩
end SemanticProduct
end
end SimpleFInftyOvergroups.SemanticEncoding
end OAI
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