| import Mathlib |
| import OAI.Probability.LatticeSurface.Geometry.CapMatching |
|
|
| namespace OAI |
|
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| noncomputable section |
|
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| namespace ExtremalEdges |
|
|
| open Finset |
|
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| variable {V I : Type*} [Fintype V] [Fintype I] |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
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| local instance capOrientDecidableEq (carrier : Sort*) : DecidableEq carrier := |
| Classical.typeDecidableEq carrier |
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| variable {V I : Type*} [Fintype V] [Fintype I] |
|
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| omit [Fintype V] [Fintype I] in |
| theorem orient_matching (e : I → V × V) (pins : Set V) |
| (hne : ∀ i, (e i).1 ≠ (e i).2) (hp : ∀ i, ¬ ((e i).1 ∈ pins ∧ (e i).2 ∈ pins)) |
| (M : Finset I) (hM : (M : Set I).Pairwise (fun i k => Disjoint (edgeEnds e i) (edgeEnds e k))) : |
| ∃ (W : Finset V) (r : V → V), W.card = M.card ∧ |
| (∀ w ∈ W, w ∉ pins) ∧ (∀ w ∈ W, r w ∉ W) ∧ |
| ∀ w ∈ W, ∃ i ∈ M, (w = (e i).1 ∧ r w = (e i).2) ∨ |
| (w = (e i).2 ∧ r w = (e i).1) := by |
| let o (i : I) : V × V := if (e i).1 ∈ pins then ((e i).2,(e i).1) else e i |
| have hof (i : I) : (o i).1 ∉ pins := by |
| by_cases ht : (e i).1 ∈ pins |
| · simpa [o,ht] using (fun hh => hp i ⟨ht,hh⟩) |
| · simp [o,ht] |
| have hoo (i : I) : (o i).1 ≠ (o i).2 := by |
| by_cases ht : (e i).1 ∈ pins |
| · simpa [o,ht] using (hne i).symm |
| · simpa [o,ht] using hne i |
| have hoEnds (i : I) : edgeEnds o i = edgeEnds e i := by |
| by_cases ht : (e i).1 ∈ pins <;> simp [edgeEnds,o,ht,Finset.pair_comm] |
| have hMo : (M : Set I).Pairwise (fun i k => Disjoint (edgeEnds o i) (edgeEnds o k)) := by |
| simpa only [hoEnds] using hM |
| let f : M → V := fun i => (o i).1 |
| let g : M → V := fun i => (o i).2 |
| have hf : Function.Injective f := by |
| intro i k hik |
| apply Subtype.ext |
| by_contra hne' |
| have hiEnd : f i ∈ edgeEnds o i := by simp [f,edgeEnds] |
| have hkEnd : f i ∈ edgeEnds o k := by rw [hik]; simp [f,edgeEnds] |
| exact Finset.disjoint_left.mp (hMo i.2 k.2 hne') hiEnd hkEnd |
| let W := Finset.univ.image f |
| let r := Function.extend f g id |
| have hr (i : M) : r (f i) = g i := hf.extend_apply g id i |
| refine ⟨W,r,?_,?_,?_,?_⟩ |
| · simp [W,Finset.card_image_of_injective _ hf] |
| · intro w hw |
| obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hw |
| exact hof i |
| · intro w hw |
| obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hw |
| rw [hr] |
| intro hgw |
| obtain ⟨k,_,hk⟩ := Finset.mem_image.mp hgw |
| by_cases hik : i = k |
| · subst k |
| exact hoo i hk |
| · have hne' : (i : I) ≠ k := fun hh => hik (Subtype.ext hh) |
| have hd := hMo i.2 k.2 hne' |
| have hiEnd : g i ∈ edgeEnds o i := by simp [g,edgeEnds] |
| have hkEnd : g i ∈ edgeEnds o k := by rw [← hk]; simp [f,edgeEnds] |
| exact Finset.disjoint_left.mp hd hiEnd hkEnd |
| · intro w hw |
| obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp hw |
| refine ⟨i,i.2,?_⟩ |
| rw [hr] |
| by_cases ht : (e i).1 ∈ pins |
| · right; simp [f,g,o,ht] |
| · left; simp [f,g,o,ht] |
|
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| end ExtremalEdges |
|
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| end |
|
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| end OAI |
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|
