File size: 2,804 Bytes
a97875e | 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 | 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
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