openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Configurations /PolygonStrandReconstruction.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonStringNormalization | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section PolygonStrandReconstruction | |
| open Classical CategoryTheory | |
| namespace PolygonObject | |
| variable {a n : ℕ} | |
| theorem representative_sub (u v : CutRing × CutRing) : | |
| orbitRepresentative (u-v)=orbitRepresentative u-orbitRepresentative v := by | |
| simp [orbitRepresentative,sub_mul] | |
| theorem trajectoryShift_translationString (U : PolygonObject a) | |
| (d : Fin (n+1) → Fin U.tracks → CutRing × CutRing) (x : U.Point) (i : Fin (n+1)) : | |
| trajectoryShift (translationString U d) ((shiftArrow U (d 0)).toEquiv x) i= | |
| orbitRepresentative (d i x.val.1-d 0 x.val.1) := by | |
| apply pointShift_unique | |
| change ((CategoryTheory.inv (shiftArrow U (d 0)) ≫ shiftArrow U (d i)).toEquiv | |
| ((shiftArrow U (d 0)).toEquiv x)).val.2 = _ | |
| simp only [arrow_comp_apply,inv_arrow_apply,Equiv.symm_apply_apply] | |
| simp only [shiftArrow_apply] | |
| rw [←translate_add,sub_add_cancel] | |
| theorem increments_translationString (U : PolygonObject a) | |
| (d : Fin (n+1) → Fin U.tracks → CutRing × CutRing) (x : U.Point) (i : Fin n) : | |
| increments (translationString U d) ((shiftArrow U (d 0)).toEquiv x) i= | |
| orbitRepresentative (d i.succ x.val.1-d i.castSucc x.val.1) := by | |
| unfold increments | |
| rw [trajectoryShift_translationString,trajectoryShift_translationString,←representative_sub] | |
| congr 1 | |
| abel | |
| noncomputable def incrementString (U : PolygonObject a) | |
| (γ : Fin U.tracks → Fin n → CutRing × CutRing) : RawString a n := | |
| translationString U (fun i j => Fin.partialSum (γ j) i) | |
| noncomputable def incrementInitial (U : PolygonObject a) | |
| (γ : Fin U.tracks → Fin n → CutRing × CutRing) : U ⟶ (incrementString U γ).obj 0 := | |
| shiftArrow U (fun j => Fin.partialSum (γ j) 0) | |
| theorem incrementInitial_positional (U : PolygonObject a) | |
| (γ : Fin U.tracks → Fin n → CutRing × CutRing) : Positional (incrementInitial U γ) := by | |
| intro x | |
| change translate a (Fin.partialSum (γ x.val.1) 0) x.val.2=x.val.2 | |
| rw [Fin.partialSum_zero,translate_zero] | |
| theorem incrementString_spec (U : PolygonObject a) | |
| (γ : Fin U.tracks → Fin n → CutRing × CutRing) (x : U.Point) (i : Fin n) : | |
| increments (incrementString U γ) ((incrementInitial U γ).toEquiv x) i= | |
| orbitRepresentative (γ x.val.1 i) := by | |
| change increments (translationString U (fun i j => Fin.partialSum (γ j) i)) | |
| ((shiftArrow U (fun j => Fin.partialSum (γ j) 0)).toEquiv x) i=_ | |
| rw [increments_translationString,Fin.partialSum_succ] | |
| congr 1 | |
| abel | |
| theorem incrementString_spec_reduced (U : PolygonObject a) | |
| (γ : Fin U.tracks → Fin n → CutRing × CutRing) | |
| (hγ : ∀j i,orbitRepresentative (γ j i)=γ j i) (x : U.Point) : | |
| increments (incrementString U γ) ((incrementInitial U γ).toEquiv x)=γ x.val.1 := by | |
| funext i | |
| rw [incrementString_spec,hγ] | |
| end PolygonObject | |
| end PolygonStrandReconstruction | |
| section PolygonIncrementFibers | |
| open Classical CategoryTheory | |
| namespace PolygonObject | |
| variable {a n : ℕ} | |
| theorem partialSum_increments (F : RawString a n) (x : (F.obj 0).Point) (i : Fin (n+1)) : | |
| Fin.partialSum (increments F x) i=trajectoryShift F x i := by | |
| induction i using Fin.induction with | |
| | zero => simp | |
| | succ i hi => | |
| rw [Fin.partialSum_succ,hi] | |
| unfold increments | |
| abel | |
| theorem trajectory_eq_of_increments {F G : RawString a n} (h : F.obj 0 ⟶ G.obj 0) | |
| (hp : Positional h) (he : ∀x,increments G (h.toEquiv x)=increments F x) | |
| (x : (F.obj 0).Point) : trajectory G (h.toEquiv x)=trajectory F x := by | |
| funext i | |
| unfold trajectory | |
| rw [pointShift_spec (baseArrow G i),pointShift_spec (baseArrow F i)] | |
| change translate a (trajectoryShift G (h.toEquiv x) i) (h.toEquiv x).val.2= | |
| translate a (trajectoryShift F x i) x.val.2 | |
| rw [hp x,←partialSum_increments,←partialSum_increments,he x] | |
| theorem extendLadder_positional_of_increments {F G : RawString a n} (h : F.obj 0 ⟶ G.obj 0) | |
| (hp : Positional h) (he : ∀x,increments G (h.toEquiv x)=increments F x) : | |
| (ladderProperty a n) (extendLadder h) := | |
| extendLadder_positional h (trajectory_eq_of_increments h hp he) | |
| noncomputable def incrementLadderEquiv (F G : StringGroupoid a n) : | |
| (F ⟶ G) ≃ {h : F.obj.obj 0 ⟶ G.obj.obj 0 // | |
| Positional h ∧ ∀x,increments G.obj (h.toEquiv x)=increments F.obj x} where | |
| toFun h := ⟨h.hom.app 0,h.property 0,ladder_preserves_increments h⟩ | |
| invFun h := ⟨extendLadder h.val, | |
| extendLadder_positional_of_increments h.val h.property.1 h.property.2⟩ | |
| left_inv h := by apply WideSubcategory.hom_ext; exact extendLadder_app_zero h.hom | |
| right_inv h := by apply Subtype.ext; exact extendLadder_zero h.val | |
| end PolygonObject | |
| end PolygonIncrementFibers | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |