| import OAI.Analysis.CircleDomains.Uniformization.CyclicCircularization | |
| namespace OAI | |
| /-! Exact restart identities for the deterministic finite circularization. -/ | |
| open Set Function Metric Filter | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity.FiniteJordanConfiguration | |
| variable {I:Type*} [Finite I] | |
| lemma iterate_add (C:FiniteJordanConfiguration I) (s:ℕ→I) (a b:ℕ) : | |
| iterate C s (a+b)=iterate (iterate C s a) (fun n => s (a+n)) b := by | |
| induction b with | |
| | zero => simp only [Nat.add_zero,iterate] | |
| | succ b ih => | |
| exact congrArg (fun D : FiniteJordanConfiguration I => | |
| (chosenStep D (s (a+b))).target) ih | |
| lemma stageMap_add (C:FiniteJordanConfiguration I) (s:ℕ→I) (a b:ℕ) : | |
| stageMap C s (a+b)=stageMap (iterate C s a) (fun n => s (a+n)) b := by | |
| exact congrArg (fun D : FiniteJordanConfiguration I => | |
| (chosenStep D (s (a+b))).map) (iterate_add C s a b) | |
| lemma cumulativeMap_add (C:FiniteJordanConfiguration I) (s:ℕ→I) (a b:ℕ) : | |
| cumulativeMap C s (a+b)=cumulativeMap (iterate C s a) (fun n => s (a+n)) b ∘ | |
| cumulativeMap C s a := by | |
| induction b with | |
| | zero => simp only [Nat.add_zero,cumulativeMap,id_comp] | |
| | succ b ih => | |
| change stageMap C s (a+b) ∘ cumulativeMap C s (a+b) = _ | |
| rw [stageMap_add C s a b,ih] | |
| rfl | |
| lemma cyclicSchedule_cycle_add (m n:ℕ) : cyclicSchedule m (m+1+n)=cyclicSchedule m n := by | |
| apply Fin.ext | |
| simp [cyclicSchedule] | |
| lemma iterate_cycle_add {m:ℕ} (C:FiniteJordanConfiguration (Fin (m+1))) (n:ℕ) : | |
| iterate C (cyclicSchedule m) (m+1+n)= | |
| iterate (iterate C (cyclicSchedule m) (m+1)) (cyclicSchedule m) n := by | |
| rw [iterate_add] | |
| congr 1 | |
| funext k | |
| exact cyclicSchedule_cycle_add m k | |
| lemma selectedCenter_cycle_add {m:ℕ} (C:FiniteJordanConfiguration (Fin (m+1))) (n:ℕ) : | |
| selectedCenter C (cyclicSchedule m) (m+1+n)= | |
| selectedCenter (iterate C (cyclicSchedule m) (m+1)) (cyclicSchedule m) n := by | |
| unfold selectedCenter | |
| rw [cyclicSchedule_cycle_add] | |
| exact congrArg (fun D : FiniteJordanConfiguration (Fin (m+1)) => | |
| (chosenStep D (cyclicSchedule m n)).center) (iterate_cycle_add C n) | |
| lemma selectedRadius_cycle_add {m:ℕ} (C:FiniteJordanConfiguration (Fin (m+1))) (n:ℕ) : | |
| selectedRadius C (cyclicSchedule m) (m+1+n)= | |
| selectedRadius (iterate C (cyclicSchedule m) (m+1)) (cyclicSchedule m) n := by | |
| unfold selectedRadius | |
| rw [cyclicSchedule_cycle_add] | |
| exact congrArg (fun D : FiniteJordanConfiguration (Fin (m+1)) => | |
| (chosenStep D (cyclicSchedule m n)).radius) (iterate_cycle_add C n) | |
| lemma visitTime_succ {m:ℕ} (n:ℕ) (i:Fin (m+1)) : | |
| visitTime (n+1) i=m+1+visitTime n i := by | |
| simp only [visitTime,Nat.add_mul,Nat.one_mul] | |
| omega | |
| end CircleDomainRigidity.FiniteJordanConfiguration | |
| end | |
| end OAI | |