| import OAI.RingTheory.DirectFiniteness.HoleExistence |
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|
| namespace OAI |
|
|
| universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 |
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|
| noncomputable section |
| open scoped Classical BigOperators |
| namespace KaplanskyCounterexample.FiniteGeometry |
|
|
| /-- In a cyclic sequence of at most five positions with unequal neighboring |
| labels, no label can occur thrice. This includes the wrap-around adjacency. -/ |
| lemma cyclic_label_count_le_two {k : ℕ} {L : Type u_1} [DecidableEq L] (lab : Fin k → L) |
| (hk : k ≤ 5) (hadj : ∀ i, lab i ≠ lab (finRotate k i)) (l : L) : |
| ((Finset.univ : Finset (Fin k)).filter (fun i => lab i = l)).card ≤ 2 := by |
| let s := (Finset.univ : Finset (Fin k)).filter (fun i => lab i = l) |
| have hd : Disjoint s (s.image (finRotate k)) := by |
| rw [Finset.disjoint_left] |
| intro i hi hir |
| obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hir |
| exact hadj j ((Finset.mem_filter.mp hj).2.trans (Finset.mem_filter.mp hi).2.symm) |
| have hc : s.card + (s.image (finRotate k)).card ≤ k := by |
| rw [← Finset.card_union_of_disjoint hd] |
| exact (Finset.card_le_card (Finset.subset_univ _)).trans_eq (Fintype.card_fin k) |
| rw [Finset.card_image_of_injective _ (finRotate k).injective] at hc |
| dsimp [s] at hc |
| omega |
|
|
| /-- A nonzero vector cannot be the sole contribution to a zero sum. -/ |
| lemma count_ne_one_of_sum_eq_zero {I : Type u_2} {L : Type u_3} {M : Type u_4} [Fintype I] [DecidableEq L] |
| [AddCommGroup M] (lab : I → L) (v : I → M) (l : L) |
| (hn : ∀ i, lab i = l → v i ≠ 0) |
| (hz : ∑ i ∈ (Finset.univ : Finset I).filter (fun i => lab i = l), v i = 0) : |
| ((Finset.univ : Finset I).filter (fun i => lab i = l)).card ≠ 1 := by |
| intro hc |
| obtain ⟨i,hi⟩ := Finset.card_eq_one.mp hc |
| have hi' : i ∈ (Finset.univ : Finset I).filter (fun i => lab i = l) := by rw [hi]; simp |
| have hil : lab i = l := (Finset.mem_filter.mp hi').2 |
| rw [hi,Finset.sum_singleton] at hz |
| exact hn i hil hz |
|
|
| lemma cyclic_label_count_eq_two {k : ℕ} {L : Type u_5} {M : Type u_6} [DecidableEq L] [AddCommGroup M] |
| (lab : Fin k → L) (v : Fin k → M) (hk : k ≤ 5) |
| (hadj : ∀ i, lab i ≠ lab (finRotate k i)) (l : L) |
| (hne : ∃ i, lab i = l) (hn : ∀ i, lab i = l → v i ≠ 0) |
| (hz : ∑ i ∈ (Finset.univ : Finset (Fin k)).filter (fun i => lab i = l), v i = 0) : |
| ((Finset.univ : Finset (Fin k)).filter (fun i => lab i = l)).card = 2 := by |
| have hle := cyclic_label_count_le_two lab hk hadj l |
| have hne1 := count_ne_one_of_sum_eq_zero lab v l hn hz |
| have hp : 0 < ((Finset.univ : Finset (Fin k)).filter (fun i => lab i = l)).card := by |
| rw [Finset.card_pos] |
| obtain ⟨i,hi⟩ := hne |
| exact ⟨i,by simp [hi]⟩ |
| omega |
|
|
| /-- Occurrence-count form of the pigeonhole bound, retaining the exact support. -/ |
| lemma sum_label_counts {I : Type u_7} {L : Type u_8} [Fintype I] [DecidableEq L] (lab : I → L) : |
| ∑ l ∈ Finset.univ.image lab, |
| ((Finset.univ : Finset I).filter (fun i => lab i = l)).card = Fintype.card I := by |
| symm |
| simpa using Finset.card_eq_sum_card_fiberwise (s := (Finset.univ : Finset I)) |
| (t := Finset.univ.image lab) (f := lab) (fun i hi => Finset.mem_image_of_mem lab hi) |
|
|
| end KaplanskyCounterexample.FiniteGeometry |
| end |
|
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| end OAI |
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|
