| import OAI.RingTheory.DirectFiniteness.RibbonDetachment |
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| namespace OAI |
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| universe u_1 |
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| noncomputable section |
| open scoped Classical |
| open scoped BigOperators |
| open Equiv Equiv.Perm |
| namespace KaplanskyCounterexample.Planar |
| open Equiv Equiv.Perm |
| variable {α : Type u_1} [Fintype α] |
| |
| omit [Fintype α] in |
| lemma Ribbon.face_step (R : Ribbon α) (x : α) : R.components.r x (R.face x) := |
| R.components.iseqv.trans (R.black_step x) (R.white_step (R.black x)) |
|
|
| lemma Ribbon.sameCycle_face_components (R : Ribbon α) {x y : α} (h : R.face.SameCycle x y) : |
| R.components.r x y := setoid_sameCycle_of_step R.components R.face R.face_step h |
|
|
| lemma Ribbon.detach_components_card (R : Ribbon α) (a : α) |
| (hb : R.black a ≠ a) (hw : R.white a ≠ a) : |
| Nat.card (Quotient (R.detach a).components) = Nat.card (Quotient R.components) + |
| if (R.detach a).components.r (R.black a) (R.white a) then 1 else 2 := by |
| let s := (R.detach a).components |
| let t := joinPair s a (R.black a) |
| have hb' : ¬ s.r a (R.black a) := fun h => hb (R.detach_isolated a h) |
| have hw' : ¬ s.r a (R.white a) := fun h => hw (R.detach_isolated a h) |
| have ht : Nat.card (Quotient t) + 1 = Nat.card (Quotient s) := joinPair_card s _ _ hb' |
| have heq : R.components = joinPair t a (R.white a) := R.components_detach a |
| by_cases h : s.r (R.black a) (R.white a) |
| · have h' : t.r a (R.white a) := (joinPair_left_iff s _ _ _).mpr (Or.inr h) |
| rw [joinPair_eq_of_related _ h'] at heq |
| change Nat.card (Quotient s) = Nat.card (Quotient R.components) + if s.r (R.black a) (R.white a) then 1 else 2 |
| simp only [h, ite_true, heq] |
| omega |
| · have h' : ¬ t.r a (R.white a) := by |
| rw [joinPair_left_iff] |
| exact not_or.mpr ⟨hw', h⟩ |
| have ht' := joinPair_card t a (R.white a) h' |
| rw [← heq] at ht' |
| change Nat.card (Quotient s) = Nat.card (Quotient R.components) + if s.r (R.black a) (R.white a) then 1 else 2 |
| simp only [h, ite_false] |
| omega |
|
|
| lemma Ribbon.detach_components_card_black_fixed (R : Ribbon α) (a : α) |
| (hb : R.black a = a) (hw : R.white a ≠ a) : |
| Nat.card (Quotient (R.detach a).components) = Nat.card (Quotient R.components) + 1 := by |
| have hn : ¬ (R.detach a).components.r a (R.white a) := fun h => hw (R.detach_isolated a h) |
| have heq := R.components_detach a |
| rw [hb, joinPair_eq_of_related _ ((R.detach a).components.iseqv.refl a)] at heq |
| have hc := joinPair_card (R.detach a).components a (R.white a) hn |
| rw [← heq] at hc |
| omega |
|
|
| lemma Ribbon.detach_components_card_white_fixed (R : Ribbon α) (a : α) |
| (hb : R.black a ≠ a) (hw : R.white a = a) : |
| Nat.card (Quotient (R.detach a).components) = Nat.card (Quotient R.components) + 1 := by |
| have hn : ¬ (R.detach a).components.r a (R.black a) := fun h => hb (R.detach_isolated a h) |
| have heq := R.components_detach a |
| rw [hw, joinPair_eq_of_related _ ((joinPair (R.detach a).components a (R.black a)).iseqv.refl a)] at heq |
| have hc := joinPair_card (R.detach a).components a (R.black a) hn |
| rw [← heq] at hc |
| omega |
|
|
| omit [Fintype α] in |
| lemma Ribbon.detach_face_black_fixed (R : Ribbon α) (a : α) (hb : R.black a = a) : |
| (R.detach a).face = unplug R.face a := by |
| rw [R.detach_face] |
| have hf : R.face a = R.white a := by simp [Ribbon.face, hb] |
| simp [hf, unplug] |
| rfl |
|
|
| omit [Fintype α] in |
| lemma Ribbon.detach_face_white_fixed (R : Ribbon α) (a : α) (hw : R.white a = a) : |
| (R.detach a).face = unplug R.face a := by |
| rw [R.detach_face, hw] |
| simp [unplug] |
| rfl |
|
|
| lemma Ribbon.detach_face_card_separating (R : Ribbon α) (a : α) |
| (hb : R.black a ≠ a) (hw : R.white a ≠ a) |
| (hsep : ¬ (R.detach a).components.r (R.black a) (R.white a)) : |
| Nat.card (Cycles (R.detach a).face) = Nat.card (Cycles R.face) + 2 := by |
| have hf : R.face a ≠ a := by |
| intro hf |
| have hs : (R.detach a).white (R.black a) = R.white a := by |
| change swap a (R.white a) (R.face a) = R.white a |
| simp [hf] |
| exact hsep (hs ▸ (R.detach a).white_step (R.black a)) |
| have hs : (R.detach a).white (R.black a) = R.face a := by |
| change swap a (R.white a) (R.face a) = R.face a |
| exact swap_apply_of_ne_of_ne hf (R.white.injective.ne hb) |
| have hnot : ¬ (R.detach a).face.SameCycle (R.white a) (R.face a) := by |
| intro h |
| have hc := (R.detach a).sameCycle_face_components h |
| have hs' := (R.detach a).white_step (R.black a) |
| rw [hs] at hs' |
| exact hsep ((R.detach a).components.iseqv.trans hs' |
| ((R.detach a).components.iseqv.symm hc)) |
| have hcard := swap_mul_cycles_card (R.detach a).face (R.white a) (R.face a) hnot |
| have heq : swap (R.white a) (R.face a) * (R.detach a).face = unplug R.face a := by |
| rw [R.detach_face_unplug a hw hf] |
| simp [← mul_assoc] |
| rw [heq, unplug_cycles_card R.face a hf] at hcard |
| omega |
|
|
| /-- Edge deletion preserves genus zero, including when deletion separates a |
| component or removes a vertex of degree one. The detached edge is represented |
| as an isolated one-edge component until restriction discards it. -/ |
| theorem Ribbon.genusZero_detach (R : Ribbon α) (hR : R.GenusZero) (a : α) : |
| (R.detach a).GenusZero := by |
| by_cases hb : R.black a = a |
| · have hblack : (R.detach a).black = R.black := by |
| simp [Ribbon.detach, unplug, hb] |
| rfl |
| by_cases hw : R.white a = a |
| · have heq : R.detach a = R := by |
| cases R |
| simp_all [Ribbon.detach, unplug] |
| rwa [heq] |
| · have hwc := unplug_cycles_card R.white a hw |
| have hcc := R.detach_components_card_black_fixed a hb hw |
| have hf : R.face a ≠ a := by simpa [Ribbon.face, hb] using hw |
| have hfc := unplug_cycles_card R.face a hf |
| rw [← R.detach_face_black_fixed a hb] at hfc |
| change Nat.card (Cycles (R.detach a).white) = _ at hwc |
| unfold Ribbon.GenusZero at hR ⊢ |
| rw [hblack, hwc, hfc, hcc] |
| simp only [Nat.mul_add, Nat.mul_one] |
| repeat rw [← Nat.add_assoc] |
| rw [Nat.add_right_comm (Nat.card (Cycles R.black) + Nat.card (Cycles R.white)) |
| 1 (Nat.card (Cycles R.face))] |
| exact congrArg (· + 2) hR |
| · have hbc := unplug_cycles_card R.black a hb |
| change Nat.card (Cycles (R.detach a).black) = _ at hbc |
| by_cases hw : R.white a = a |
| · have hwhite : (R.detach a).white = R.white := by |
| simp [Ribbon.detach, unplug, hw] |
| rfl |
| have hcc := R.detach_components_card_white_fixed a hb hw |
| have hf : R.face a ≠ a := by |
| intro hf |
| apply hb |
| apply R.white.injective |
| simpa [Ribbon.face, hw] using hf |
| have hfc := unplug_cycles_card R.face a hf |
| rw [← R.detach_face_white_fixed a hw] at hfc |
| unfold Ribbon.GenusZero at hR ⊢ |
| rw [hwhite, hbc, hfc, hcc] |
| simp only [Nat.mul_add, Nat.mul_one] |
| repeat rw [← Nat.add_assoc] |
| rw [Nat.add_right_comm (Nat.card (Cycles R.black)) 1 (Nat.card (Cycles R.white)), |
| Nat.add_right_comm (Nat.card (Cycles R.black) + Nat.card (Cycles R.white)) |
| 1 (Nat.card (Cycles R.face))] |
| exact congrArg (· + 2) hR |
| · have hwc := unplug_cycles_card R.white a hw |
| change Nat.card (Cycles (R.detach a).white) = _ at hwc |
| have hcc := R.detach_components_card a hb hw |
| by_cases hs : (R.detach a).components.r (R.black a) (R.white a) |
| · rw [ite_eq_left hs] at hcc |
| have hfc := R.detach_face_card_le a hw |
| have hbound := (R.detach a).euler_le |
| unfold Ribbon.GenusZero at hR ⊢ |
| apply le_antisymm hbound |
| rw [hbc, hwc, hcc] |
| simp only [Nat.mul_add, Nat.mul_one] |
| repeat rw [← Nat.add_assoc] |
| rw [Nat.add_right_comm (Nat.card (Cycles R.black)) 1 (Nat.card (Cycles R.white)), |
| Nat.add_right_comm (Nat.card (Cycles R.black) + Nat.card (Cycles R.white) + 1) |
| 1 (Nat.card (Cycles (R.detach a).face)), |
| Nat.add_right_comm (Nat.card (Cycles R.black) + Nat.card (Cycles R.white)) |
| 1 (Nat.card (Cycles (R.detach a).face)), ← hR] |
| exact Nat.add_le_add_right (Nat.add_le_add_left hfc _) 2 |
| · rw [ite_eq_right hs] at hcc |
| have hfc := R.detach_face_card_separating a hb hw hs |
| unfold Ribbon.GenusZero at hR ⊢ |
| rw [hbc, hwc, hfc, hcc] |
| simp only [Nat.mul_add] |
| repeat rw [← Nat.add_assoc] |
| rw [Nat.add_right_comm (Nat.card (Cycles R.black)) 1 (Nat.card (Cycles R.white)), |
| Nat.add_right_comm (Nat.card (Cycles R.black) + Nat.card (Cycles R.white) + 1) |
| 1 (Nat.card (Cycles R.face)), |
| Nat.add_right_comm (Nat.card (Cycles R.black) + Nat.card (Cycles R.white)) |
| 1 (Nat.card (Cycles R.face))] |
| exact congrArg (· + 4) hR |
|
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| end KaplanskyCounterexample.Planar |
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| end |
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| end OAI |
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