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8e4d533 | 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 | import Mathlib
import OAI.Geometry.Cannon.PlanarGraphs.Next
namespace OAI
section
section
section
noncomputable section
namespace CannonCycleOrder
theorem eq_of_steps {n : ℕ} (hn : 0<n) {X : Type*} (z : Fin n → X)
(hstep : ∀ i,z i=z (next hn i)) (i j : Fin n) : z i=z j := by
have hc (k : ℕ) (hk : k<n) : z ⟨k,hk⟩=z ⟨0,hn⟩ := by
induction k with
| zero => rfl
| succ k ih =>
have hnx : next hn ⟨k,by omega⟩=(⟨k+1,hk⟩ : Fin n) := by
simp only [next,dite_eq_left hk]
have he : z (⟨k+1,hk⟩ : Fin n)=z ⟨k,by omega⟩ := by
rw [←hnx]
exact (hstep ⟨k,by omega⟩).symm
exact he.trans (ih (by omega))
exact (hc i.val i.isLt).trans (hc j.val j.isLt).symm
theorem eq_of_steps_except {n : ℕ} (hn : 0 < n) {X : Type*} (z : Fin n → X)
(o : Fin n) (hstep : ∀ i,i≠o → next hn i≠o → z i=z (next hn i))
{i j : Fin n} (hi : i≠o) (hj : j≠o) : z i=z j := by
let rot (k : ℕ) : Fin n := ⟨(o.val+1+k)%n,Nat.mod_lt _ hn⟩
have hrnext (k : ℕ) : rot (k+1)=next hn (rot k) := by
apply Fin.ext
rw [next_mod]
dsimp [rot]
rw [←Nat.add_assoc (o.val+1) k 1, Nat.add_mod (o.val+1+k) 1,
Nat.add_mod ((o.val+1+k)%n) 1, Nat.mod_mod]
have hrne (k : ℕ) (hk : k < n-1) : rot k≠o := by
intro hh
have he := congrArg Fin.val hh
change (o.val+1+k)%n=o.val at he
by_cases h : o.val+1+k < n
· rw [Nat.mod_eq_of_lt h] at he; omega
· have hb : o.val+1+k < 2*n := by omega
have hm : (o.val+1+k)%n=o.val+1+k-n := Nat.mod_eq_sub_mod (by omega) |>.trans (Nat.mod_eq_of_lt (by omega))
rw [hm] at he
omega
have hcover (i : Fin n) (hi : i≠o) : ∃ k,k < n-1 ∧ rot k=i := by
have hne : i.val≠o.val := fun h=>hi (Fin.ext h)
by_cases h : o.val < i.val
· refine ⟨i.val-o.val-1,by omega,?_⟩
apply Fin.ext
change (o.val+1+(i.val-o.val-1))%n=i.val
rw [show o.val+1+(i.val-o.val-1)=i.val by omega,Nat.mod_eq_of_lt i.isLt]
· refine ⟨n-o.val-1+i.val,by omega,?_⟩
apply Fin.ext
change (o.val+1+(n-o.val-1+i.val))%n=i.val
rw [show o.val+1+(n-o.val-1+i.val)=n+i.val by omega,Nat.add_mod]
simp [Nat.mod_eq_of_lt i.isLt]
have hconst (k : ℕ) (hk : k < n-1) : z (rot k)=z (rot 0) := by
induction k with
| zero => rfl
| succ k ih =>
rw [hrnext]
exact (hstep (rot k) (hrne k (by omega)) (by rw [←hrnext]; exact hrne (k+1) hk)).symm.trans (ih (by omega))
obtain ⟨a,ha,hra⟩ := hcover i hi
obtain ⟨b,hb,hrb⟩ := hcover j hj
rw [←hra,←hrb]
exact (hconst a ha).trans (hconst b hb).symm
end CannonCycleOrder
end
end
end
end
end OAI
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