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[@DecidableRel.{u + 1, u + 1} V V (@SimpleGraph.Adj.{u} V H)] →\n        @DecidableRel.{u + 1, u + 1} V V\n          (@SimpleGraph.Adj.{u} V (@SDiff.sdiff.{u} (SimpleGraph.{u} V) (@SimpleGraph.sdiff.{u} V) G 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[@DecidableRel.{u + 1, u + 1} V V (@SimpleGraph.Adj.{u} V H)] →\n        @DecidableRel.{u + 1, u + 1} V V\n          (@SimpleGraph.Adj.{u} V (@Max.max.{u} (SimpleGraph.{u} V) (@SimpleGraph.instMax.{u} V) G 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(@Top.top.{u} (SimpleGraph.{u} V)\n          (@BooleanAlgebra.toTop.{u} (SimpleGraph.{u} V)\n            (@CompleteBooleanAlgebra.toBooleanAlgebra.{u} (SimpleGraph.{u} V)\n              (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n                (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V))))))","statement_sha256":"16bad8021fd7b316427b56d2e815e52212562ce9aa53025122c392460cdb9450","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-b04aae30bd7f3dee758e767b2f63f05fa6ae0a50afb4397ab1d74cb76b06d93a","qualified_name":"SimpleGraph.adj_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":6488,"column":16,"line":170},"start":{"byte":6416,"column":0,"line":169}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) (u v : V), Iff (@SimpleGraph.Adj.{u} V G u v) (@SimpleGraph.Adj.{u} V G v 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Iff (@SimpleGraph.Adj.{u} V G u v) (@SimpleGraph.Adj.{u} V G w x)","statement_sha256":"d02cf30792415a931984d232076304bd2b11af4b81101c4c4a5e09e926d3a419","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ff7f35ac884c998097ddab2a412f0d984891e989e3befa0df6c266a6b5559a2b","qualified_name":"SimpleGraph.adj_iff_exists_edge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":21863,"column":25,"line":594},"start":{"byte":21289,"column":0,"line":584}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G : SimpleGraph.{u} V} {v w : V},\n  Iff (@SimpleGraph.Adj.{u} V G v w)\n    (And (@Ne.{u + 1} V v w)\n      (∃ e,\n        And\n          (@Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V))\n            (@SimpleGraph.edgeSet.{u} V G) e)\n          (And\n            (@Membership.mem.{u, u} V (Sym2.{u} V)\n              (@SetLike.instMembership.{u, u} (Sym2.{u} V) V (@Sym2.instSetLike.{u} V)) e v)\n            (@Membership.mem.{u, u} V (Sym2.{u} V)\n              (@SetLike.instMembership.{u, u} (Sym2.{u} V) V (@Sym2.instSetLike.{u} V)) e w))))","statement_sha256":"329131561c3bc07489ab31d31661755c8584a76411692e1d47942b15a5a764da","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e092527b10eb633c1c090c64f676458ec4097b8343dcde2e26d9f8138881afa0","qualified_name":"SimpleGraph.adj_iff_exists_edge_coe","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":22024,"column":70,"line":597},"start":{"byte":21865,"column":0,"line":596}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G : SimpleGraph.{u} V} {a b : V},\n  Iff (@SimpleGraph.Adj.{u} V G a b)\n    (∃ e,\n      @Eq.{u + 1} (Sym2.{u} V)\n        (@Subtype.val.{u + 1} (Sym2.{u} V)\n          (fun x =>\n            @Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V))\n              (@SimpleGraph.edgeSet.{u} V G) x)\n          e)\n        (@Sym2.mk.{u} V a b))","statement_sha256":"2c890ac2ead9e2ca58a531b44e93d63e984ef84363a3d5364361d4df8a107f4d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4d814a1f300c90baa165aa7e695926d5ad4d8318961f534ee9095ce9f8dd2b88","qualified_name":"SimpleGraph.adj_incidenceSet_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":32172,"column":49,"line":875},"start":{"byte":31701,"column":0,"line":867}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) {v : V} {e : Sym2.{u} V},\n  @Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V))\n      (@SimpleGraph.edgeSet.{u} V G) e →\n    ∀\n      (h :\n        @Membership.mem.{u, u} V (Sym2.{u} V) (@SetLike.instMembership.{u, u} (Sym2.{u} V) V (@Sym2.instSetLike.{u} V))\n          e v),\n      @Eq.{u + 1} (Set.{u} (Sym2.{u} V))\n        (@Inter.inter.{u} (Set.{u} (Sym2.{u} V)) (@Set.instInter.{u} (Sym2.{u} V)) (@SimpleGraph.incidenceSet.{u} V G v)\n          (@SimpleGraph.incidenceSet.{u} V G (@Sym2.Mem.other.{u} V v e h)))\n        (@Singleton.singleton.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instSingletonSet.{u} (Sym2.{u} V)) e)","statement_sha256":"9745125e0919a1f6c6b4c49d74f149a35d177af2cd4a16e59cd6faf3700a3831","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a02ea597c8f09bd451ff58a88782b4a020b116122bb0aa553c52659d8f3be9ae","qualified_name":"SimpleGraph.adj_inj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":7277,"column":22,"line":198},"start":{"byte":7180,"column":0,"line":196}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G H : SimpleGraph.{u} V},\n  Iff (@Eq.{u + 1} (V → V → Prop) (@SimpleGraph.Adj.{u} V G) (@SimpleGraph.Adj.{u} V H))\n    (@Eq.{u + 1} (SimpleGraph.{u} V) G H)","statement_sha256":"d2746679b353f3324780bedd5789c9178c516cc07c21987f09ee1e6f6b47fb95","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-78b87352ccc7059643a6e1dd922ba7bf54338fab0cc4080d0870fc2975d17538","qualified_name":"SimpleGraph.adj_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":7178,"column":28,"line":194},"start":{"byte":7070,"column":0,"line":193}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u}, @Function.Injective.{u + 1, u + 1} (SimpleGraph.{u} V) (V → V → Prop) (@SimpleGraph.Adj.{u} V)","statement_sha256":"bcaa57369255036f33002d22b5ee0a0a7401e921983d82b40ec64f6ba95debca","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-dd00a1570e4e4b87e795792b8854dfd0e5e03f663d168a5e288a1c935dcea03a","qualified_name":"SimpleGraph.adj_of_mem_incidenceSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":29365,"column":87,"line":798},"start":{"byte":29104,"column":0,"line":795}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) {a b : V} {e : Sym2.{u} V},\n  @Ne.{u + 1} V a b →\n    @Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V))\n        (@SimpleGraph.incidenceSet.{u} V G a) e →\n      @Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V))\n          (@SimpleGraph.incidenceSet.{u} V G b) e →\n        @SimpleGraph.Adj.{u} V G a b","statement_sha256":"1bd25572da9c0b2ca7b1995caab4216c8c8b804e11ddeab64af5b2d44de85305","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e349384a7e57b97002df1148f73ef5c5d20ff00216fdd6e811ad43c111957776","qualified_name":"SimpleGraph.adj_symm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":6565,"column":19,"line":174},"start":{"byte":6490,"column":0,"line":172}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) {u v : V}, @SimpleGraph.Adj.{u} V G u v → @SimpleGraph.Adj.{u} V G v u","statement_sha256":"b26eaf40253871e8ec12c467622ab39d7389485fd990af77b083d08a0b77cadd","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-949b328181252d4a98006f874fbb49b292330cfe63ba78188b0d8a7e08ec79fa","qualified_name":"SimpleGraph.bot_adj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":12891,"column":9,"line":343},"start":{"byte":12794,"column":0,"line":341}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (v w : V),\n  Iff\n    (@SimpleGraph.Adj.{u} V\n      (@Bot.bot.{u} (SimpleGraph.{u} V)\n        (@BooleanAlgebra.toBot.{u} 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False","statement_sha256":"08ccf1da51f906b9f39717b1433898d79e5f28e6d2b3bcf26aaa1d01f0720e67","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-9b6b792b95d012924a895b91e0995f62fba9f19e3317969fb1e866869bee79fa","qualified_name":"SimpleGraph.card_neighborSet_union_compl_neighborSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":33055,"column":63,"line":895},"start":{"byte":32675,"column":0,"line":891}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} [inst : Fintype.{u} V] (G : SimpleGraph.{u} V) (v : V)\n  [inst_1 :\n    Fintype.{u}\n      (@Set.Elem.{u} V\n        (@Union.union.{u} (Set.{u} V) (@Set.instUnion.{u} V) (@SimpleGraph.neighborSet.{u} V G v)\n          (@SimpleGraph.neighborSet.{u} V (@Compl.compl.{u} (SimpleGraph.{u} V) (@SimpleGraph.instCompl.{u} V) G) v)))],\n  @Eq.{1} Nat\n    (@Finset.card.{u} V\n      (@Set.toFinset.{u} V\n        (@Union.union.{u} (Set.{u} V) (@Set.instUnion.{u} V) (@SimpleGraph.neighborSet.{u} V G v)\n          (@SimpleGraph.neighborSet.{u} V (@Compl.compl.{u} (SimpleGraph.{u} V) (@SimpleGraph.instCompl.{u} V) G) v))\n        inst_1))\n    (@HSub.hSub.{0, 0, 0} Nat Nat Nat (@instHSub.{0} Nat instSubNat) (@Fintype.card.{u} V inst)\n      (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))","statement_sha256":"63b1953a0bc9f86e31d3f014836b64b039c97e4478972af0f39a26dd48140d2e","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-962e51e829171c5fff0c8184b9986dac84d90b7e3a5ca17d0828831899da1f79","qualified_name":"SimpleGraph.commonNeighbors","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":33880,"column":37,"line":922},"start":{"byte":33662,"column":0,"line":919}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{V : Type u} → SimpleGraph.{u} V → V → V → Set.{u} 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(@Inter.inter.{u} (Set.{u} V) (@Set.instInter.{u} V) (@SimpleGraph.neighborSet.{u} V G v)\n      (@SimpleGraph.neighborSet.{u} V G 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SimpleGraph.{u} V) (v w : V),\n  @LE.le.{u} (Set.{u} V) (@Set.instLE.{u} V) (@SimpleGraph.commonNeighbors.{u} V G v w)\n    (@SimpleGraph.neighborSet.{u} V G w)","statement_sha256":"0405f9c59f19a9f60ff6bf99ceea79f11815274950382f897a925f41e39386ea","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ea85153fcc25fee754b80ab4544cac8077eb05f01b7744b2be42686658f77976","qualified_name":"SimpleGraph.commonNeighbors_symm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":34213,"column":20,"line":931},"start":{"byte":34103,"column":0,"line":930}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) (v w : V),\n  @Eq.{u + 1} (Set.{u} V) (@SimpleGraph.commonNeighbors.{u} V G v w) (@SimpleGraph.commonNeighbors.{u} V G w v)","statement_sha256":"a98acd5670be5430ff6676347c6030f8ca8aebd6096595789e44f584d5574daf","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-91956134b83889773380bca386a177c33bee6b6649ae19d35c4861c65ceacc08","qualified_name":"SimpleGraph.commonNeighbors_top_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":35343,"column":41,"line":960},"start":{"byte":35179,"column":0,"line":957}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {v w : V},\n  @Eq.{u + 1} (Set.{u} V)\n    (@SimpleGraph.commonNeighbors.{u} V\n      (@Top.top.{u} (SimpleGraph.{u} V)\n        (@BooleanAlgebra.toTop.{u} (SimpleGraph.{u} V)\n          (@CompleteBooleanAlgebra.toBooleanAlgebra.{u} (SimpleGraph.{u} V)\n            (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n              (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V)))))\n      v w)\n    (@SDiff.sdiff.{u} (Set.{u} V) (@Set.instSDiff.{u} V) (@Set.univ.{u} V)\n      (@Insert.insert.{u, u} V (Set.{u} V) (@Set.instInsert.{u} V) v\n        (@Singleton.singleton.{u, u} V (Set.{u} V) (@Set.instSingletonSet.{u} V) w)))","statement_sha256":"f0f87c840134c17477a57429c0a7e768021dc2ac373f3e18199d186524200a37","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-28fda237e2c7dbe5bcdb4b3393c523e3a5ea926c34594ae68802e8281ec70511","qualified_name":"SimpleGraph.compl_adj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":9676,"column":9,"line":257},"start":{"byte":9565,"column":0,"line":255}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) (v w : V),\n  Iff (@SimpleGraph.Adj.{u} V (@Compl.compl.{u} (SimpleGraph.{u} V) 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: SimpleGraph.{u} V) (v : V),\n  @Disjoint.{u} (Set.{u} V)\n    (@SemilatticeInf.toPartialOrder.{u} (Set.{u} V)\n      (@Lattice.toSemilatticeInf.{u} (Set.{u} V)\n        (@GeneralizedCoheytingAlgebra.toLattice.{u} (Set.{u} V)\n          (@CoheytingAlgebra.toGeneralizedCoheytingAlgebra.{u} (Set.{u} V)\n            (@BiheytingAlgebra.toCoheytingAlgebra.{u} (Set.{u} V)\n              (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} V) (@Set.instBooleanAlgebra.{u} V)))))))\n    (@HeytingAlgebra.toOrderBot.{u} (Set.{u} V)\n      (@BiheytingAlgebra.toHeytingAlgebra.{u} (Set.{u} V)\n        (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} V) (@Set.instBooleanAlgebra.{u} V))))\n    (@SimpleGraph.neighborSet.{u} V G v)\n    (@SimpleGraph.neighborSet.{u} V (@Compl.compl.{u} (SimpleGraph.{u} V) (@SimpleGraph.instCompl.{u} V) G) v)","statement_sha256":"afef610913448f20a5fc27693420a32299b309881640a1220315f9b8e0c04e9f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-786630a8568d30a602df576f05307d181956e5bd0881578e48fae9d0d030e5e5","qualified_name":"SimpleGraph.completeAtomicBooleanAlgebra","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":12412,"column":51,"line":328},"start":{"byte":11631,"column":0,"line":311}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{V : Type u} → CompleteAtomicBooleanAlgebra.{u} (SimpleGraph.{u} V)","statement_sha256":"4f56f266eed000770b5dd247975fa89c6729561449b04107eb3322581a030969","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-0d4fcb1450a59aa6118ee8e9a499e9cfb0970cd00b57348110e85d1f6743f38e","qualified_name":"SimpleGraph.completeGraph","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":12586,"column":56,"line":332},"start":{"byte":12414,"column":0,"line":330}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"(V : Type u) → SimpleGraph.{u} 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 (@BooleanAlgebra.toTop.{u} (SimpleGraph.{u} V)\n        (@CompleteBooleanAlgebra.toBooleanAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n            (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V)))))","statement_sha256":"410e5dde822e97de6b61e2b49ec0936790440e7ce5f3ae41f60ba65d91e92ae2","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-002723bd0e7864cb6034df952c16f6a0222d3f490cf598c772babb7d7d9bafe8","qualified_name":"SimpleGraph.decidableMemCommonNeighbors","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":34921,"column":91,"line":949},"start":{"byte":34708,"column":0,"line":947}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{V : Type u} →\n  (G : SimpleGraph.{u} V) →\n    [@DecidableRel.{u + 1, u + 1} V V (@SimpleGraph.Adj.{u} V G)] →\n      (v w : V) →\n        @DecidablePred.{u + 1} V fun x =>\n          @Membership.mem.{u, u} V (Set.{u} V) (@Set.instMembership.{u} V) (@SimpleGraph.commonNeighbors.{u} V G v w) x","statement_sha256":"6d852f6a85865c756cdb4a804af16fe515f4facf2dd5f7a9ae3c0b92b20d7722","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-50bab85328013252ee931c657845e1e4e7225debf02acb579ff2d25495330778","qualified_name":"SimpleGraph.decidableMemEdgeSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":22678,"column":35,"line":616},"start":{"byte":22555,"column":0,"line":615}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{V : Type u} →\n  (G : SimpleGraph.{u} V) →\n    [@DecidableRel.{u + 1, u + 1} V V (@SimpleGraph.Adj.{u} V G)] →\n      @DecidablePred.{u + 1} (Sym2.{u} V) fun x =>\n        @Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V))\n          (@SimpleGraph.edgeSet.{u} V G) x","statement_sha256":"16f1ed00719c1732304821e40a327420659140fc428e567ec7eb34ac74d1ee34","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-368c31d1af52ac70a5ecf967942b3b7dff639faa4b394d1520d2d4d1cf444a51","qualified_name":"SimpleGraph.decidableMemIncidenceSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":29841,"column":71,"line":808},"start":{"byte":29642,"column":0,"line":806}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{V : Type u} →\n  (G : SimpleGraph.{u} V) →\n    [DecidableEq.{u + 1} V] →\n      [@DecidableRel.{u + 1, u + 1} V V (@SimpleGraph.Adj.{u} V G)] →\n        (v : V) →\n          @DecidablePred.{u + 1} (Sym2.{u} V) fun x =>\n            @Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V))\n              (@SimpleGraph.incidenceSet.{u} V G v) x","statement_sha256":"38389bed06c13f4d5f083142c66af1d7b8f41b1cdea30583e820ca9ba4f72ea3","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-17dd4978b7bcf1d15e3b39381b3689f5969caec4e2073f32cd104aa6964ba937","qualified_name":"SimpleGraph.default_def","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":13570,"column":7,"line":369},"start":{"byte":13565,"column":2,"line":369}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (V : Type u),\n  @Eq.{u + 1} (SimpleGraph.{u} V) (@Inhabited.default.{u + 1} (SimpleGraph.{u} V) (SimpleGraph.instInhabited.{u} V))\n    (@Bot.bot.{u} (SimpleGraph.{u} V)\n      (@BooleanAlgebra.toBot.{u} (SimpleGraph.{u} V)\n        (@CompleteBooleanAlgebra.toBooleanAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n            (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V)))))","statement_sha256":"7972032e240100cb7f71e2274a924a10b05e0c05379c489d16203948491ca856","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-81046dc6e32561f9304fa4cb92a89a075fd4427e7c846cc0db47ef3ef62a16d6","qualified_name":"SimpleGraph.disjoint_commonNeighbors","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":35177,"column":7,"line":955},"start":{"byte":34941,"column":0,"line":952}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G H : SimpleGraph.{u} V},\n  Iff\n    (∀ (u v : V),\n      @Disjoint.{u} (Set.{u} V)\n        (@SemilatticeInf.toPartialOrder.{u} (Set.{u} V)\n          (@Lattice.toSemilatticeInf.{u} (Set.{u} V)\n            (@GeneralizedCoheytingAlgebra.toLattice.{u} (Set.{u} V)\n              (@CoheytingAlgebra.toGeneralizedCoheytingAlgebra.{u} (Set.{u} V)\n                (@BiheytingAlgebra.toCoheytingAlgebra.{u} (Set.{u} V)\n                  (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} V) (@Set.instBooleanAlgebra.{u} V)))))))\n        (@HeytingAlgebra.toOrderBot.{u} (Set.{u} V)\n          (@BiheytingAlgebra.toHeytingAlgebra.{u} (Set.{u} V)\n            (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} V) (@Set.instBooleanAlgebra.{u} V))))\n        (@SimpleGraph.commonNeighbors.{u} V G u v) (@SimpleGraph.commonNeighbors.{u} V H u v))\n    (@Disjoint.{u} (SimpleGraph.{u} V) (@SimpleGraph.instPartialOrder.{u} V)\n      (@HeytingAlgebra.toOrderBot.{u} (SimpleGraph.{u} V)\n        (@Order.Frame.toHeytingAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteDistribLattice.toFrame.{u} (SimpleGraph.{u} V)\n            (@CompleteBooleanAlgebra.toCompleteDistribLattice.{u} (SimpleGraph.{u} V)\n              (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n                (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V))))))\n      G H)","statement_sha256":"86332b6086820f247039fac09a75110ceacf17752bb5e3c236ce1cb1ab9f82a7","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-1f4eda9f12fad633b324c13919dade0d9fa42fc8710d496dfcc381475a836011","qualified_name":"SimpleGraph.disjoint_edgeSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":20490,"column":104,"line":565},"start":{"byte":20289,"column":0,"line":564}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G₁ G₂ : SimpleGraph.{u} V},\n  Iff\n    (@Disjoint.{u} (Set.{u} (Sym2.{u} V))\n      (@SemilatticeInf.toPartialOrder.{u} (Set.{u} (Sym2.{u} V))\n        (@Lattice.toSemilatticeInf.{u} (Set.{u} (Sym2.{u} V))\n          (@GeneralizedCoheytingAlgebra.toLattice.{u} (Set.{u} (Sym2.{u} V))\n            (@CoheytingAlgebra.toGeneralizedCoheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n              (@BiheytingAlgebra.toCoheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                  (@Set.instBooleanAlgebra.{u} (Sym2.{u} V))))))))\n      (@HeytingAlgebra.toOrderBot.{u} (Set.{u} (Sym2.{u} V))\n        (@BiheytingAlgebra.toHeytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n          (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} (Sym2.{u} V)) (@Set.instBooleanAlgebra.{u} (Sym2.{u} V)))))\n      (@SimpleGraph.edgeSet.{u} V G₁) (@SimpleGraph.edgeSet.{u} V G₂))\n    (@Disjoint.{u} (SimpleGraph.{u} V) (@SimpleGraph.instPartialOrder.{u} V)\n      (@HeytingAlgebra.toOrderBot.{u} (SimpleGraph.{u} V)\n        (@Order.Frame.toHeytingAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteDistribLattice.toFrame.{u} (SimpleGraph.{u} V)\n            (@CompleteBooleanAlgebra.toCompleteDistribLattice.{u} (SimpleGraph.{u} V)\n              (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n                (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V))))))\n      G₁ G₂)","statement_sha256":"0e790dbd6a41a81ba8ddd7f8718e9801084b0a23f5f524c7fed152faf10dc07e","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f661744f53ff84c234ade42951e4fc90213307a1235983e62e62f2faf6922557","qualified_name":"SimpleGraph.disjoint_fromEdgeSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":26924,"column":38,"line":739},"start":{"byte":26682,"column":0,"line":736}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) (s : Set.{u} (Sym2.{u} V)),\n  Iff\n    (@Disjoint.{u} (SimpleGraph.{u} V) (@SimpleGraph.instPartialOrder.{u} V)\n      (@HeytingAlgebra.toOrderBot.{u} (SimpleGraph.{u} V)\n        (@Order.Frame.toHeytingAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteDistribLattice.toFrame.{u} (SimpleGraph.{u} V)\n            (@CompleteBooleanAlgebra.toCompleteDistribLattice.{u} (SimpleGraph.{u} V)\n              (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n                (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V))))))\n      G (@SimpleGraph.fromEdgeSet.{u} V s))\n    (@Disjoint.{u} (Set.{u} (Sym2.{u} V))\n      (@SemilatticeInf.toPartialOrder.{u} (Set.{u} (Sym2.{u} V))\n        (@Lattice.toSemilatticeInf.{u} (Set.{u} (Sym2.{u} V))\n          (@GeneralizedCoheytingAlgebra.toLattice.{u} (Set.{u} (Sym2.{u} V))\n            (@CoheytingAlgebra.toGeneralizedCoheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n              (@BiheytingAlgebra.toCoheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                  (@Set.instBooleanAlgebra.{u} (Sym2.{u} V))))))))\n      (@HeytingAlgebra.toOrderBot.{u} (Set.{u} (Sym2.{u} V))\n        (@BiheytingAlgebra.toHeytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n          (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} (Sym2.{u} V)) (@Set.instBooleanAlgebra.{u} (Sym2.{u} V)))))\n      (@SimpleGraph.edgeSet.{u} V G) s)","statement_sha256":"1e976eb89b2035bea45dac06484020e8939eb1fa99ad02b21a4af1f37255fb7f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-81dfc28ceb2084398ae82a45340da0379e327a21e154e48f1dd1b92da534ce36","qualified_name":"SimpleGraph.disjoint_incidenceSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":28599,"column":34,"line":783},"start":{"byte":28303,"column":0,"line":779}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G H : SimpleGraph.{u} V},\n  Iff\n    (∀ (v : V),\n      @Disjoint.{u} (Set.{u} (Sym2.{u} V))\n        (@SemilatticeInf.toPartialOrder.{u} (Set.{u} (Sym2.{u} V))\n          (@Lattice.toSemilatticeInf.{u} (Set.{u} (Sym2.{u} V))\n            (@GeneralizedCoheytingAlgebra.toLattice.{u} (Set.{u} (Sym2.{u} V))\n              (@CoheytingAlgebra.toGeneralizedCoheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                (@BiheytingAlgebra.toCoheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                  (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                    (@Set.instBooleanAlgebra.{u} (Sym2.{u} V))))))))\n        (@HeytingAlgebra.toOrderBot.{u} (Set.{u} (Sym2.{u} V))\n          (@BiheytingAlgebra.toHeytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n            (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} (Sym2.{u} V)) (@Set.instBooleanAlgebra.{u} (Sym2.{u} V)))))\n        (@SimpleGraph.incidenceSet.{u} V G v) (@SimpleGraph.incidenceSet.{u} V H v))\n    (@Disjoint.{u} (SimpleGraph.{u} V) (@SimpleGraph.instPartialOrder.{u} V)\n      (@HeytingAlgebra.toOrderBot.{u} 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H)","statement_sha256":"8ff65b2cb8cc20340af3dc686f5ebf7da94b5546b3af91992da6f64bfa49fbef","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-c3903566fb58f568638e1e6ae40052c44052f6117dcfd89941aa3b59a81f5d24","qualified_name":"SimpleGraph.disjoint_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":27386,"column":61,"line":751},"start":{"byte":27221,"column":0,"line":750}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G H : SimpleGraph.{u} V},\n  Iff\n    (@Disjoint.{u} (SimpleGraph.{u} V) (@SimpleGraph.instPartialOrder.{u} V)\n      (@HeytingAlgebra.toOrderBot.{u} (SimpleGraph.{u} V)\n        (@Order.Frame.toHeytingAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteDistribLattice.toFrame.{u} (SimpleGraph.{u} V)\n            (@CompleteBooleanAlgebra.toCompleteDistribLattice.{u} (SimpleGraph.{u} V)\n              (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n                (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V))))))\n      G H)\n    (∀ (x y : V), @SimpleGraph.Adj.{u} V G x y → Not (@SimpleGraph.Adj.{u} V H x y))","statement_sha256":"3c0b0ba388b9e3decac790a65ec6c495bb59bb5e73cadaf8f934000c12de518e","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-62ab55c941c5281d74aefc2e249e67aa8276108da5d682beeb2e759f1c977d20","qualified_name":"SimpleGraph.disjoint_neighborSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":30577,"column":94,"line":831},"start":{"byte":30371,"column":0,"line":829}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G H : SimpleGraph.{u} V},\n  Iff\n    (∀ (v : V),\n      @Disjoint.{u} (Set.{u} V)\n        (@SemilatticeInf.toPartialOrder.{u} (Set.{u} V)\n          (@Lattice.toSemilatticeInf.{u} (Set.{u} V)\n            (@GeneralizedCoheytingAlgebra.toLattice.{u} (Set.{u} V)\n              (@CoheytingAlgebra.toGeneralizedCoheytingAlgebra.{u} (Set.{u} V)\n                (@BiheytingAlgebra.toCoheytingAlgebra.{u} (Set.{u} V)\n                  (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} V) (@Set.instBooleanAlgebra.{u} V)))))))\n        (@HeytingAlgebra.toOrderBot.{u} (Set.{u} V)\n          (@BiheytingAlgebra.toHeytingAlgebra.{u} (Set.{u} V)\n            (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} V) (@Set.instBooleanAlgebra.{u} V))))\n        (@SimpleGraph.neighborSet.{u} V G v) (@SimpleGraph.neighborSet.{u} V H v))\n    (@Disjoint.{u} (SimpleGraph.{u} V) (@SimpleGraph.instPartialOrder.{u} V)\n      (@HeytingAlgebra.toOrderBot.{u} (SimpleGraph.{u} V)\n        (@Order.Frame.toHeytingAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteDistribLattice.toFrame.{u} (SimpleGraph.{u} 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H)","statement_sha256":"6725034f94352462056d6bccf090f134b7489670f89f806e588fcc7621cabc7d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-7476237a2d3273b18eee804e9ec0861b5a39ef9a586a9613265e0037d99741e6","qualified_name":"SimpleGraph.disjoint_of_disjoint_support","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":20714,"column":21,"line":570},"start":{"byte":20492,"column":0,"line":567}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G : SimpleGraph.{u} V} (H : SimpleGraph.{u} V),\n  @Disjoint.{u} (Set.{u} V)\n      (@SemilatticeInf.toPartialOrder.{u} (Set.{u} V)\n        (@Lattice.toSemilatticeInf.{u} (Set.{u} V)\n          (@GeneralizedCoheytingAlgebra.toLattice.{u} (Set.{u} V)\n            (@CoheytingAlgebra.toGeneralizedCoheytingAlgebra.{u} (Set.{u} V)\n              (@BiheytingAlgebra.toCoheytingAlgebra.{u} (Set.{u} V)\n                (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} V) (@Set.instBooleanAlgebra.{u} V)))))))\n      (@HeytingAlgebra.toOrderBot.{u} (Set.{u} V)\n        (@BiheytingAlgebra.toHeytingAlgebra.{u} (Set.{u} V)\n          (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} V) (@Set.instBooleanAlgebra.{u} V))))\n      (@SimpleGraph.support.{u} V G) (@SimpleGraph.support.{u} V H) →\n    @Disjoint.{u} (SimpleGraph.{u} V) (@SimpleGraph.instPartialOrder.{u} V)\n      (@HeytingAlgebra.toOrderBot.{u} (SimpleGraph.{u} V)\n        (@Order.Frame.toHeytingAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteDistribLattice.toFrame.{u} (SimpleGraph.{u} V)\n            (@CompleteBooleanAlgebra.toCompleteDistribLattice.{u} (SimpleGraph.{u} V)\n              (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n                (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V))))))\n      G H","statement_sha256":"e91bf44c2dd06dd47e11884aa442b0cc2cde767908aa61e9a717aacf485bfa78","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"definition","object_contract_version":1,"object_id":"object-890eacf97a31b7d3029a58cfaa8819b7031a57e7896ed7c7ddfff21c27796281","qualified_name":"SimpleGraph.distribLattice","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":11629,"column":76,"line":309},"start":{"byte":11493,"column":0,"line":308}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{V : Type u} → DistribLattice.{u} (SimpleGraph.{u} 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(@Bot.bot.{u} (SimpleGraph.{u} V)\n        (@BooleanAlgebra.toBot.{u} (SimpleGraph.{u} V)\n          (@CompleteBooleanAlgebra.toBooleanAlgebra.{u} (SimpleGraph.{u} V)\n            (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n              (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V))))))\n    (@EmptyCollection.emptyCollection.{u} (Set.{u} (Sym2.{u} V)) (@Set.instEmptyCollection.{u} (Sym2.{u} V)))","statement_sha256":"5f6ce2affba4dfbf55bc0b1321584361843563a84e9555dc349d16c3fef7390d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-340e3aa59678a7d9058d1662bf6e7fac6c4c9a372d9499bc1c095b36c69ef56b","qualified_name":"SimpleGraph.edgeSet_eq_empty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":20816,"column":100,"line":572},"start":{"byte":20716,"column":0,"line":572}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} {G : SimpleGraph.{u} V},\n  Iff\n    (@Eq.{u + 1} (Set.{u} (Sym2.{u} V)) (@SimpleGraph.edgeSet.{u} V G)\n      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V))))))","statement_sha256":"5cd8bd2e3af2bd88c8b7f6410d1d7cb1ce042ba1e75a21e17397d56d9f15c6a5","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-6faf1c7179c01ce95bb6f2721fa943e029ed75cf99220d80604155be949f7c47","qualified_name":"SimpleGraph.edgeSet_eq_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":24951,"column":43,"line":678},"start":{"byte":24754,"column":0,"line":676}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) (s : Set.{u} (Sym2.{u} V)),\n  Iff (@Eq.{u + 1} (Set.{u} (Sym2.{u} V)) 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G₂))","statement_sha256":"7095bd8e0bb23ad3ac02d43cec4accf495b889026496c04e756734605223f22c","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f3c9715bb2cd3afeda6b57600f051c09a8d503a749bd14b6749146ffb7c9739c","qualified_name":"SimpleGraph.edgeSet_sdiff_sdiff_isDiag","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":21287,"column":53,"line":582},"start":{"byte":20945,"column":0,"line":577}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) (s : Set.{u} (Sym2.{u} V)),\n  @Eq.{u + 1} (Set.{u} (Sym2.{u} V))\n    (@SDiff.sdiff.{u} (Set.{u} (Sym2.{u} V)) (@Set.instSDiff.{u} (Sym2.{u} V)) (@SimpleGraph.edgeSet.{u} V G)\n      (@SDiff.sdiff.{u} (Set.{u} (Sym2.{u} V)) (@Set.instSDiff.{u} (Sym2.{u} V)) s (@Sym2.diagSet.{u} V)))\n    (@SDiff.sdiff.{u} (Set.{u} (Sym2.{u} V)) (@Set.instSDiff.{u} (Sym2.{u} V)) (@SimpleGraph.edgeSet.{u} V G) 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      (@PartialOrder.toPreorder.{u} (SimpleGraph.{u} V) (@SimpleGraph.instPartialOrder.{u} V)))\n      G₁ G₂ →\n    @LT.lt.{u} (Set.{u} (Sym2.{u} V))\n      (@Preorder.toLT.{u} (Set.{u} (Sym2.{u} V))\n        (@PartialOrder.toPreorder.{u} (Set.{u} (Sym2.{u} V))\n          (@SemilatticeInf.toPartialOrder.{u} (Set.{u} (Sym2.{u} V))\n            (@Lattice.toSemilatticeInf.{u} (Set.{u} (Sym2.{u} V))\n              (@GeneralizedCoheytingAlgebra.toLattice.{u} (Set.{u} (Sym2.{u} V))\n                (@CoheytingAlgebra.toGeneralizedCoheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                  (@BiheytingAlgebra.toCoheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                    (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                      (@Set.instBooleanAlgebra.{u} (Sym2.{u} V))))))))))\n      (@SimpleGraph.edgeSet.{u} V G₁) (@SimpleGraph.edgeSet.{u} V G₂)","statement_sha256":"3733bcc7758cc626f6e36665061f75e351c25cdafb0e59c98fba6d7688bc1d97","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f9f9d9edeb4570e6d14c258352e1d604239b2e59510b574116ff6b834c67702c","qualified_name":"SimpleGraph.edgeSet_subset_compl_diagSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":19253,"column":88,"line":524},"start":{"byte":19082,"column":0,"line":522}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V),\n  @LE.le.{u} (Set.{u} (Sym2.{u} V)) (@Set.instLE.{u} (Sym2.{u} V)) (@SimpleGraph.edgeSet.{u} V G)\n    (@Compl.compl.{u} (Set.{u} (Sym2.{u} V)) (@Set.instCompl.{u} (Sym2.{u} V)) (@Sym2.diagSet.{u} 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a)","statement_sha256":"ba51d8ec91d0ceb84faec93ec9acd9e53a58ae4b9850b1813fc1800389c20951","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-eedf4bb83af4be41ef8d94aa3a5f417c64e79db7a769945bd454af97ab3320a2","qualified_name":"SimpleGraph.edge_other_incident_set","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":35904,"column":47,"line":977},"start":{"byte":35704,"column":0,"line":974}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) [inst : DecidableEq.{u + 1} V] {v : V} {e : Sym2.{u} V}\n  (h :\n    @Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V))\n      (@SimpleGraph.incidenceSet.{u} V G v) e),\n  @Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V))\n    (@SimpleGraph.incidenceSet.{u} V G (@SimpleGraph.otherVertexOfIncident.{u} V G inst v e h)) 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(@BooleanAlgebra.toBot.{u} (SimpleGraph.{u} V)\n        (@CompleteBooleanAlgebra.toBooleanAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n            (@SimpleGraph.completeAtomicBooleanAlgebra.{u} 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(And\n      (@Membership.mem.{u, u} (Sym2.{u} V) (Set.{u} (Sym2.{u} V)) (@Set.instMembership.{u} (Sym2.{u} V)) s\n        (@Sym2.mk.{u} V v w))\n      (@Ne.{u + 1} V v 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(@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n                  (@Set.instBooleanAlgebra.{u} (Sym2.{u} V))))))))\n      (@HeytingAlgebra.toOrderBot.{u} (Set.{u} (Sym2.{u} V))\n        (@BiheytingAlgebra.toHeytingAlgebra.{u} (Set.{u} (Sym2.{u} V))\n          (@BooleanAlgebra.toBiheytingAlgebra.{u} (Set.{u} (Sym2.{u} V)) (@Set.instBooleanAlgebra.{u} (Sym2.{u} V)))))\n      s (@SimpleGraph.edgeSet.{u} V 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(@EmptyCollection.emptyCollection.{u} (Set.{u} (Sym2.{u} V)) (@Set.instEmptyCollection.{u} (Sym2.{u} V))))\n    (@Bot.bot.{u} (SimpleGraph.{u} V)\n      (@BooleanAlgebra.toBot.{u} (SimpleGraph.{u} V)\n        (@CompleteBooleanAlgebra.toBooleanAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n            (@SimpleGraph.completeAtomicBooleanAlgebra.{u} 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(@Top.top.{u} (SimpleGraph.{u} V)\n      (@BooleanAlgebra.toTop.{u} (SimpleGraph.{u} V)\n        (@CompleteBooleanAlgebra.toBooleanAlgebra.{u} (SimpleGraph.{u} V)\n          (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n            (@SimpleGraph.completeAtomicBooleanAlgebra.{u} 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w))","statement_sha256":"a5d66ffc7b99b023406e50edafbd4e4bf69e2c7a611b6e7e38633ef39cd99823","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-6ba109efe35c9bddd729152eecf7c066bf017207b2edaf0feecd9e3a6a7a2956","qualified_name":"SimpleGraph.incidenceSet_inter_incidenceSet_of_adj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":29102,"column":87,"line":793},"start":{"byte":28787,"column":0,"line":789}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) {a b : V},\n  @SimpleGraph.Adj.{u} V G a b →\n    @Eq.{u + 1} 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DecidableEq.{u + 1} V] {v w : V}\n  (h : @Membership.mem.{u, u} V (Set.{u} V) (@Set.instMembership.{u} V) (@SimpleGraph.neighborSet.{u} V G v) w),\n  @Eq.{u + 1} V (@SimpleGraph.otherVertexOfIncident.{u} V G inst v (@Sym2.mk.{u} V v w) ⋯) 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(SimpleGraph.{u} V))\n        (@SimpleGraph.mk'.{u} V) x)\n      v w)\n    (@Eq.{1} Bool\n      (@Subtype.val.{u + 1} (V → V → Bool)\n        (fun adj =>\n          And (∀ (x y : V), @Eq.{1} Bool (adj x y) (adj y x)) (∀ (x : V), Not (@Eq.{1} Bool (adj x x) Bool.true)))\n        x v w)\n      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b","statement_sha256":"43097bcbceaca0a854bd77b9495179f4178083dd78edcf6d712f2274a8c08087","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-7b017c80dc05d98ee526157fd28762914c1b12f71564d3a71fe4047cb73824c3","qualified_name":"SimpleGraph.ne_of_adj_of_not_adj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Combinatorics.SimpleGraph.Basic","path":"Mathlib/Combinatorics/SimpleGraph/Basic.lean","range":{"end":{"byte":7068,"column":15,"line":191},"start":{"byte":6954,"column":0,"line":190}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {V : Type u} (G : SimpleGraph.{u} V) {v w x : V},\n  @SimpleGraph.Adj.{u} V G v x → Not (@SimpleGraph.Adj.{u} V G w x) → @Ne.{u 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(SimpleGraph.{u} V)\n        (@BooleanAlgebra.toTop.{u} (SimpleGraph.{u} V)\n          (@CompleteBooleanAlgebra.toBooleanAlgebra.{u} (SimpleGraph.{u} V)\n            (@CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u} (SimpleGraph.{u} V)\n              (@SimpleGraph.completeAtomicBooleanAlgebra.{u} V))))))\n    (∃ a b, And (@Ne.{u + 1} V a b) (Not (@SimpleGraph.Adj.{u} V G a 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