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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Configurations.ReserveAtom
import OAI.GroupTheory.SimpleAmenable.CentralCovers.InitialCoverSystem
import OAI.GroupTheory.SimpleAmenable.CentralCovers.PrimitiveCopies
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section CommonTranslatedCopies
structure CommonFrame (a : ℕ) (r : CutRing) (m : ℕ) (hm : 2 ≤ m)
(I : Finset (Fin (m+1))) (u : CutRing × CutRing) where
k : Multiplicative (FreeAbelianGroup (Fin m × Fin 2))
d : Fin (m+1) → CutRing × CutRing
projection : sourceLatticeFullMap a r m hm k = trackTranslation d
common : ∀ i ∈ I, d i = u
noncomputable def commonFrame (a : ℕ) (r : CutRing) (m : ℕ) (hm : 2 ≤ m)
(I : Finset (Fin (m+1))) (b : Fin (m+1)) (hb : b ∉ I) (u : CutRing × CutRing) :
CommonFrame a r m hm I u := by
let h := sourceLatticeFullMap_prescribed a r m hm I b hb (fun _ => u)
exact {
k := h.choose
d := h.choose_spec.choose
projection := h.choose_spec.choose_spec.1
common := h.choose_spec.choose_spec.2 }
noncomputable def spatialTranslate {a : ℕ} (u : CutRing × CutRing)
(U : polygonAlgebra a) : polygonAlgebra a :=
⟨translate a (-u) ⁻¹' U.val,polygon_preimage_translate (-u) U.property⟩
namespace CommonFrame
variable {a m : ℕ} {r : CutRing} {hm : 2 ≤ m}
{I : Finset (Fin (m+1))} {u : CutRing × CutRing}
(F : CommonFrame a r m hm I u)
theorem conditional_projection (U : polygonAlgebra a) (s : alternatingGroup I) :
sourceLatticeMap a r m hm F.k * conditionalAlternatingHom U (subtypeAlternatingHom I s) *
(sourceLatticeMap a r m hm F.k)⁻¹ =
conditionalAlternatingHom (spatialTranslate u U) (subtypeAlternatingHom I s) := by
apply Subtype.ext
change sourceLatticeFullMap a r m hm F.k * conditionalHom U (subtypeAlternatingHom I s).val *
(sourceLatticeFullMap a r m hm F.k)⁻¹ = _
rw [F.projection]
apply (conditional_translation_conjugate U (subtypeAlternatingHom I s).val F.d u ?_).symm
intro i hi
exact F.common i ((Equiv.Perm.mem_range_ofSubtype_iff.mp ⟨s.val,rfl⟩)
(Equiv.Perm.mem_support.mpr hi))
end CommonFrame
namespace InitialCoverSystem
variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m}
(B : InitialCoverSystem a r m hm M)
theorem commonTranslatedAlphabet_projection {I : Finset (Fin (m+1))}
{u : CutRing × CutRing} (F : CommonFrame a r m hm I u) (j : Fin 5) :
(coverMap M (alternatingGenerator a r m hm)).comp (B.translatedAlphabet I j F.k) =
(conditionalAlternatingHom (spatialTranslate u (initialTest a r j))).comp
(subtypeAlternatingHom I) := by
apply MonoidHom.ext
intro s
change coverMap M (alternatingGenerator a r m hm)
(B.t F.k * B.initialConditional j (subtypeAlternatingHom I s) * (B.t F.k)⁻¹) = _
rw [map_mul,map_mul,map_inv]
rw [show coverMap M (alternatingGenerator a r m hm) (B.t F.k) =
sourceLatticeMap a r m hm F.k from DFunLike.congr_fun B.t_projection F.k]
rw [show coverMap M (alternatingGenerator a r m hm)
(B.initialConditional j (subtypeAlternatingHom I s)) =
conditionalAlternatingHom (initialTest a r j) (subtypeAlternatingHom I s) from
DFunLike.congr_fun (B.initialConditional_projection j) (subtypeAlternatingHom I s)]
exact F.conditional_projection _ s
theorem commonTranslatedAlphabet_independent {I : Finset (Fin (m+1))}
{u v : CutRing × CutRing} (F : CommonFrame a r m hm I u)
(G : CommonFrame a r m hm I v) (j : Fin 5)
(huv : spatialTranslate u (initialTest a r j) = spatialTranslate v (initialTest a r j)) :
B.translatedAlphabet I j F.k = B.translatedAlphabet I j G.k := by
apply B.initialAlphabet_orbit_independent
intro s
rw [F.conditional_projection,G.conditional_projection,huv]
noncomputable def primitiveCopy (I : Finset (Fin (m+1)))
(b : Fin (m+1)) (hb : b ∉ I) (p : Fin 5 × (CutRing × CutRing)) :
alternatingGroup I →* BoundedRelationCover M (alternatingGenerator a r m hm) :=
B.translatedAlphabet I p.1 (commonFrame a r m hm I b hb p.2).k
theorem primitiveCopy_projection (I : Finset (Fin (m+1)))
(b : Fin (m+1)) (hb : b ∉ I) (p : Fin 5 × (CutRing × CutRing)) :
(coverMap M (alternatingGenerator a r m hm)).comp (B.primitiveCopy I b hb p) =
(conditionalAlternatingHom (spatialTranslate p.2 (initialTest a r p.1))).comp
(subtypeAlternatingHom I) :=
B.commonTranslatedAlphabet_projection (commonFrame a r m hm I b hb p.2) p.1
theorem primitiveCopy_aligned (I : Finset (Fin (m+1)))
(b : Fin (m+1)) (hb : b ∉ I) (p : Fin 5 × (CutRing × CutRing)) (s : alternatingGroup I) :
B.primitiveCopy I b hb p s ∈ sourceAlignedGroup a r m hm M B.t I :=
B.translatedAlphabet_aligned I p.1 _ s
end InitialCoverSystem
end CommonTranslatedCopies
end SimpleAmenable
end
end
end OAI
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