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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 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundFiniteWordLaws
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem finiteWordMapGraph_injection
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(separation : SeparationSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{A B e : ZFSet.{u}} (hA : A ∈ M.sets) (hB : B ∈ M.sets)
(code : InjectionCode A B e) :
InjectionCode (M.finiteWords A) (M.finiteWords B) (M.finiteWordMapGraph A B e) := by
let s := SetMembers.presentation A
let t := SetMembers.presentation B
let map := s.decodeFunction t e code.function
have graph : s.graph t map = e := s.graph_decodeFunction t e code.function
rw [← graph]
change InjectionCode (M.finiteWords s.carrier) (M.finiteWords t.carrier)
(M.finiteWordMapGraph s.carrier t.carrier (s.graph t map))
rw [M.finiteWordMapGraph_eq_graph pairing unions separation infinite s t map hA hB]
exact (finiteWordPresentation s M pairing unions separation infinite hA).graph_injection
(finiteWordPresentation t M pairing unions separation infinite hB) (List.map map)
(List.map_injective_iff.2 code.decoded_injective)
theorem finiteWordMapGraph_ground_injection
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{A B e : ZFSet.{u}} (hA : A ∈ M.sets) (hB : B ∈ M.sets) (he : e ∈ M.sets)
(code : InjectionCode A B e) :
M.finiteWordMapGraph A B e ∈ M.sets ∧
InjectionCode (M.finiteWords A) (M.finiteWords B) (M.finiteWordMapGraph A B e) :=
⟨M.finiteWordMapGraph_mem pairing unions powers separation infinite hA hB he,
M.finiteWordMapGraph_injection pairing unions separation infinite hA hB code⟩
theorem groundEmbeddable_finiteWords
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{A B : ZFSet.{u}} (hA : A ∈ M.sets) (hB : B ∈ M.sets)
(injection : M.groundEmbeddable A B) :
M.groundEmbeddable (M.finiteWords A) (M.finiteWords B) := by
obtain ⟨e, he, code⟩ := injection
exact ⟨M.finiteWordMapGraph A B e,
M.finiteWordMapGraph_ground_injection pairing unions powers separation infinite hA hB he code⟩
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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