| import OAI.SetTheory.PartitionPrinciple.Geometry.ExternalTranslation |
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| namespace OAI |
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| section |
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| namespace PartitionPilot.Compression.RecursiveMaps.TileRequests.Family |
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|
| 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] |
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|
| 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) |
|
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| 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] |
|
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| end PartitionPilot.Compression.RecursiveMaps.TileRequests.Family |
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| end |
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| end OAI |
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|
