File size: 3,322 Bytes
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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.SetTheory.PartitionPrinciple.Geometry.ExternalTranslation
namespace OAI
section
namespace PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
open Attachment CosetGeometry
universe u
variable {A J R : Type u} [Group J] {P : MeetOrder A}
variable (f : Family P R) (a : OrderAction J P)
theorem translate_one (z : f.Node a) : f.translate a 1 z = z := by
rcases f.copy_cover a z with ⟨w, x, rfl⟩
rw [f.translate_copy, one_mul]
theorem translate_mul (g h : External J f.Letters) (z : f.Node a) :
f.translate a (g * h) z = f.translate a g (f.translate a h z) := by
rcases f.copy_cover a z with ⟨w, x, rfl⟩
rw [f.translate_copy, f.translate_copy, f.translate_copy, mul_assoc]
theorem translate_left_inverse (g : External J f.Letters) (z : f.Node a) :
f.translate a g⁻¹ (f.translate a g z) = z := by
rw [← f.translate_mul, inv_mul_cancel, f.translate_one]
theorem translate_right_inverse (g : External J f.Letters) (z : f.Node a) :
f.translate a g (f.translate a g⁻¹ z) = z := by
rw [← f.translate_mul, mul_inv_cancel, f.translate_one]
theorem translate_monotone (g : External J f.Letters) (x y : f.Node a)
(h : (f.gluedOrder a).le x y) :
(f.gluedOrder a).le (f.translate a g x) (f.translate a g y) := by
rcases f.comparison_in_copy a x y h with ⟨w, b, c, rfl, rfl, hbc⟩
rw [f.translate_copy, f.translate_copy]
exact ((f.copy a (g * w)).le_iff b c).mpr hbc
theorem translate_le_iff (g : External J f.Letters) (x y : f.Node a) :
(f.gluedOrder a).le (f.translate a g x) (f.translate a g y) ↔ (f.gluedOrder a).le x y := by
constructor
· intro h
have back := f.translate_monotone a g⁻¹ _ _ h
rw [f.translate_left_inverse, f.translate_left_inverse] at back
exact back
· exact f.translate_monotone a g x y
noncomputable def externalIso (g : External J f.Letters) : MeetIso (f.gluedOrder a) (f.gluedOrder a) :=
isoOfOrder (f.translate a g) (f.translate a g⁻¹)
(f.translate_left_inverse a g) (f.translate_right_inverse a g) (f.translate_le_iff a g)
noncomputable def externalAction : OrderAction (External J f.Letters) (f.gluedOrder a) where
iso := f.externalIso a
one := f.translate_one a
mul := f.translate_mul a
theorem externalAction_copy (g w : External J f.Letters) (x : A) :
((f.externalAction a).iso g).toFun ((f.copy a w).toFun x) =
(f.copy a (g * w)).toFun x := f.translate_copy a g w x
theorem externalAction_meet (g : External J f.Letters) (x y : f.Node a) :
((f.externalAction a).iso g).toFun ((f.gluedOrder a).meet x y) =
(f.gluedOrder a).meet (((f.externalAction a).iso g).toFun x)
(((f.externalAction a).iso g).toFun y) :=
((f.externalAction a).iso g).map_meet x y
theorem extends_old_action (j : J) (x : A) :
((f.externalAction a).iso (old J f.Letters j)).toFun ((f.copy a 1).toFun x) =
(f.copy a 1).toFun ((a.iso j).toFun x) := by
rw [f.externalAction_copy, mul_one]
simpa only [one_mul] using f.copy_coordinate a 1 j x
theorem generator_on_root (s : f.Letters) (x : A) :
((f.externalAction a).iso (generator J f.Letters s)).toFun ((f.copy a 1).toFun x) =
(f.copy a (generator J f.Letters s)).toFun x := by
rw [f.externalAction_copy, mul_one]
end PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
end
end OAI
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