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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.Geometry.Cannon.PlanarGraphs.Next
namespace OAI
section
section
section
noncomputable section
namespace CannonCycleOrder
def cutOrder {n : ℕ} (hn : 0<n) (o : Fin n) (k : ℕ) : Fin n :=
⟨(o.val+1+k)%n,Nat.mod_lt _ hn⟩
lemma cutOrder_next {n : ℕ} (hn : 0<n) (o : Fin n) (k : ℕ) :
cutOrder hn o (k+1)=next hn (cutOrder hn o k) := by
apply Fin.ext
rw [next_mod]
dsimp [cutOrder]
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]
lemma cutOrder_last {n : ℕ} (hn : 0<n) (o : Fin n) : cutOrder hn o (n-1)=o := by
apply Fin.ext
change (o.val+1+(n-1))%n=o.val
rw [show o.val+1+(n-1)=n+o.val by omega,Nat.add_mod]
simp [Nat.mod_eq_of_lt o.isLt]
lemma cutOrder_first {n : ℕ} (hn : 0<n) (o : Fin n) : cutOrder hn o 0=next hn o := by
apply Fin.ext
rw [next_mod]
simp [cutOrder]
lemma cutOrder_ne {n : ℕ} (hn : 0<n) (o : Fin n) {k : ℕ} (hk : k<n-1) :
cutOrder hn o 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 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
lemma cutOrder_inj {n : ℕ} (hn : 0<n) (o : Fin n) {k l : ℕ} (hk : k<n) (hl : l<n)
(he : cutOrder hn o k=cutOrder hn o l) : k=l := by
have hh := congrArg Fin.val he
change (o.val+1+k)%n=(o.val+1+l)%n at hh
have hm (j : ℕ) (hj : j<n) (hh : n≤o.val+1+j) :
(o.val+1+j)%n=o.val+1+j-n := Nat.mod_eq_sub_mod hh |>.trans (Nat.mod_eq_of_lt (by omega))
by_cases h1 : o.val+1+k<n <;> by_cases h2 : o.val+1+l<n
· rw [Nat.mod_eq_of_lt h1,Nat.mod_eq_of_lt h2] at hh; omega
· rw [Nat.mod_eq_of_lt h1,hm l hl (by omega)] at hh; omega
· rw [hm k hk (by omega),Nat.mod_eq_of_lt h2] at hh; omega
· rw [hm k hk (by omega),hm l hl (by omega)] at hh; omega
lemma cutOrder_cover {n : ℕ} (hn : 0<n) (o i : Fin n) (hi : i≠o) :
∃ k,k<n-1 ∧ cutOrder hn o 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]
lemma sum_cutOrder {n : ℕ} (hn : 0<n) (o : Fin n) (f : Fin n → ℝ) (ho : f o=0) :
∑ k∈Finset.range (n-1),f (cutOrder hn o k)=∑ i,f i := by
classical
have he : (∑ k∈Finset.range (n-1),f (cutOrder hn o k))=∑ i∈Finset.univ.erase o,f i := by
apply Finset.sum_bij (fun k _=>cutOrder hn o k)
· intro k hk; exact Finset.mem_erase.mpr ⟨cutOrder_ne hn o (Finset.mem_range.mp hk),Finset.mem_univ _⟩
· intro k hk l hl he
exact cutOrder_inj hn o (by have := Finset.mem_range.mp hk; omega) (by have := Finset.mem_range.mp hl; omega) he
· intro i hi
obtain ⟨k,hk,he⟩:=cutOrder_cover hn o i (Finset.mem_erase.mp hi).1
exact ⟨k,Finset.mem_range.mpr hk,he⟩
· intro k hk; rfl
rw [he]
have hh:=Finset.sum_erase_add Finset.univ f (Finset.mem_univ o)
rw [ho,add_zero] at hh
exact hh
end CannonCycleOrder
end
end
end
end
end OAI
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