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e3e0a0d | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | import OAI.SetTheory.PartitionPrinciple.Forcing.ExternalWordInverseEvaluation
import OAI.SetTheory.PartitionPrinciple.Models.GroundQuotientBinaryOperations
import OAI.SetTheory.PartitionPrinciple.Models.GroundQuotientUnaryOperations
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
open Compression.RecursiveMaps.CosetGeometry
universe u
variable (M : TransitiveGround.{u})
variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0)
variable (unions : Realize M.sets SetFormula.union Fin.elim0)
variable (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
variable (infinite : Realize M.sets infinity Fin.elim0)
variable {J L : Type u} [Group J] (s : Presentation J) (t : Presentation L)
variable (hG : s.carrier ∈ M.sets) (hS : t.carrier ∈ M.sets)
noncomputable def externalGroupPresentation : Presentation (External J L) :=
(M.externalWordQuotientPresentation pairing unions powers separation infinite s t hG hS).reindex
(ExternalWordRewrite.quotientEquiv J L).symm.toEquiv
theorem externalGroupPresentation_carrier :
(M.externalGroupPresentation pairing unions powers separation infinite s t hG hS).carrier =
equivalenceClasses (M.finiteWords (externalGeneratorAlphabet s.carrier t.carrier))
(M.externalWordEquivalence s.carrier t.carrier
((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1)) := rfl
theorem externalGroupPresentation_encode_evalWord (xs : List (J ⊕ (L × Bool))) :
(M.externalGroupPresentation pairing unions powers separation infinite s t hG hS).encode
(ExternalWordRewrite.evalWord J L xs) =
equivalenceClass (M.finiteWords (externalGeneratorAlphabet s.carrier t.carrier))
(M.externalWordEquivalence s.carrier t.carrier
((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1))
(finiteWordGraph (externalGeneratorAlphabetPresentation s t) xs) := by
change (M.externalWordQuotientPresentation pairing unions powers separation infinite s t hG hS).encode
(ExternalWordRewrite.fromExternal J L (ExternalWordRewrite.evalWord J L xs)) = _
rw [ExternalWordRewrite.fromExternal_evalWord]
exact M.externalWordQuotientPresentation_encode_mk pairing unions powers separation infinite s t hG hS xs
theorem externalGroupClassifier_eq_graph :
equivalenceClassifier (M.finiteWords (externalGeneratorAlphabet s.carrier t.carrier))
(M.externalWordEquivalence s.carrier t.carrier
((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1)) =
(finiteWordPresentation (externalGeneratorAlphabetPresentation s t) M pairing unions separation infinite
(M.externalGeneratorAlphabet_mem pairing unions powers separation hG hS)).graph
(M.externalGroupPresentation pairing unions powers separation infinite s t hG hS)
(ExternalWordRewrite.evalWord J L) := by
let words := finiteWordPresentation (externalGeneratorAlphabetPresentation s t) M
pairing unions separation infinite
(M.externalGeneratorAlphabet_mem pairing unions powers separation hG hS)
let quotient := M.externalWordQuotientPresentation pairing unions powers separation infinite s t hG hS
refine (M.externalWordClassifier_eq_graph pairing unions powers separation infinite s t hG hS).trans ?_
refine Eq.trans ?_ (quotient.graph_reindex
(ExternalWordRewrite.quotientEquiv J L).symm.toEquiv words (ExternalWordRewrite.evalWord J L)).symm
exact congrArg (words.graph quotient)
(funext fun xs => (ExternalWordRewrite.fromExternal_evalWord J L xs).symm)
theorem externalGroupPresentation_ground
(hm : (s.product s).graph s (fun p => p.1 * p.2) ∈ M.sets) :
(M.externalGroupPresentation pairing unions powers separation infinite s t hG hS).carrier ∈ M.sets ∧
(finiteWordPresentation (externalGeneratorAlphabetPresentation s t) M pairing unions separation infinite
(M.externalGeneratorAlphabet_mem pairing unions powers separation hG hS)).graph
(M.externalGroupPresentation pairing unions powers separation infinite s t hG hS)
(ExternalWordRewrite.evalWord J L) ∈ M.sets := by
have ground := M.externalWordQuotient_ground pairing unions powers separation infinite s t hG hS hm
exact ⟨ground.1, (M.externalGroupClassifier_eq_graph pairing unions powers separation infinite s t hG hS) ▸ ground.2⟩
end PartitionPilot.Forcing.TransitiveGround
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
open Compression.RecursiveMaps.CosetGeometry
universe u
variable (M : TransitiveGround.{u})
noncomputable def externalGroupCarrier (G S m one : ZFSet.{u}) : ZFSet.{u} :=
equivalenceClasses (M.finiteWords (externalGeneratorAlphabet G S))
(M.externalWordEquivalence G S m one)
noncomputable def externalGroupProjection (G S m one : ZFSet.{u}) : ZFSet.{u} :=
equivalenceClassifier (M.finiteWords (externalGeneratorAlphabet G S))
(M.externalWordEquivalence G S m one)
noncomputable def externalGroupMultiplication (G S m one : ZFSet.{u}) : ZFSet.{u} :=
quotientBinaryGraph (M.finiteWords (externalGeneratorAlphabet G S))
(M.externalGroupCarrier G S m one) (M.externalGroupProjection G S m one)
(M.finiteWordAppendGraph (externalGeneratorAlphabet G S))
noncomputable def externalGroupInverse (G S m inv one : ZFSet.{u}) : ZFSet.{u} :=
quotientUnaryGraph (M.finiteWords (externalGeneratorAlphabet G S))
(M.externalGroupCarrier G S m one) (M.externalGroupProjection G S m one)
(M.externalInverseWordGraph G S inv)
noncomputable def externalGroupIdentity (G S m one : ZFSet.{u}) : ZFSet.{u} :=
equivalenceClass (M.finiteWords (externalGeneratorAlphabet G S))
(M.externalWordEquivalence G S m one) ∅
variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0)
variable (unions : Realize M.sets SetFormula.union Fin.elim0)
variable (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
variable (infinite : Realize M.sets infinity Fin.elim0)
include pairing unions powers separation infinite in
theorem externalGroupOperations_ground {G S m inv one : ZFSet.{u}}
(hG : G ∈ M.sets) (hS : S ∈ M.sets) (hm : m ∈ M.sets)
(hInv : inv ∈ M.sets) (hOne : one ∈ M.sets) :
M.externalGroupCarrier G S m one ∈ M.sets ∧
M.externalGroupMultiplication G S m one ∈ M.sets ∧
M.externalGroupInverse G S m inv one ∈ M.sets ∧
M.externalGroupIdentity G S m one ∈ M.sets ∧
M.externalGroupProjection G S m one ∈ M.sets := by
have hAlphabet := M.externalGeneratorAlphabet_mem pairing unions powers separation hG hS
have hWords := M.finiteWords_mem pairing unions powers separation infinite hAlphabet
have hE := M.externalWordEquivalence_mem pairing unions powers separation infinite hG hS hm hOne
have hQ := M.equivalenceClasses_mem powers separation hWords hE
have hProjection := M.equivalenceClassifier_mem pairing unions powers separation hWords hE
exact ⟨hQ,
M.quotientBinaryGraph_mem pairing unions powers separation hWords hQ hProjection
(M.finiteWordAppendGraph_mem pairing unions powers separation infinite hAlphabet),
M.quotientUnaryGraph_mem pairing unions powers separation hWords hQ hProjection
(M.externalInverseWordGraph_mem pairing unions powers separation infinite hG hS hInv),
M.equivalenceClass_mem separation hWords hE (M.empty_mem ⟨G, hG⟩), hProjection⟩
variable {J L : Type u} [Group J] (s : Presentation J) (t : Presentation L)
variable (hG : s.carrier ∈ M.sets) (hS : t.carrier ∈ M.sets)
theorem externalGroupMultiplication_eq_graph :
let q := M.externalGroupPresentation pairing unions powers separation infinite s t hG hS
M.externalGroupMultiplication s.carrier t.carrier
((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1) =
(q.product q).graph q (fun p => p.1 * p.2) := by
dsimp only
unfold externalGroupMultiplication externalGroupProjection
rw [M.externalGroupClassifier_eq_graph pairing unions powers separation infinite s t hG hS]
have hAlphabet := M.externalGeneratorAlphabet_mem pairing unions powers separation hG hS
have append := M.finiteWordAppendGraph_eq_graph pairing unions separation infinite
(externalGeneratorAlphabetPresentation s t) hAlphabet
change M.finiteWordAppendGraph (externalGeneratorAlphabet s.carrier t.carrier) = _ at append
rw [append]
exact quotientBinaryGraph_eq_graph
(finiteWordPresentation (externalGeneratorAlphabetPresentation s t) M pairing unions separation infinite hAlphabet)
(M.externalGroupPresentation pairing unions powers separation infinite s t hG hS)
(ExternalWordRewrite.evalWord J L) List.append
(fun x y => x * y) (ExternalWordRewrite.evalWord_surjective J L)
(fun x y => (ExternalWordRewrite.evalWord J L).map_mul x y)
theorem externalGroupInverse_eq_graph :
let q := M.externalGroupPresentation pairing unions powers separation infinite s t hG hS
M.externalGroupInverse s.carrier t.carrier
((s.product s).graph s (fun p => p.1 * p.2)) (s.graph s Inv.inv) (s.encode 1) =
q.graph q Inv.inv := by
dsimp only
unfold externalGroupInverse externalGroupProjection
rw [M.externalGroupClassifier_eq_graph pairing unions powers separation infinite s t hG hS]
have hAlphabet := M.externalGeneratorAlphabet_mem pairing unions powers separation hG hS
rw [M.externalInverseWordGraph_eq_graph pairing unions separation infinite s t hAlphabet]
exact quotientUnaryGraph_eq_graph
(finiteWordPresentation (externalGeneratorAlphabetPresentation s t) M pairing unions separation infinite hAlphabet)
(M.externalGroupPresentation pairing unions powers separation infinite s t hG hS)
(ExternalWordRewrite.evalWord J L) externalWordInverse Inv.inv
(ExternalWordRewrite.evalWord_surjective J L) (ExternalWordRewrite.evalWord_inverse J L)
theorem externalGroupIdentity_eq_encode :
M.externalGroupIdentity s.carrier t.carrier
((s.product s).graph s (fun p => p.1 * p.2)) (s.encode 1) =
(M.externalGroupPresentation pairing unions powers separation infinite s t hG hS).encode 1 := by
have unit : ExternalWordRewrite.evalWord J L [] = 1 := (ExternalWordRewrite.evalWord J L).map_one
have encoded := M.externalGroupPresentation_encode_evalWord pairing unions powers separation infinite
s t hG hS []
rw [unit, finiteWordGraph_nil] at encoded
exact encoded.symm
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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