| 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) |
|
|
| / |
| 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 |
|
|
