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fcab081 | 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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import Mathlib.Algebra.Group.Fin.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Prod
namespace OAI
/-! # Component Indices -/
noncomputable section
namespace HadwigerCounterexample
abbrev Tag := Fin
abbrev Comp (g : ℕ) := { p : Tag g × Tag g // p.1 < p.2 }
variable {g : ℕ}
def edge (l d : Tag g) (h : l ≠ d) : Comp g :=
if hl : l < d then ⟨(l,d),hl⟩ else ⟨(d,l),lt_of_le_of_ne (le_of_not_gt hl) h.symm⟩
theorem edge_eq {l d : Tag g} (h : l < d) : edge l d (ne_of_lt h) = ⟨(l,d),h⟩ := by
simp [edge,h]
theorem edge_symm {l d : Tag g} (h : l ≠ d) : edge l d h = edge d l h.symm := by
rcases lt_or_gt_of_ne h with e|e
· simp [edge,e,not_lt.mpr e.le]
· simp [edge,e,not_lt.mpr e.le]
def inc (e : Comp g) (l : Tag g) : Prop := e.val.1 = l ∨ e.val.2 = l
instance (e : Comp g) (l : Tag g) : Decidable (inc e l) := inferInstanceAs (Decidable (_ ∨ _))
theorem inc_edge_fst {l d : Tag g} (h : l ≠ d) : inc (edge l d h) l := by
by_cases e : l < d <;> simp [inc,edge,e]
theorem inc_edge_iff {l d x : Tag g} (h : l ≠ d) : inc (edge l d h) x ↔ (l=x ∨ d=x) := by
by_cases e : l < d <;> simp [inc,edge,e,or_comm]
abbrev TagStar (l : Tag g) := {e : Comp g // inc e l}
def starEquiv (l : Tag g) : {d : Tag g // l ≠ d} ≃ TagStar l := by
let f (d : {d : Tag g // l ≠ d}) : TagStar l := ⟨edge l d d.property,inc_edge_fst _⟩
apply Equiv.ofBijective f
constructor
· intro i j h
have hi : inc (edge l i i.property) i := (inc_edge_iff _).mpr (Or.inr rfl)
change inc (f i).val i at hi
rw [h] at hi
exact Subtype.ext (((inc_edge_iff j.property).mp hi).resolve_left i.property).symm
· rintro ⟨⟨⟨i,j⟩,hij⟩,he⟩
rcases he with g|g
· change i = l at g
subst l
refine ⟨⟨j,hij.ne⟩,Subtype.ext ?_⟩
exact edge_eq hij
· change j = l at g
subst l
refine ⟨⟨i,hij.ne.symm⟩,Subtype.ext ?_⟩
dsimp only [f]
rw [edge_symm,edge_eq hij]
theorem card_star (l : Tag g) : Fintype.card (TagStar l) = g - 1 := by
rw [← Fintype.card_congr (starEquiv l)]
exact (Fintype.card_subtype_compl (fun d => l=d)).trans (by simp)
/-- l in I_d. -/
def allowed (d l : Tag g) :=
0 < (l-d).val ∧ (l-d).val ≤ (g-1)/2
instance (d l : Tag g) : Decidable (allowed d l) :=
inferInstanceAs (Decidable (_ ∧ _))
theorem not_allowed (d : Tag g) : ¬allowed d d := by
have : NeZero g := ⟨by have := d.isLt; omega⟩
simp [allowed]
theorem card_allowed (d : Tag g) : Fintype.card {l : Tag g // allowed d l} = (g-1)/2 := by
have : NeZero g := ⟨by have := d.isLt; omega⟩
let T := {l : Tag g // allowed d l}
let f (i : Fin ((g-1)/2)) : Tag g := ⟨i.val+1,by have := i.isLt; omega⟩
have hf (i) : allowed d (f i + d) := by
have := i.isLt
simp only [allowed,add_sub_cancel_right,f]
omega
let h : Fin ((g-1)/2) → T := fun i => ⟨f i + d,hf i⟩
have hh : Function.Bijective h := by
constructor
· intro i j hij
have he : f i = f j := add_right_cancel (congrArg Subtype.val hij)
have he' := congrArg Fin.val he
exact Fin.ext (Nat.add_right_cancel he')
· intro l
have hl := l.property
unfold allowed at hl
let i : Fin ((g-1)/2) := ⟨(l.val-d).val-1,by omega⟩
have hi : f i = l.val-d := Fin.ext (by dsimp only [i,f]; omega)
refine ⟨i,Subtype.ext ?_⟩
change f i + d = _
rw [hi,sub_add_cancel]
simpa using (Fintype.card_congr (Equiv.ofBijective h hh)).symm
theorem card_allowed' (l : Tag g) :
Fintype.card {d : Tag g // allowed d l} = (g-1)/2 := by
have : NeZero g := ⟨by have := l.isLt; omega⟩
let h : {d : Tag g // allowed d l} ≃ {x : Tag g // allowed 0 x} := {
toFun := fun d => ⟨l-d.val, by simpa only [allowed,sub_zero] using d.property⟩
invFun := fun x => ⟨l-x.val, by simpa only [allowed,sub_sub_cancel,sub_zero] using
x.property⟩
left_inv := fun d => Subtype.ext (sub_sub_cancel _ _)
right_inv := fun x => Subtype.ext (sub_sub_cancel _ _) }
exact (Fintype.card_congr h).trans (card_allowed _)
end HadwigerCounterexample
end
end OAI
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