@@ -30,7 +30,6 @@ open SimpleGraph
3030
3131variable {V : Type } {G : SimpleGraph V}
3232
33-
3433@[category undergraduate, AMS 5]
3534lemma completeGraphIsClique (s : Finset V) : (⊤ : SimpleGraph V).IsClique s :=
3635 Pairwise.set_pairwise (fun _ _ a ↦ a) _
@@ -40,7 +39,7 @@ variable [Fintype V]
4039@[category undergraduate, AMS 5]
4140lemma completeGraph_cliqueSet :
4241 (⊤ : SimpleGraph V).cliqueSet (Fintype.card V) = {Set.univ.toFinset} := by
43- simp only [SimpleGraph. cliqueSet, SimpleGraph. isNClique_iff ⊤, completeGraphIsClique, true_and,
42+ simp only [cliqueSet, isNClique_iff ⊤, completeGraphIsClique, true_and,
4443 Set.toFinset_univ]
4544 exact (Set.Sized.univ_mem_iff fun ⦃x⦄ a ↦ a).mp rfl
4645
@@ -71,9 +70,9 @@ The triangle is an example with 3 vertices satisfying the condition.
7170theorem triangle_locallyLinear_and_nonEdgesAreDiagonals : (completeGraph (Fin 3 )).LocallyLinear ∧
7271 NonEdgesAreDiagonals (completeGraph (Fin 3 )) := by
7372 constructor
74- · simp [SimpleGraph. LocallyLinear]
73+ · simp [LocallyLinear]
7574 constructor
76- · simp only [SimpleGraph. EdgeDisjointTriangles, Set.Pairwise]
75+ · simp only [EdgeDisjointTriangles, Set.Pairwise]
7776 intro x hx y hy hxy
7877 have := @completeGraph_cliqueSet (Fin 3 ) _
7978 rw [Fintype.card_fin] at this
@@ -101,15 +100,14 @@ theorem conway9_nonEdgesAreDiagonals : NonEdgesAreDiagonals Conway9 := by
101100 exact Fintype.ofFinite ↑(Conway9.neighborSet i)
102101 have : ∀ i j, ((Conway9.neighborFinset i) ∩ Conway9.neighborFinset j).card =
103102 (Conway9.neighborSet i ∩ Conway9.neighborSet j).ncard := by
104- simp only [SimpleGraph. neighborFinset]
103+ simp only [neighborFinset]
105104 intros
106105 rw [← Set.toFinset_inter, Set.ncard_eq_toFinset_card']
107106 simp only [← this]
108107 intro x y
109108 have ⟨x1, x2⟩ := x
110109 have ⟨y1, y2⟩ := y
111- simp only [Conway9, SimpleGraph.completeGraph_eq_top,
112- SimpleGraph.boxProd_adj, SimpleGraph.top_adj, SimpleGraph.neighborFinset_boxProd]
110+ simp only [Conway9, completeGraph_eq_top, boxProd_adj, top_adj, neighborFinset_boxProd]
113111 fin_cases x1 <;> fin_cases x2 <;> fin_cases y1 <;> fin_cases y2 <;> decide
114112
115113@[category API, AMS 5]
@@ -121,9 +119,9 @@ lemma completeGraph_boxProd_completeGraph_cliqueSet :
121119
122120@[category test, AMS 5]
123121theorem conway9_locallyLinear : Conway9.LocallyLinear := by
124- dsimp [SimpleGraph. LocallyLinear]
122+ dsimp [LocallyLinear]
125123 constructor
126- · simp only [SimpleGraph. EdgeDisjointTriangles, Set.Pairwise]
124+ · simp only [EdgeDisjointTriangles, Set.Pairwise]
127125 intro x hx y hy hxy
128126 simp only [Conway9, completeGraph_boxProd_completeGraph_cliqueSet] at hx hy
129127 rcases hx with hx | hx <;>
@@ -142,15 +140,15 @@ theorem conway9_locallyLinear : Conway9.LocallyLinear := by
142140 intro h
143141 use {(x1, x2), (y1, y2), (x1 + x1 + y1 + y1, x2 + x2 + y2 + y2)}
144142 simp only [Finset.mem_insert, Prod.mk.injEq, true_or, or_true, and_true]
145- simp only [Conway9, SimpleGraph.completeGraph_eq_top, SimpleGraph.boxProd_adj,
146- SimpleGraph.top_adj, ne_eq] at h ⊢
143+ simp only [Conway9, completeGraph_eq_top, boxProd_adj, top_adj, ne_eq] at h ⊢
147144 constructor
148- · dsimp [SimpleGraph. IsClique, ]
145+ · dsimp [IsClique]
149146 fin_cases x1 <;> fin_cases x2 <;> fin_cases y1 <;> fin_cases y2 <;>
150147 simp only [Fin.reduceFinMk, not_true_eq_false, Fin.reduceEq, or_true, Fin.reduceAdd,
151- Finset.coe_insert, Finset.coe_singleton, SimpleGraph.isClique_insert,
152- Set.pairwise_singleton, Set.mem_singleton_iff, ne_eq, SimpleGraph.boxProd_adj,
153- SimpleGraph.top_adj, forall_eq, Prod.mk.injEq, and_false, imp_self, Set.mem_insert_iff,forall_eq_or_imp, or_false, and_true, not_false_eq_true] at h ⊢
148+ Finset.coe_insert, Finset.coe_singleton, isClique_insert,
149+ Set.pairwise_singleton, Set.mem_singleton_iff, ne_eq, boxProd_adj,
150+ top_adj, forall_eq, Prod.mk.injEq, and_false, imp_self, Set.mem_insert_iff,
151+ forall_eq_or_imp, or_false, and_true, not_false_eq_true] at h ⊢
154152 · fin_cases x1 <;> fin_cases x2 <;> fin_cases y1 <;> fin_cases y2 <;>
155153 simp only [not_true_eq_false, or_self, and_true] at h <;>
156154 decide
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