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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.ResolvedBoolean
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section FinitePrimitiveResolution
variable {X ι : Type*}
theorem finite_resolving_subfamily (U : ι → Set X) {V : Set X}
(hV : V ∈ BooleanSubalgebra.closure (Set.range U)) :
∃ S : Finset ι, ResolvedBy (fun i : S => U i.val) V := by
classical
induction hV using BooleanSubalgebra.closure_bot_sup_induction with
| mem V h =>
obtain ⟨i,rfl⟩ := h
exact ⟨{i},fun x y he => he ⟨i,Finset.mem_singleton_self i⟩⟩
| bot => exact ⟨∅,fun _ _ _ => Iff.rfl⟩
| sup V W _ _ ihV ihW =>
obtain ⟨S,hS⟩ := ihV
obtain ⟨T,hT⟩ := ihW
refine ⟨S ∪ T,fun x y he => or_congr (hS x y ?_) (hT x y ?_)⟩
· intro i
exact he ⟨i.val,Finset.mem_union_left _ i.property⟩
· intro i
exact he ⟨i.val,Finset.mem_union_right _ i.property⟩
| compl V _ ih =>
obtain ⟨S,hS⟩ := ih
exact ⟨S,fun x y he => not_congr (hS x y he)⟩
noncomputable def translatedPrimitiveAlgebra (a : ℕ) (r : CutRing) :
BooleanSubalgebra (Set (GenericSquare a)) :=
BooleanSubalgebra.closure (Set.range fun p : Fin 5 × (CutRing × CutRing) =>
(spatialTranslate p.2 (initialTest a r p.1)).val)
theorem translatedPrimitiveAlgebra_test (a : ℕ) (r : CutRing)
(p : Fin 5 × (CutRing × CutRing)) :
(spatialTranslate p.2 (initialTest a r p.1)).val ∈ translatedPrimitiveAlgebra a r :=
BooleanSubalgebra.subset_closure ⟨p,rfl⟩
theorem translatedPrimitiveAlgebra_translate (a : ℕ) (r : CutRing)
(u : CutRing × CutRing) {V : Set (GenericSquare a)}
(hV : V ∈ translatedPrimitiveAlgebra a r) :
translate a u ⁻¹' V ∈ translatedPrimitiveAlgebra a r := by
induction hV using BooleanSubalgebra.closure_bot_sup_induction with
| mem V h =>
obtain ⟨p,rfl⟩ := h
have he : translate a u ⁻¹' (spatialTranslate p.2 (initialTest a r p.1)).val =
(spatialTranslate (p.2-u) (initialTest a r p.1)).val := by
ext x
simp only [spatialTranslate,Set.mem_preimage,← translate_add]
abel_nf
rw [he]
exact translatedPrimitiveAlgebra_test a r (p.1,p.2-u)
| bot => exact BooleanSubalgebra.bot_mem
| sup V W _ _ ihV ihW => exact BooleanSubalgebra.sup_mem ihV ihW
| compl V _ ih => exact BooleanSubalgebra.compl_mem ih
theorem translatedPrimitiveAlgebra_initial (a : ℕ) (r : CutRing) (j : Fin 5) :
(initialTest a r j).val ∈ translatedPrimitiveAlgebra a r := by
have h := translatedPrimitiveAlgebra_test a r (j,0)
have he : (spatialTranslate (0:CutRing × CutRing) (initialTest a r j)).val =
(initialTest a r j).val := by
ext p
simp [spatialTranslate,translate_zero]
rwa [he] at h
theorem polygon_finite_primitive_resolution {a : ℕ} (r : CutRing)
(hr : 0 < ordinary r ∧ ordinary r < 1/2) (V : polygonAlgebra a) :
∃ S : Finset (Fin 5 × (CutRing × CutRing)),
ResolvedBy (fun p : S => (spatialTranslate p.val.2 (initialTest a r p.val.1)).val) V.val := by
apply finite_resolving_subfamily (fun p : Fin 5 × (CutRing × CutRing) =>
(spatialTranslate p.2 (initialTest a r p.1)).val)
let B := translatedPrimitiveAlgebra a r
have ht := fun u U hU => translatedPrimitiveAlgebra_translate a r u (V := U) hU
have hcoord (j : Fin 2) (z : CutRing) :
halfPlane a (Fin.castLE (by omega) j) z ∈ B := by
apply coordinate_cuts_generated B j ht
fin_cases j
· exact translatedPrimitiveAlgebra_initial a r 1
· exact translatedPrimitiveAlgebra_initial a r 2
apply polygon_induction (P := fun U => U ∈ B) ?_ BooleanSubalgebra.bot_mem
(fun _ _ hU hV => BooleanSubalgebra.sup_mem hU hV)
(fun _ hU => BooleanSubalgebra.compl_mem hU) V.property
intro j z
fin_cases j
· exact hcoord 0 z
· exact hcoord 1 z
· exact slope_cuts_generated r hr B ht hcoord 2 (Or.inl rfl)
(translatedPrimitiveAlgebra_initial a r 3) z
· exact slope_cuts_generated r hr B ht hcoord 3 (Or.inr rfl)
(translatedPrimitiveAlgebra_initial a r 4) z
end FinitePrimitiveResolution
end SimpleAmenable
end
end
end OAI
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