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(@Finset.instOrderBot.{u_1} α) s t) {i : α}\n  (hi :\n    @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s i)\n  (hi' :\n    @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α))\n      (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t) i),\n  @Eq.{u_1 + 1} (Sum.{u_1, u_1} ↥s ↥t)\n    (@DFunLike.coe.{u_1 + 1, u_1 + 1, u_1 + 1}\n      (Equiv.{u_1 + 1, u_1 + 1} (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t))\n        (Sum.{u_1, u_1} ↥s ↥t))\n      (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t)) (fun x => Sum.{u_1, u_1} ↥s ↥t)\n      (@EquivLike.toFunLike.{u_1 + 1, u_1 + 1, u_1 + 1}\n        (Equiv.{u_1 + 1, u_1 + 1} (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t))\n          (Sum.{u_1, u_1} ↥s ↥t))\n        (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t)) (Sum.{u_1, u_1} ↥s ↥t)\n        (@Equiv.instEquivLike.{u_1 + 1, u_1 + 1}\n          (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t)) (Sum.{u_1, u_1} ↥s ↥t)))\n      (@Equiv.symm.{u_1 + 1, u_1 + 1} (Sum.{u_1, u_1} ↥s ↥t)\n        (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t))\n        (@Equiv.Finset.union.{u_1} α inst s t h))\n      (@Subtype.mk.{u_1 + 1} α\n        (fun x =>\n          @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n            (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α))\n            (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t) x)\n        i hi'))\n    (@Sum.inl.{u_1, u_1} (↥s) (↥t)\n      (@Subtype.mk.{u_1 + 1} α\n        (fun x =>\n          @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n            (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s x)\n        i hi))","statement_sha256":"15dafb7d049994c316741c46f236127f91dd86e4af17ce6dd561bfe27844e72a","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-3f17c31cd125ab2fb782cab771ac45c3fbbabe87581343fb8e8b5484ec0f0841","qualified_name":"Equiv.Finset.union_symm_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":21565,"column":28,"line":607},"start":{"byte":21353,"column":0,"line":604}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α}\n  (h : @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s t) {i : α}\n  (hi :\n    @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) t i)\n  (hi' :\n    @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α))\n      (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t) i),\n  @Eq.{u_1 + 1} (Sum.{u_1, u_1} ↥s ↥t)\n    (@DFunLike.coe.{u_1 + 1, u_1 + 1, u_1 + 1}\n      (Equiv.{u_1 + 1, u_1 + 1} (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t))\n        (Sum.{u_1, u_1} ↥s ↥t))\n      (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t)) (fun x => Sum.{u_1, u_1} ↥s ↥t)\n      (@EquivLike.toFunLike.{u_1 + 1, u_1 + 1, u_1 + 1}\n        (Equiv.{u_1 + 1, u_1 + 1} (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t))\n          (Sum.{u_1, u_1} ↥s ↥t))\n        (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t)) (Sum.{u_1, u_1} ↥s ↥t)\n        (@Equiv.instEquivLike.{u_1 + 1, u_1 + 1}\n          (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t)) (Sum.{u_1, u_1} ↥s ↥t)))\n      (@Equiv.symm.{u_1 + 1, u_1 + 1} (Sum.{u_1, u_1} ↥s ↥t)\n        (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t))\n        (@Equiv.Finset.union.{u_1} α inst s t h))\n      (@Subtype.mk.{u_1 + 1} α\n        (fun x =>\n          @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n            (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α))\n            (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t) x)\n        i hi'))\n    (@Sum.inr.{u_1, u_1} (↥s) (↥t)\n      (@Subtype.mk.{u_1 + 1} α\n        (fun x =>\n          @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n            (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) t x)\n        i hi))","statement_sha256":"1d0ba685d67a2a736464c9b21f78f87714155e626dbc3402be17f14fce6b991e","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-caebe61b574ecec837bcb3cb4b3d31f918db225f35f3c60408327a199ebbb044","qualified_name":"Equiv.piFinsetUnion","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":23172,"column":102,"line":641},"start":{"byte":22653,"column":0,"line":636}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{ι : Type u_4} →\n  [inst : DecidableEq.{u_4 + 1} ι] →\n    (α : ι → Type u_3) →\n      {s t : Finset.{u_4} ι} →\n        @Disjoint.{u_4} (Finset.{u_4} ι) (@Finset.instPartialOrder.{u_4} ι) (@Finset.instOrderBot.{u_4} ι) s t →\n          Equiv.{(max u_3 u_4) + 1, max (u_3 + 1) (u_4 + 1)}\n            (Prod.{max u_3 u_4, max u_3 u_4}\n              ((i : ↥s) →\n                α\n                  (@Subtype.val.{u_4 + 1} ι\n                    (fun x =>\n                      @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                        (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n                    i))\n              ((i : ↥t) →\n                α\n                  (@Subtype.val.{u_4 + 1} ι\n                    (fun x =>\n                      @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                        (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n                    i)))\n            ((i : ↥(@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t)) →\n              α\n                (@Subtype.val.{u_4 + 1} ι\n                  (fun x =>\n                    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n                      (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n                  i))","statement_sha256":"c3198dce5b5dafa65f483d3b77c7306edb913495a58548c4743be2b6f164a760","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-9ee6db5f467cfc8d1a8a77b441820d04e403380dd8c2341b90cade2253b1cb26","qualified_name":"Equiv.piFinsetUnion_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":23603,"column":5,"line":649},"start":{"byte":23231,"column":0,"line":644}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_4} [inst : DecidableEq.{u_4 + 1} ι] (α : ι → Type u_3) {s t : Finset.{u_4} ι}\n  (h : @Disjoint.{u_4} (Finset.{u_4} ι) (@Finset.instPartialOrder.{u_4} ι) (@Finset.instOrderBot.{u_4} ι) s t)\n  {f :\n    (i : ↥s) →\n      α\n        (@Subtype.val.{u_4 + 1} ι\n          (fun x =>\n            @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n              (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n          i)}\n  {g :\n    (i : ↥t) →\n      α\n        (@Subtype.val.{u_4 + 1} ι\n          (fun x =>\n            @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n              (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n          i)}\n  {i : ι}\n  (hi :\n    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s i)\n  (hi' :\n    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n      (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} 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→\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n              i))\n        ((i : ↥t) →\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n              i)))\n      (fun x =>\n        (i : ↥(@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t)) →\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n                  (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n              i))\n      (@EquivLike.toFunLike.{max (u_3 + 1) (u_4 + 1), max (u_3 + 1) (u_4 + 1), max (u_3 + 1) (u_4 + 1)}\n        (Equiv.{(max u_3 u_4) + 1, max (u_3 + 1) (u_4 + 1)}\n          (Prod.{max u_3 u_4, max u_3 u_4}\n            ((i : ↥s) →\n              α\n                (@Subtype.val.{u_4 + 1} ι\n                  (fun x =>\n                    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n                  i))\n            ((i : ↥t) →\n              α\n                (@Subtype.val.{u_4 + 1} ι\n                  (fun x =>\n                    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n                  i)))\n          ((i : ↥(@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t)) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n                    (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n                i)))\n        (Prod.{max u_3 u_4, max u_3 u_4}\n          ((i : ↥s) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n                i))\n          ((i : ↥t) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n                i)))\n        ((i : ↥(@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t)) →\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n                  (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n              i))\n        (@Equiv.instEquivLike.{max (u_3 + 1) (u_4 + 1), max (u_3 + 1) (u_4 + 1)}\n          (Prod.{max u_3 u_4, max u_3 u_4}\n            ((i : ↥s) →\n              α\n                (@Subtype.val.{u_4 + 1} ι\n                  (fun x =>\n                    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n                  i))\n            ((i : ↥t) →\n              α\n                (@Subtype.val.{u_4 + 1} ι\n                  (fun x =>\n                    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n                  i)))\n          ((i : ↥(@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t)) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n                    (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n                i))))\n      (@Equiv.piFinsetUnion.{u_3, u_4} ι inst α s t h)\n      (@Prod.mk.{max u_3 u_4, max u_3 u_4}\n        ((i : ↥s) →\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n              i))\n        ((i : ↥t) →\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n              i))\n        f g)\n      (@Subtype.mk.{u_4 + 1} ι\n        (fun x =>\n          @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n            (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n            (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n        i hi'))\n    (f\n      (@Subtype.mk.{u_4 + 1} ι\n        (fun x =>\n          @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n            (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n        i hi))","statement_sha256":"393b108d4486a34c0bd358abf0df313d24df25ffc29bd07200f918036ccd0f9f","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-fc335c7feddcbde8cdce3627eb9f22e4151387449124df6678805ab04f03cbfb","qualified_name":"Equiv.piFinsetUnion_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":24042,"column":5,"line":657},"start":{"byte":23662,"column":0,"line":652}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_4} [inst : DecidableEq.{u_4 + 1} ι] (α : ι → Type u_3) {s t : Finset.{u_4} ι}\n  (h : @Disjoint.{u_4} (Finset.{u_4} ι) (@Finset.instPartialOrder.{u_4} ι) (@Finset.instOrderBot.{u_4} ι) s t)\n  {f :\n    (i : ↥s) →\n      α\n        (@Subtype.val.{u_4 + 1} ι\n          (fun x =>\n            @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n              (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n          i)}\n  {g :\n    (i : ↥t) →\n      α\n        (@Subtype.val.{u_4 + 1} ι\n          (fun x =>\n            @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n              (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n          i)}\n  {i : ι}\n  (hi :\n    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t i)\n  (hi' :\n    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n      (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) i),\n  @Eq.{u_3 + 1}\n    (α\n      (@Subtype.val.{u_4 + 1} ι\n        (fun x =>\n          @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n            (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n            (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n        (@Subtype.mk.{u_4 + 1} ι\n          (fun x =>\n            @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n              (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n              (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n          i hi')))\n    (@DFunLike.coe.{max (u_3 + 1) (u_4 + 1), max (u_3 + 1) (u_4 + 1), max (u_3 + 1) (u_4 + 1)}\n      (Equiv.{(max u_3 u_4) + 1, max (u_3 + 1) (u_4 + 1)}\n        (Prod.{max u_3 u_4, max u_3 u_4}\n          ((i : ↥s) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n                i))\n          ((i : ↥t) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n                i)))\n        ((i : ↥(@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t)) →\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n                  (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n              i)))\n      (Prod.{max u_3 u_4, max u_3 u_4}\n        ((i : ↥s) 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inst) s t) x)\n              i))\n      (@EquivLike.toFunLike.{max (u_3 + 1) (u_4 + 1), max (u_3 + 1) (u_4 + 1), max (u_3 + 1) (u_4 + 1)}\n        (Equiv.{(max u_3 u_4) + 1, max (u_3 + 1) (u_4 + 1)}\n          (Prod.{max u_3 u_4, max u_3 u_4}\n            ((i : ↥s) →\n              α\n                (@Subtype.val.{u_4 + 1} ι\n                  (fun x =>\n                    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n                  i))\n            ((i : ↥t) →\n              α\n                (@Subtype.val.{u_4 + 1} ι\n                  (fun x =>\n                    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n                  i)))\n          ((i : ↥(@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t)) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n                    (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n                i)))\n        (Prod.{max u_3 u_4, max u_3 u_4}\n          ((i : ↥s) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n                i))\n          ((i : ↥t) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n                i)))\n        ((i : ↥(@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t)) →\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n                  (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n              i))\n        (@Equiv.instEquivLike.{max (u_3 + 1) (u_4 + 1), max (u_3 + 1) (u_4 + 1)}\n          (Prod.{max u_3 u_4, max u_3 u_4}\n            ((i : ↥s) →\n              α\n                (@Subtype.val.{u_4 + 1} ι\n                  (fun x =>\n                    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n                  i))\n            ((i : ↥t) →\n              α\n                (@Subtype.val.{u_4 + 1} ι\n                  (fun x =>\n                    @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                      (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n                  i)))\n          ((i : ↥(@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t)) →\n            α\n              (@Subtype.val.{u_4 + 1} ι\n                (fun x =>\n                  @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                    (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n                    (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n                i))))\n      (@Equiv.piFinsetUnion.{u_3, u_4} ι inst α s t h)\n      (@Prod.mk.{max u_3 u_4, max u_3 u_4}\n        ((i : ↥s) →\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) s x)\n              i))\n        ((i : ↥t) →\n          α\n            (@Subtype.val.{u_4 + 1} ι\n              (fun x =>\n                @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n                  (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n              i))\n        f g)\n      (@Subtype.mk.{u_4 + 1} ι\n        (fun x =>\n          @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n            (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι))\n            (@Union.union.{u_4} (Finset.{u_4} ι) (@Finset.instUnion.{u_4} ι inst) s t) x)\n        i hi'))\n    (g\n      (@Subtype.mk.{u_4 + 1} ι\n        (fun x =>\n          @Membership.mem.{u_4, u_4} ι (Finset.{u_4} ι)\n            (@SetLike.instMembership.{u_4, u_4} (Finset.{u_4} ι) ι (@Finset.instSetLike.{u_4} ι)) t x)\n        i hi))","statement_sha256":"388167f0c541dcfbe6733bde9240378873df135f5904922005861601378750bf","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-f558b8b873247ff2314f39f1d8be4290df8eeac54b3f7303ed838902e302b895","qualified_name":"Finset.Nonempty.attach","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":10628,"column":63,"line":292},"start":{"byte":10513,"column":0,"line":291}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s : Finset.{u_1} α},\n  @Finset.Nonempty.{u_1} α s → @Finset.Nonempty.{u_1} (↥s) (@Finset.attach.{u_1} α s)","statement_sha256":"0237b72b2f26f97a373bf9deae7e3da477bed57cc787aab1fda42cfed7bb80cb","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-c7334045781378a30cac92e836f712e11c98f80d43fe96fad252d65aec1c57ec","qualified_name":"Finset.Nonempty.not_disjoint","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":2658,"column":71,"line":88},"start":{"byte":2587,"column":0,"line":88}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α},\n  @Finset.Nonempty.{u_1} α (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} 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Finset.{u_1} α},\n  @Finset.Nonempty.{u_1} α s → Not (@Eq.{1} Bool (@List.isEmpty.{u_1} α (@Finset.toList.{u_1} α s)) Bool.true)","statement_sha256":"f076b8e91dcd8c63b6718453f87fd4eeadae1c3b4669a79483da905aa82691b1","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-16245bde00535b357147bdafab75e59c3b943b9a4aca3904c51bf197437c3b93","qualified_name":"Finset.Nonempty.toList_ne_nil","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":19174,"column":33,"line":535},"start":{"byte":19055,"column":0,"line":534}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s : Finset.{u_1} α},\n  @Finset.Nonempty.{u_1} α s → @Ne.{u_1 + 1} (List.{u_1} α) (@Finset.toList.{u_1} α s) (@List.nil.{u_1} α)","statement_sha256":"41aaa38c5b9f543e9b1e3ee5748c9f81d27e43420dd859a7583cff1a815d8634","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-9aac4db119af9adc6517f7e3619afe7be80e1282b17f8b6a580c8e9ce5b10fe1","qualified_name":"Finset.Nontrivial.erase_nonempty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":3733,"column":37,"line":120},"start":{"byte":3608,"column":0,"line":119}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Finset.{u_1} α} {a : α},\n  @Finset.Nontrivial.{u_1} α s → @Finset.Nonempty.{u_1} α (@Finset.erase.{u_1} α inst s a)","statement_sha256":"c293884349963db5b0321007317ee5e2e924864e3aa09fa03fe8e6a7fd71005e","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-0043f186461afcd86a86c8c9e2050f0fdf091680ef90c26a6422145107adda98","qualified_name":"Finset.Nontrivial.exists_cons_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":6810,"column":86,"line":189},"start":{"byte":6448,"column":0,"line":184}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s : Finset.{u_1} α},\n  @Finset.Nontrivial.{u_1} α s →\n    ∃ t a,\n      ∃ (ha : a ∉ t),\n        ∃ b,\n          ∃ (hb : b ∉ t) (hab : Not (@Eq.{u_1 + 1} α a b)),\n            @Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.cons.{u_1} α a (@Finset.cons.{u_1} α b t hb) ⋯) s","statement_sha256":"43c6e92a2b18b6b89aa6c96fa2f80350a5d78ae46e533c92c75c67da3d8b5403","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-12f48084bcabd65866a9b6c7ab552d07337e20abd9d89c3a21fe9d1b1d0e71a8","qualified_name":"Finset.attach_empty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":10392,"column":5,"line":285},"start":{"byte":10322,"column":0,"line":283}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1},\n  @Eq.{u_1 + 1}\n    (Finset.{u_1} ↥(@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))\n    (@Finset.attach.{u_1} α\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))\n    (@EmptyCollection.emptyCollection.{u_1}\n      (Finset.{u_1} ↥(@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))\n      (@Finset.instEmptyCollection.{u_1}\n        ↥(@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α))))","statement_sha256":"932e34dbf0d0b9dac8af6111784794084c170f15e71a461683c6df7c210e3d28","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-4c9a3470eac1cca6a14e32e1cc60fe3a499bd7daa96346b92eff40a8737a1904","qualified_name":"Finset.attach_eq_empty_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":10752,"column":35,"line":296},"start":{"byte":10630,"column":0,"line":294}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s : Finset.{u_1} α},\n  Iff\n    (@Eq.{u_1 + 1} (Finset.{u_1} ↥s) (@Finset.attach.{u_1} α s)\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} ↥s) (@Finset.instEmptyCollection.{u_1} ↥s)))\n    (@Eq.{u_1 + 1} (Finset.{u_1} α) s\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))","statement_sha256":"e7eaf5fd0427700072e8a4e8eb15c318c74b48ffde0ba3a0b2f63c3b526ee3c6","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-3ecb38a3a0fdcb9fdcc028eafcdf0c38cf1c5c7b863aa87e54a9736868017e77","qualified_name":"Finset.attach_nonempty_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":10511,"column":24,"line":289},"start":{"byte":10394,"column":0,"line":287}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s : Finset.{u_1} α},\n  Iff (@Finset.Nonempty.{u_1} (↥s) (@Finset.attach.{u_1} α s)) (@Finset.Nonempty.{u_1} α s)","statement_sha256":"3ed5064974f08c2a80c1e938fbc5f8d7b601dd7691f5e9ebab4dea24f0ba8a87","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-b3c548da204c5d3e194a4904472b5c8a9b81ed7abcf057e97ecfffe95d9a6952","qualified_name":"Finset.choose","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":19936,"column":16,"line":557},"start":{"byte":19693,"column":0,"line":554}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u_1} →\n  (p : α → Prop) →\n    [@DecidablePred.{u_1 + 1} α p] →\n      (l : Finset.{u_1} α) →\n        (@ExistsUnique.{u_1 + 1} α fun a =>\n            And\n              (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n                (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l a)\n              (p a)) →\n          α","statement_sha256":"c22759acc0c6b54e77c7e541c1eb4405d9d6ec61896107822810309156d34a0c","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-26c90d022387e31ba33a9feae9cb8c266f8775c29b0f30f0c41d079a47f68c35","qualified_name":"Finset.chooseX","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":19691,"column":20,"line":552},"start":{"byte":19412,"column":0,"line":549}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u_1} →\n  (p : α → Prop) →\n    [@DecidablePred.{u_1 + 1} α p] →\n      (l : Finset.{u_1} α) →\n        (@ExistsUnique.{u_1 + 1} α fun a =>\n            And\n              (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n                (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l a)\n              (p a)) →\n          @Subtype.{u_1 + 1} α fun a =>\n            And\n              (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n                (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l a)\n              (p a)","statement_sha256":"5c7eb1c5144bbc31d668823945ca8342f5bd2dde815f474e61034337b632cbda","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-3edf8adc74054fa085eb8853bead4f182f289c4fb08036ca159267a77bf6cdbb","qualified_name":"Finset.choose_eq_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":20490,"column":26,"line":573},"start":{"byte":20359,"column":0,"line":572}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] (l : Finset.{u_1} α)\n  (hp :\n    @ExistsUnique.{u_1 + 1} α fun a =>\n      And\n        (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n          (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l a)\n        (p a))\n  {a : α},\n  Iff (@Eq.{u_1 + 1} α (@Finset.choose.{u_1} α p inst l hp) a)\n    (And\n      (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n        (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l a)\n      (p a))","statement_sha256":"6164cb85ddc59658bb3ee9b9c6843472e32782fd6507c91dded5f38c9e6f6505","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-914bf2812a326f0d7b2c97ddb851514329f293782e095d86321f4cd49efeda5f","qualified_name":"Finset.choose_mem","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":20163,"column":23,"line":563},"start":{"byte":20065,"column":0,"line":562}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] (l : Finset.{u_1} α)\n  (hp :\n    @ExistsUnique.{u_1 + 1} α fun a =>\n      And\n        (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n          (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l a)\n        (p a)),\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n    (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l\n    (@Finset.choose.{u_1} α p inst l hp)","statement_sha256":"caee163254467c41b5362b8fdb23a578243df36828803735e56092e31ec0753f","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-465fc66b05e6a73b7dc31bcb3f2f7dbd568264563bd5b76b082b75c495eb8344","qualified_name":"Finset.choose_property","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":20309,"column":23,"line":568},"start":{"byte":20208,"column":0,"line":567}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] (l : Finset.{u_1} α)\n  (hp :\n    @ExistsUnique.{u_1 + 1} α fun a =>\n      And\n        (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n          (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l a)\n        (p a)),\n  p (@Finset.choose.{u_1} α p inst l hp)","statement_sha256":"eeaed7900bd207585bb3eeddd3fa89badf387a02ea0bef8751779efa1886d3aa","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-1be1ee31a0298e1d28c644dd3650c1acc02a6bf5d6c76fd434b40d20a66f6783","qualified_name":"Finset.choose_spec","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":20063,"column":27,"line":560},"start":{"byte":19938,"column":0,"line":559}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] (l : Finset.{u_1} α)\n  (hp :\n    @ExistsUnique.{u_1 + 1} α fun a =>\n      And\n        (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n          (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l a)\n        (p a)),\n  And\n    (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) l\n      (@Finset.choose.{u_1} α p inst l hp))\n    (p (@Finset.choose.{u_1} α p inst l hp))","statement_sha256":"ef085138b8319bc126e6830c9ec1a148f13511c413896e730ecd7aa02b931e7f","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-a7d06ba598b8ebeed2fbb4ea46ac98e98b7874b10f583489a4ca872a65b20942","qualified_name":"Finset.disjUnion_eq_union","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":2108,"column":78,"line":73},"start":{"byte":2022,"column":0,"line":72}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s t : Finset.{u_1} α)\n  (h : @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) 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t)","statement_sha256":"08875c839ad6088abb6ccd54220a32f98193518828a7e6a154a74759f6f4bb54","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-b59c142f14547a24e913145c20217bfa3421e000b883057585c3a34a9acaabe5","qualified_name":"Finset.disjoint_disjUnion_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":12719,"column":62,"line":341},"start":{"byte":12503,"column":0,"line":338}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s t u : Finset.{u_1} α}\n  (h : @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s t),\n  Iff\n    (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α)\n      (@Finset.disjUnion.{u_1} α s t h) u)\n    (And (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s u)\n      (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) t u))","statement_sha256":"4eccc2cfeac112815a3518a54800a5dc438afed3138b9d0ca567d2ed9f240f55","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-364e3ebf9d0c8bfdca7cbb96b39777ffa675d7353c52ed39d40d0b42b0c7c009","qualified_name":"Finset.disjoint_disjUnion_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":12939,"column":63,"line":346},"start":{"byte":12721,"column":0,"line":343}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s t u : Finset.{u_1} α}\n  (h : @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) t u),\n  Iff\n    (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s\n      (@Finset.disjUnion.{u_1} α t u h))\n    (And (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s t)\n      (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s u))","statement_sha256":"b5850723cad09937f186f658c59e8fb34e6909af5cc8751927ecb07e80eea608","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-3b250d2182c4dd7307633f1a94aa54fa0233f97ca1bb177d39b10f948327066b","qualified_name":"Finset.disjoint_erase_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":7536,"column":41,"line":210},"start":{"byte":7409,"column":0,"line":209}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  Iff\n    (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α)\n      (@Finset.erase.{u_1} α inst s a) t)\n    (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s\n      (@Finset.erase.{u_1} α inst t a))","statement_sha256":"d090e91790b7e7ff924f229cdfea7bde431263b800fa190eb32640c6cc17a20a","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-4931761fb71ae2735d97e0386127b6f2cf9e98e8bafb209484c35ebedf5a95c5","qualified_name":"Finset.disjoint_erase_insert","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":7834,"column":47,"line":216},"start":{"byte":7685,"column":0,"line":215}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  a ∉ s →\n    Iff\n      (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α)\n        (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a s)\n        (@Finset.erase.{u_1} α inst t a))\n      (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s t)","statement_sha256":"df1317906f450202dce216c86964e47651efd8c9c94e702b870878d820f612e9","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-9f69918d1da74e543cb142ed42c7150b7ea703232544c65ab99407f3165dbc2e","qualified_name":"Finset.disjoint_filter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":11560,"column":50,"line":315},"start":{"byte":11336,"column":0,"line":313}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s : Finset.{u_1} α} {p q : α → Prop} [inst : @DecidablePred.{u_1 + 1} α p]\n  [inst_1 : @DecidablePred.{u_1 + 1} α q],\n  Iff\n    (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α)\n      (@Finset.filter.{u_1} α p inst s) (@Finset.filter.{u_1} α q inst_1 s))\n    (∀ (x : α),\n      @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n          (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s x →\n        p x → Not (q x))","statement_sha256":"cceb1743195fe6d03863f5178ef753d76e1010c43270c64f85aca337805ef52d","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-94ee2d73c0fc1ab7b6085e3b91fc51f64728eec92902f17222f0d76381c80089","qualified_name":"Finset.disjoint_filter_filter'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":11865,"column":26,"line":323},"start":{"byte":11562,"column":0,"line":317}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (s t : Finset.{u_1} α) {p q : α → Prop} [inst : @DecidablePred.{u_1 + 1} α p]\n  [inst_1 : @DecidablePred.{u_1 + 1} α q],\n  @Disjoint.{u_1} (α → Prop) (@Pi.partialOrder.{u_1, 0} α (fun a => Prop) fun i => «Prop».partialOrder)\n      (@Pi.instOrderBot.{u_1, 0} α (fun a => Prop) (fun i => «Prop».le) fun i =>\n        @HeytingAlgebra.toOrderBot.{0} Prop «Prop».instHeytingAlgebra)\n      p q →\n    @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α)\n      (@Finset.filter.{u_1} α p inst s) (@Finset.filter.{u_1} α q inst_1 t)","statement_sha256":"cc6711471ba75b7e449c74d047100efe52e0d6cde1275e2f3fd53e77dfbe6142","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-2cf779b1055a8e489fda57eeed4505a7db2eaef82736a7b83d2c3356588722cb","qualified_name":"Finset.disjoint_filter_filter_not","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":12094,"column":50,"line":328},"start":{"byte":11867,"column":0,"line":325}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (s t : Finset.{u_1} α) (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p]\n  [inst_1 : (x : α) → Decidable (Not (p x))],\n  @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α)\n    (@Finset.filter.{u_1} α p inst s) ({a ∈ t | Not (p a)})","statement_sha256":"c243b1ca7723f89789ddfbcdeb56d6e1ac7adf2bd0de0e906779c63ab1eb783d","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-a9329a24cf4ab9aa15109e1c8115ffba608e3fd5165df755f74b2436619951af","qualified_name":"Finset.disjoint_insert_erase","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":7683,"column":43,"line":213},"start":{"byte":7538,"column":0,"line":212}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  a ∉ t →\n    Iff\n      (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α)\n        (@Finset.erase.{u_1} α inst s a)\n        (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a t))\n      (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s t)","statement_sha256":"f96b261a646acd03dc9c3dd81e53064923d199fd1931533e4c150f36376a102a","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-a8717fe78206ba80751e03cfa442091123165dedfaaf35eaa522fbbfa0761a28","qualified_name":"Finset.disjoint_of_erase_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":8044,"column":35,"line":220},"start":{"byte":7836,"column":0,"line":218}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  a ∉ t →\n    @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α)\n        (@Finset.erase.{u_1} α inst s a) t →\n      @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s t","statement_sha256":"237b07a4ceff325b21ed5b5cbe32efb9025c27616825c163cdad2e31545ee1fe","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-3d5fbace186279bdcd6df96c2fab1ef6031a4edd9d6c8115f84fde1b097c71a7","qualified_name":"Finset.disjoint_of_erase_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":8250,"column":35,"line":224},"start":{"byte":8046,"column":0,"line":222}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  a ∉ s →\n    @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s\n        (@Finset.erase.{u_1} α inst t a) →\n      @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s t","statement_sha256":"c68e2c3e60174fc5205183170bc6e7a2626b1c56a22e7ec770eacfb28f8ba3e6","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-cdb5d1e8a50ba1cf2a965b982dba1cfff4b1218df7a662f4ba1f1dcf3e0bd8b0","qualified_name":"Finset.disjoint_of_subset_iff_left_eq_empty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":2966,"column":34,"line":97},"start":{"byte":2839,"column":0,"line":95}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s t : Finset.{u_1} α},\n  @LE.le.{u_1} (Finset.{u_1} α)\n      (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n        (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n      s t →\n    Iff (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s t)\n      (@Eq.{u_1 + 1} (Finset.{u_1} α) s\n        (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))","statement_sha256":"f0ac78b17dcc980bd62758efb4fcd1c89d80393100478fc8c88b75cd52fe1467","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-781c122396e2924ac078bfa478daf2d2976b4540bd6f208910718f47ac7907b2","qualified_name":"Finset.disjoint_or_nonempty_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":2812,"column":12,"line":92},"start":{"byte":2660,"column":0,"line":90}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s t : Finset.{u_1} α),\n  Or (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s t)\n    (@Finset.Nonempty.{u_1} α (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) s t))","statement_sha256":"c48fa816003abe27861684453effbe21195f5510f991cbdf2f943e365c42319e","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-1860f11d443b70cba0682868d574d8d37983a8a14ab8171003a4e7bcb822b47e","qualified_name":"Finset.disjoint_sdiff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":10149,"column":21,"line":274},"start":{"byte":10081,"column":0,"line":273}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α},\n  @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α) s\n    (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) t s)","statement_sha256":"834874a782cb1e760faecb3cf90f8dc746c965b21d2cd6fd5e53723a0d637a68","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-098ee0fd74e0124fd4525f5c553bdc5f8b5e72c6198a7511bf537dba019c9ed5","qualified_name":"Finset.disjoint_sdiff_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":10290,"column":60,"line":277},"start":{"byte":10151,"column":0,"line":276}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s t : Finset.{u_1} α),\n  @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) 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u))","statement_sha256":"32381575e34fa0efbf59b05ccb55f6ecc8f5f69c86535cd956895f9e640c744f","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-9672773524870b11488d77eb03e699d890671784e2cc1332fa410a9f1ea7b116","qualified_name":"Finset.disjoint_union_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":2428,"column":59,"line":81},"start":{"byte":2269,"column":0,"line":79}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t u : Finset.{u_1} α},\n  Iff\n    (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) 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u))","statement_sha256":"20d9a644f85235b6c64f14cde8a8ad72a00c79326a2bdd4e41fd6c06da4d3d32","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-2ad9f32018b0e2bab1224ff9b23ea451fcf0b77d5120c792f9c2a7db15af217a","qualified_name":"Finset.empty_toList","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":18964,"column":78,"line":528},"start":{"byte":18886,"column":0,"line":528}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s : Finset.{u_1} α},\n  Iff (@Eq.{1} Bool (@List.isEmpty.{u_1} α (@Finset.toList.{u_1} α s)) Bool.true)\n    (@Eq.{u_1 + 1} (Finset.{u_1} α) s\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))","statement_sha256":"df9bca2e48b0561a42721660bf3da04fe5b531554fe302770f7e509a146f80ad","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-1a68349e9baeb0c3de2e7882ad708d5cfb33dfd9cab22c0bd858d034ac740da0","qualified_name":"Finset.equivToSet","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":24248,"column":38,"line":662},"start":{"byte":24044,"column":0,"line":659}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u_1} →\n  (s : Finset.{u_1} α) →\n    Equiv.{u_1 + 1, u_1 + 1} (↥s)\n      (@Set.Elem.{u_1} α (@SetLike.coe.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α) s))","statement_sha256":"a08b4ea96e4d3f39090c0c2587d62512c3aa5050061c53dca206b2587452a62f","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-346f0e109e214a1895bf5e5dba8469ce4c13b16d3021078fa33b4c90dc433433","qualified_name":"Finset.erase_cons","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":5645,"column":96,"line":163},"start":{"byte":5549,"column":0,"line":163}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Finset.{u_1} α} {a : α} (h : a ∉ s),\n  @Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.erase.{u_1} α inst (@Finset.cons.{u_1} α a s h) a) s","statement_sha256":"2b4688364babf46882ec127b82c4890878e41752cd52357acc9a122b3f99d26b","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-62b9e985a7799d28befb0a5fcc764061dd4620055eef8d09e6aac07a2df95a48","qualified_name":"Finset.erase_cons_of_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":4601,"column":92,"line":140},"start":{"byte":4425,"column":0,"line":139}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {a b : α} {s : Finset.{u_1} α} (ha : a ∉ s),\n  @Ne.{u_1 + 1} α a b →\n    @Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.erase.{u_1} α inst (@Finset.cons.{u_1} α a s ha) b)\n      (@Finset.cons.{u_1} α a (@Finset.erase.{u_1} α inst s b) ⋯)","statement_sha256":"bede91209976d3e00e36cf90e7bd4ab6a8a36d63f64a793278765a14f17524cf","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-4e72c3bd65c1df8eb143a71f2bcc505c662a683e6a7de2fa977202085bc421d0","qualified_name":"Finset.erase_empty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":3606,"column":5,"line":117},"start":{"byte":3541,"column":0,"line":115}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (a : α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Finset.erase.{u_1} α inst\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)) a)\n    (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α))","statement_sha256":"f0de09cbf8021c3ddfe764413fc7a7572836d437d80cdcf11145dbe5428ca32e","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-2bafa45be929537cc262e04ad8788fb6b723201d87ed969fdd528b76f5fafa74","qualified_name":"Finset.erase_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":7407,"column":37,"line":207},"start":{"byte":7303,"column":0,"line":206}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s : Finset.{u_1} α) (a : α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.erase.{u_1} α inst s a)\n    (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s\n      (@Singleton.singleton.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instSingleton.{u_1} α) a))","statement_sha256":"04d132ad22d1bfb9b26f6d3f2c2e7ceec0e163b6922802ddf872f9f44f2847f3","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-439a3a6fe03ab83a1ef6a3a35633aad4545919a475591bcd47a57e26f0502d00","qualified_name":"Finset.erase_eq_empty_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":9923,"column":84,"line":267},"start":{"byte":9739,"column":0,"line":266}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s : Finset.{u_1} α) (a : α),\n  Iff\n    (@Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.erase.{u_1} α inst s a)\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))\n    (Or\n      (@Eq.{u_1 + 1} (Finset.{u_1} α) s\n        (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))\n      (@Eq.{u_1 + 1} (Finset.{u_1} α) s\n        (@Singleton.singleton.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instSingleton.{u_1} α) a)))","statement_sha256":"f010e6ae8e4c62231cb6a68272b53b264e430b3b5509eee2042efcabf906e061","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-ecc05a6aedf34f6f70031ccf14b7faf591a95892dde40de327c84c6df46de25a","qualified_name":"Finset.erase_eq_iff_eq_insert","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":4873,"column":7,"line":147},"start":{"byte":4766,"column":0,"line":146}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s a →\n    a ∉ t →\n      Iff (@Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.erase.{u_1} α inst s a) t)\n        (@Eq.{u_1 + 1} (Finset.{u_1} α) s\n          (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a t))","statement_sha256":"fbad8bf4fb1fc55ea59d4d8bb3ec8d04364194e89e126807298ceacdbe7e7583","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-45ee7c01c4366fb51089de23bee8dd8ac82ca00e3a342346bbd28f04baa6d4c4","qualified_name":"Finset.erase_injOn'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":6435,"column":90,"line":180},"start":{"byte":6257,"column":0,"line":179}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (a : α),\n  @Set.InjOn.{u_1, u_1} (Finset.{u_1} α) (Finset.{u_1} α) (fun s => @Finset.erase.{u_1} α inst s a)\n    (@Set.ofPred.{u_1} (Finset.{u_1} α) fun s =>\n      @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n        (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s a)","statement_sha256":"f1c8967931e777fef6cffc76533b5ef624eedd3f7e633663d75bd24836e167cc","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-eaa3502db3a1a7f40f90cf07ac0a671d778529b58535de8b4b4fbff20629864a","qualified_name":"Finset.erase_insert","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":4292,"column":98,"line":134},"start":{"byte":4194,"column":0,"line":134}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {a : α} {s : Finset.{u_1} α},\n  a ∉ s →\n    @Eq.{u_1 + 1} (Finset.{u_1} α)\n      (@Finset.erase.{u_1} α inst (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a s)\n        a)\n      s","statement_sha256":"58d1b8afdb17df5cc9ff56faf711a417b7db0d3e8e425d80a10a62e1b9911f65","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-3ab7b8f2f1cbfb9ae5cc424b04154dbc937d0bfe74595f96e31d02b9ba6c6010","qualified_name":"Finset.erase_insert_eq_erase","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":4192,"column":101,"line":132},"start":{"byte":4083,"column":0,"line":131}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s : Finset.{u_1} α) (a : α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Finset.erase.{u_1} α inst (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a s) a)\n    (@Finset.erase.{u_1} α inst s a)","statement_sha256":"662bef7af0fd7a9f2f5033e3d4f352a27eb06d1a877dbe6d297da3b7ba0ea001","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-f56ef70659d23748dede7fbd531dd754ed318b138866eb34d5a029e2035cc5cd","qualified_name":"Finset.erase_insert_of_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":4423,"column":59,"line":137},"start":{"byte":4294,"column":0,"line":136}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {a b : α} {s : Finset.{u_1} α},\n  @Ne.{u_1 + 1} α a b →\n    @Eq.{u_1 + 1} (Finset.{u_1} α)\n      (@Finset.erase.{u_1} α inst (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a s)\n        b)\n      (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a\n        (@Finset.erase.{u_1} α inst s 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 (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n    (@Finset.erase.{u_1} α inst (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a s) a)\n    s","statement_sha256":"c852be62798eddc351291181b580f568adbdd93f967a11142788fe85c5ba1985","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-8e0f5bf8e4203286d8ad975a8597731f9b4860b9e1263afe8db4afb04437f7d1","qualified_name":"Finset.erase_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":8454,"column":96,"line":229},"start":{"byte":8350,"column":0,"line":228}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (a : α) (s t : Finset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Inter.inter.{u_1} (Finset.{u_1} α) 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a)","statement_sha256":"62127ae47479939c4771bce2c9370c7a6894fd4d208a41e230a5e7ad65fc2a6a","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-304f4297a314f337c2c45d6d8c75c7ac0d7d14cff1cb88038bab61812ae093ea","qualified_name":"Finset.erase_sdiff_distrib","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":9004,"column":7,"line":242},"start":{"byte":8894,"column":0,"line":241}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s t : Finset.{u_1} α) (a : α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Finset.erase.{u_1} α inst 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a))","statement_sha256":"11a34cc88a179df42841f018c3a1e25a3eb7c9ce99d46f8d6f3a9e64e8e81868","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-bedbf7c3b5fb79072b1058ca443ca5f49eb2892b5bf4375e0a6f745c0ebadcfa","qualified_name":"Finset.erase_sdiff_erase","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":7005,"column":12,"line":199},"start":{"byte":6902,"column":0,"line":198}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Finset.{u_1} α} {a b : α},\n  @Ne.{u_1 + 1} α a b →\n    @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n        (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s b →\n      @Eq.{u_1 + 1} (Finset.{u_1} α)\n        (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) (@Finset.erase.{u_1} α inst s a)\n          (@Finset.erase.{u_1} α inst s b))\n        (@Singleton.singleton.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instSingleton.{u_1} α) 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(Finset.{u_1} α) (@Finset.instSingleton.{u_1} α) a)\n      a)\n    (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α))","statement_sha256":"7baf218f45b1a52cf06912fb0ce2183dea25547a1720bf037289616a3353c3e3","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-c5447bb670f99755bc7a19dc6573f603cfa140dd78cfd11e369f7e52d4069d5b","qualified_name":"Finset.erase_ssubset","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":5152,"column":90,"line":153},"start":{"byte":5062,"column":0,"line":153}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {a : α} {s : Finset.{u_1} α},\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s a →\n    @LT.lt.{u_1} (Finset.{u_1} α)\n      (@Preorder.toLT.{u_1} (Finset.{u_1} α)\n        (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n      (@Finset.erase.{u_1} α inst s a) s","statement_sha256":"0c19f6b804d82b7679a1fe015c49347ebb2e4f947da71113f8ac9089bd3a49db","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-c9f5c2c711b19aa32651ec0824fb30091111fcf85345ba1d76a133eaf2f3c6a7","qualified_name":"Finset.erase_ssubset_insert","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":5547,"column":92,"line":161},"start":{"byte":5371,"column":0,"line":160}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s : Finset.{u_1} α) (a : α),\n  @LT.lt.{u_1} (Finset.{u_1} α)\n    (@Preorder.toLT.{u_1} (Finset.{u_1} α)\n      (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n    (@Finset.erase.{u_1} α inst s a)\n    (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a s)","statement_sha256":"5e783d383931073b740b79cf17f844fcf5861ee16d34a2ae8f85ab019aa4dd42","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-ec90d7c4d0dfc63c544f682e935cba12ab6e7e37bb52aa13f20e42ffaf33144c","qualified_name":"Finset.erase_subset_iff_of_mem","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":6255,"column":48,"line":177},"start":{"byte":6125,"column":0,"line":176}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) t a →\n    Iff\n      (@LE.le.{u_1} (Finset.{u_1} α)\n        (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n          (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n        (@Finset.erase.{u_1} α inst s a) t)\n      (@LE.le.{u_1} (Finset.{u_1} α)\n        (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n          (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n        s t)","statement_sha256":"60462529414b9788566a4bc8e0b0527f4da075dda324f8345deebb70c348ff60","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-01862ad539bb2fe2dc89a64120b93d1ab18ea804b9ae7bc9ffec5bf21f0ebf1d","qualified_name":"Finset.erase_union_distrib","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":8770,"column":7,"line":236},"start":{"byte":8656,"column":0,"line":235}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s t : Finset.{u_1} α) (a : α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Finset.erase.{u_1} α inst 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a))","statement_sha256":"c5cec172b552c9e8730a8b21d9899d4cc9cb4181b51a5c9860a4d2573ebecbe2","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-69502a8427e41dccb64371e87b075aa1bb402c05f7e56441f0d8fd7095de2df9","qualified_name":"Finset.erase_union_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":5253,"column":99,"line":155},"start":{"byte":5154,"column":0,"line":155}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (a : α) (s : Finset.{u_1} α),\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s a →\n    @Eq.{u_1 + 1} (Finset.{u_1} α)\n      (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) (@Finset.erase.{u_1} α inst s a)\n        (@Singleton.singleton.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instSingleton.{u_1} α) a))\n      s","statement_sha256":"dd25c4ddeb6d9d085cb8f652372b0b987f4fc6c2f28aa662e293c6a4555e8824","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-0cdc11230d4486802e2a167ea8031d4ec0191b10fdc1311e521b2bf920b4cb4d","qualified_name":"Finset.erase_union_of_mem","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":9105,"column":7,"line":245},"start":{"byte":9006,"column":0,"line":244}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {t : Finset.{u_1} α} {a : α},\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) t a →\n    ∀ (s : Finset.{u_1} α),\n      @Eq.{u_1 + 1} (Finset.{u_1} α)\n        (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) (@Finset.erase.{u_1} α inst s a) t)\n        (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t)","statement_sha256":"7767cbf6a3a76761a0352763b62279cd90498dff21fd5110a1c5dcc04adf7f48","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-035f7e50a776fba1af18acceff48360b61f5efaa3a76599225b08bb5ce8348e0","qualified_name":"Finset.filter_and","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":14245,"column":7,"line":380},"start":{"byte":14135,"column":0,"line":379}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p q : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : @DecidablePred.{u_1 + 1} α q]\n  [inst_2 : DecidableEq.{u_1 + 1} α] (s : Finset.{u_1} 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s))","statement_sha256":"e056c880043b4267dd70af923ba160bccac17d3b5f1048ab5f03c67c0171919d","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-597af1f4ec358bccb053f092f9124c04f5aa220d1cdc3c52ab88c5660074b8d9","qualified_name":"Finset.filter_and_not","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":14514,"column":77,"line":386},"start":{"byte":14342,"column":0,"line":385}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s : Finset.{u_1} α) (p q : α → Prop)\n  [inst_1 : @DecidablePred.{u_1 + 1} α p] [inst_2 : @DecidablePred.{u_1 + 1} α q],\n  @Eq.{u_1 + 1} (Finset.{u_1} α) ({a ∈ s | And (p a) (Not (q a))})\n    (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) (@Finset.filter.{u_1} α p inst_1 s)\n      (@Finset.filter.{u_1} α q inst_2 s))","statement_sha256":"97f4f52500d229150b6155d96c02727960a874fc3a60ff6faa1048260efd6317","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-8b9318f538ecd5ae56f0bad45aaec7c0bb46904474904e0e101dcb9fe5e2d0b8","qualified_name":"Finset.filter_cons","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":12501,"column":95,"line":336},"start":{"byte":12315,"column":0,"line":334}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] {a : α} (s : Finset.{u_1} α) (ha : a ∉ s),\n  @Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.filter.{u_1} α p inst (@Finset.cons.{u_1} α a s ha))\n    (@ite.{u_1 + 1} (Finset.{u_1} α) (p a) (inst a) (@Finset.cons.{u_1} α a (@Finset.filter.{u_1} α p inst s) ⋯)\n      (@Finset.filter.{u_1} α p inst s))","statement_sha256":"12246187aa0af686b7a1ad86a3cb71e3df53f2a38cf82dfaaa7005fdc787b518","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-185a7b754726e5fb3f5ee205c9f02c607074380f562ece3143e847c2c43cd30a","qualified_name":"Finset.filter_cons_of_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":11334,"column":42,"line":311},"start":{"byte":11167,"column":0,"line":309}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] (a : α) (s : Finset.{u_1} α) (ha : a ∉ s),\n  Not (p a) →\n    @Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.filter.{u_1} α p inst (@Finset.cons.{u_1} α a s ha))\n      (@Finset.filter.{u_1} α p inst s)","statement_sha256":"a85999631504fcb8138530dae04a3b5e5b784d6c11f401fcf359e87b59fd5604","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-9b0c6ab5ef6496eba386bde3d9de03a9a4ef8c44939f1a0a52b3526d540ba255","qualified_name":"Finset.filter_cons_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":11165,"column":42,"line":307},"start":{"byte":10961,"column":0,"line":305}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] (a : α) (s : Finset.{u_1} α) (ha : a ∉ s),\n  p a →\n    @Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.filter.{u_1} α p inst (@Finset.cons.{u_1} α a s ha))\n      (@Finset.cons.{u_1} α a (@Finset.filter.{u_1} α p inst s) ⋯)","statement_sha256":"85f9d2faeec88e280e656159451e1b3055b44cda7f1cc5b0b8cef73000ee14ad","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-5942b677f54b0db621d17d9831f3aabc93811112504a9211d90720b8b9581775","qualified_name":"Finset.filter_disjUnion","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":12313,"column":40,"line":332},"start":{"byte":12096,"column":0,"line":330}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] (s t : Finset.{u_1} α)\n  (h : @Disjoint.{u_1} (Finset.{u_1} α) 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⋯)","statement_sha256":"70d57cbfda1afa0bb923ce3e8ea00c106df0ffeffbb2e35004ac08c89ca9524d","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-9bfa3e105e5db4daad9bf86679d811103e4118a3fdd81360db3bf6f5016c71fe","qualified_name":"Finset.filter_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":15491,"column":55,"line":408},"start":{"byte":15202,"column":0,"line":403}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {β : Type u_2} [inst : DecidableEq.{u_2 + 1} β] (s : Finset.{u_2} β) (b : β),\n  @Eq.{u_2 + 1} (Finset.{u_2} β) (@Finset.filter.{u_2} β (@Eq.{u_2 + 1} β b) (inst b) s)\n    (@ite.{u_2 + 1} (Finset.{u_2} β)\n      (@Membership.mem.{u_2, u_2} β (Finset.{u_2} β)\n        (@SetLike.instMembership.{u_2, u_2} (Finset.{u_2} β) β (@Finset.instSetLike.{u_2} β)) s b)\n      (@Finset.decidableMem.{u_2} β inst b s)\n      (@Singleton.singleton.{u_2, u_2} β (Finset.{u_2} β) (@Finset.instSingleton.{u_2} β) b)\n      (@EmptyCollection.emptyCollection.{u_2} (Finset.{u_2} β) (@Finset.instEmptyCollection.{u_2} β)))","statement_sha256":"9786e86e4fa5d0c7871266f71219246911156e110bee2add2e988ac0cc462bd7","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-1c36dc74b38a8447591986d5f30d25de95facfcb8e5824da8ccbb74c8aa63279","qualified_name":"Finset.filter_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":15792,"column":65,"line":415},"start":{"byte":15493,"column":0,"line":410}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {β : Type u_2} [inst : DecidableEq.{u_2 + 1} β] (s : Finset.{u_2} β) (b : β),\n  @Eq.{u_2 + 1} (Finset.{u_2} β) ({a ∈ s | @Eq.{u_2 + 1} β a b})\n    (@ite.{u_2 + 1} 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β)))","statement_sha256":"0cae5471c500f9a05313bef4f608b96478e20f2189503f1843e418715de10164","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-1f5dcaa2a7f3f93cab2a24e92fc2087571d7d2aeb0b6d92704386a7516825fa4","qualified_name":"Finset.filter_erase","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":14022,"column":7,"line":374},"start":{"byte":13917,"column":0,"line":373}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : DecidableEq.{u_1 + 1} α] (a : α)\n  (s : Finset.{u_1} α),\n  @Eq.{u_1 + 1} 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u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : DecidableEq.{u_1 + 1} α] (a : α)\n  (s : Finset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Finset.filter.{u_1} α p inst\n      (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst_1) a s))\n    (@ite.{u_1 + 1} (Finset.{u_1} α) (p a) (inst a)\n      (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst_1) a\n        (@Finset.filter.{u_1} α p inst s))\n      (@Finset.filter.{u_1} α p inst s))","statement_sha256":"ad829da6c2cf2e845e103555c9668e856dfb8c7a6af2675c0423c4f509baa674","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-645bcbcb6cae3863dba3dab239c76aa854e57f45d8cef386393b0d0a4a959f80","qualified_name":"Finset.filter_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":13683,"column":90,"line":366},"start":{"byte":13593,"column":0,"line":366}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : DecidableEq.{u_1 + 1} α]\n  (s t : Finset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst_1) (@Finset.filter.{u_1} α p inst s) t)\n    (@Finset.filter.{u_1} α p inst (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst_1) s t))","statement_sha256":"35e6c7b3b6d0bf2d79f66813372deae9c84d035273d3439acc8dd9088009e73f","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-9075db716bc6917b04139b0cb65167999386701a6f3d2e4dc29e1a5970bc36e6","qualified_name":"Finset.filter_inter_distrib","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":13591,"column":7,"line":364},"start":{"byte":13482,"column":0,"line":363}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : DecidableEq.{u_1 + 1} α]\n  (s t : Finset.{u_1} α),\n  @Eq.{u_1 + 1} 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t))","statement_sha256":"d07bd286117fa083265508571a7009deec82cdcdf26b2a37201579a01d46f9e1","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-7be05d167b84262fb356ca158b00d65ebfb4631da53b8a9e112b558ebf9634ed","qualified_name":"Finset.filter_mem_eq_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":13347,"column":53,"line":358},"start":{"byte":13217,"column":0,"line":357}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α}\n  [inst_1 :\n    (i : α) →\n      Decidable\n        (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n          (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) t i)],\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    ({i ∈ s |\n      @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n        (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) t i})\n    (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) s 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t)","statement_sha256":"c75baa0b4c76f48719a74c5b6af6dc6725f75066605a013320d5cc2a7ece6e1c","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-3227a4ffd2a02579e67d95e6f6e6b6b51d369be5d5b0f07a33dcd1375b7bffa3","qualified_name":"Finset.filter_or","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":14133,"column":7,"line":377},"start":{"byte":14024,"column":0,"line":376}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p q : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : @DecidablePred.{u_1 + 1} α q]\n  [inst_2 : DecidableEq.{u_1 + 1} α] (s : Finset.{u_1} 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s))","statement_sha256":"2f83cd1e697c4cb0909673f7411015ede86c126148be8bc82961825f5f319aba","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-ba66593e61ee6b2b05ecc15b6e248c7f984cdff6aef13306098bb3960004a2da","qualified_name":"Finset.filter_singleton","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":10959,"column":87,"line":303},"start":{"byte":10872,"column":0,"line":303}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] (a : α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Finset.filter.{u_1} α p inst\n      (@Singleton.singleton.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instSingleton.{u_1} α) a))\n    (@ite.{u_1 + 1} (Finset.{u_1} α) (p a) (inst a)\n      (@Singleton.singleton.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instSingleton.{u_1} α) a)\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))","statement_sha256":"391d409b4ae730b62d851bc0b98cc56d6e0fe4af506ed368f2908717e687c7c1","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-59e735f83208122bd374a30c317e816cc14ab7832a8a36b0e0971740e57839c2","qualified_name":"Finset.filter_union","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":13095,"column":7,"line":352},"start":{"byte":12976,"column":0,"line":351}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : DecidableEq.{u_1 + 1} α]\n  (s₁ s₂ : Finset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} 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s₂))","statement_sha256":"f114de4f5021c403413c5e734be3d0774ba31b24e167b313b3ca58d403ffa218","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-af31c1c74b9ae9f60248befcdcb9bfe2c464f95063f51ff2c5e7ab7e0ae16c24","qualified_name":"Finset.filter_union_filter_not_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":16503,"column":73,"line":429},"start":{"byte":16298,"column":0,"line":427}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : DecidableEq.{u_1 + 1} α]\n  [inst_2 : (x : α) → Decidable (Not (p x))] 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s","statement_sha256":"f00c1c9c00fff0175c62502441ee9f1a9f300902a5df890eb6d611cfde2e31cb","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-eea51da815b1dafd0af68355eeca32f353a7e1158dd215182a99366b1973f3e3","qualified_name":"Finset.filter_union_filter_of_codisjoint","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":16296,"column":82,"line":425},"start":{"byte":16096,"column":0,"line":423}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p q : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : @DecidablePred.{u_1 + 1} α q]\n  [inst_2 : DecidableEq.{u_1 + 1} 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s","statement_sha256":"7e706e7b31c0cebf8eae26082f8a4c9b96caa5e52e49e064c5b0c57ab99a3d27","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-1e165973f42d16e12c4650b31ebd51dec534248c9dd54a2b4d2b475f080274ce","qualified_name":"Finset.filter_union_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":13215,"column":73,"line":355},"start":{"byte":13097,"column":0,"line":354}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p q : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : @DecidablePred.{u_1 + 1} α q]\n  [inst_2 : DecidableEq.{u_1 + 1} α] (s : 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x)})","statement_sha256":"9ad2daba13bd69a3836032150f976cc46fdcd2ed3fdbaa8527fbf882076e24b1","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-c299aa72e6e857b961fcd44ae8ef914f53fd7d931ab37b4a0a81ec54a64e0d17","qualified_name":"Finset.insert_erase","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":4764,"column":73,"line":144},"start":{"byte":4691,"column":0,"line":144}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Finset.{u_1} α} {a : α},\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, 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s","statement_sha256":"787225996460602ffd0d671cf250c676556059d2929882a3343e7ac388e7ac54","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-e696d98657933e048e395484b4317677c6e013a674cb6980f08001cd704f3837","qualified_name":"Finset.insert_erase_eq_insert","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":4689,"column":86,"line":142},"start":{"byte":4603,"column":0,"line":142}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Finset.{u_1} α} {a : α},\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Insert.insert.{u_1, u_1} α (Finset.{u_1} 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(@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n    s\n    (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a (@Finset.erase.{u_1} α inst s 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t))","statement_sha256":"113e56b9961b7fbb8fadf874a12a04446b1a26cafc56923045fe4a34383e79bc","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-5e7e28e480c48a8d110df736778d63f0cda947174b6c48a3a43710d4a314ac06","qualified_name":"Finset.inter_erase","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":8348,"column":96,"line":226},"start":{"byte":8252,"column":0,"line":226}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (a : α) (s t : Finset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) s (@Finset.erase.{u_1} α inst t a))\n    (@Finset.erase.{u_1} α inst (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) s t) a)","statement_sha256":"f386d8c2caaa25ca77596cab497a3623d2e6e846d000d0258a04ea854c4f0dc0","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-a6bdb3e7e88a67f7498bf2f349a5b948a717782c381eeace442698b167f527fc","qualified_name":"Finset.inter_filter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":13775,"column":90,"line":368},"start":{"byte":13685,"column":0,"line":368}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (p : α → Prop) [inst : @DecidablePred.{u_1 + 1} α p] [inst_1 : DecidableEq.{u_1 + 1} α]\n  (s t : Finset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst_1) s (@Finset.filter.{u_1} α p inst t))\n    (@Finset.filter.{u_1} α p inst (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst_1) s t))","statement_sha256":"3bd85dc5f5b58fc0a1c32c909f6066515d38eb79860b32872168489e769893f8","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-72f892831d4c9da739680a9babfbc617edfb202188c0d15c1f7f51dc105bcdcd","qualified_name":"Finset.isDirected_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":3369,"column":83,"line":106},"start":{"byte":3286,"column":0,"line":106}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1},\n  @IsDirectedOrder.{u_1} (Finset.{u_1} α)\n    (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n      (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))","statement_sha256":"35689f26f67825c7a20192d9c6a079774182939975a73240a79fa47eb7b1c2fe","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-0ceface301a7d1025b2d3475e581acddd1002b38dff996dc6173e2fc478bb06e","qualified_name":"Finset.isDirected_subset","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":3450,"column":80,"line":107},"start":{"byte":3370,"column":0,"line":107}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1},\n  IsDirected.{u_1 + 1} (Finset.{u_1} α) fun x1 x2 =>\n    @LE.le.{u_1} (Finset.{u_1} α)\n      (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n        (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n      x1 x2","statement_sha256":"f1968863a89ca5fb917af488599854fb77314cba7b3de52032782028ac836f32","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-f230b47574309333447e8aaf043366bd04cfb58183a436e72da8c534a1ef1724","qualified_name":"Finset.isEmpty_of_forall_eq_empty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":25102,"column":54,"line":695},"start":{"byte":24964,"column":0,"line":694}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_3},\n  (∀ (s : Finset.{u_3} α),\n      @Eq.{u_3 + 1} (Finset.{u_3} α) s\n        (@EmptyCollection.emptyCollection.{u_3} (Finset.{u_3} α) (@Finset.instEmptyCollection.{u_3} α))) →\n    IsEmpty.{u_3 + 1} α","statement_sha256":"66127c26c5d15185ed38efc5ebc78ba893f0b162cbe99b429bb72232774c961d","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-796230debf8f32eeed027e61e499994461eb988ffd8019d26962e88ccc733f20","qualified_name":"Finset.mem_union_of_disjoint","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":24781,"column":7,"line":688},"start":{"byte":24566,"column":0,"line":683}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_3} [inst : DecidableEq.{u_3 + 1} α] {s t : Finset.{u_3} α},\n  @Disjoint.{u_3} (Finset.{u_3} α) (@Finset.instPartialOrder.{u_3} α) 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x))","statement_sha256":"911ffe6ace29c78c3d567bc67a8ef9a2f758e36addcd8b08eb31c550b86b23e5","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-a57c5abd4c3e84f46319bd0ed60ba95cd31b434260aa69363bc3465ad5eed739","qualified_name":"Finset.not_disjoint_iff_nonempty_inter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":2585,"column":53,"line":86},"start":{"byte":2449,"column":0,"line":85}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α},\n  Iff (Not (@Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} 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[inst : DecidableEq.{u_1 + 1} α] {ι : Type u_3} {s : Set.{u_3} ι} {f : ι → Finset.{u_1} α},\n  Iff\n    (@Set.PairwiseDisjoint.{u_1, u_3} (Finset.{u_1} α) ι (@Finset.instPartialOrder.{u_1} α)\n      (@Finset.instOrderBot.{u_1} α) s f)\n    (∀ ⦃i : ι⦄,\n      @Membership.mem.{u_3, u_3} ι (Set.{u_3} ι) (@Set.instMembership.{u_3} ι) s i →\n        ∀ ⦃j : ι⦄,\n          @Membership.mem.{u_3, u_3} ι (Set.{u_3} ι) (@Set.instMembership.{u_3} ι) s j →\n            @Finset.Nonempty.{u_1} α\n                (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) (f i) (f j)) →\n              @Eq.{u_3 + 1} ι i j)","statement_sha256":"cf377e362cd54a86c0bd8446eaa065a1a2b3136c2bb3e74662edff9446de6fe1","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-fe5a944f24fd0b129da955874a608f5bcc26758a36277ccfd2108ac0c4a420aa","qualified_name":"Finset.range_filter_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":16700,"column":104,"line":445},"start":{"byte":16588,"column":0,"line":444}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n m : Nat},\n  @Eq.{1} (Finset.{0} Nat) ({x ∈ Finset.range n | @Eq.{1} Nat x m})\n    (@ite.{1} (Finset.{0} Nat) (@LT.lt.{0} Nat instLTNat m n) (m.decLt n)\n      (@Singleton.singleton.{0, 0} Nat (Finset.{0} Nat) (@Finset.instSingleton.{0} Nat) m)\n      (@EmptyCollection.emptyCollection.{0} (Finset.{0} Nat) (@Finset.instEmptyCollection.{0} Nat)))","statement_sha256":"46c6344a5fde9a8610d398a9162bf01544b66916d83bc156545062ac0e95b9d1","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-521d63102fe7e5eafbe3b98d002cc3154ae2980453676065f2791f8d3b7a7500","qualified_name":"Finset.range_inter_range","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":16803,"column":93,"line":448},"start":{"byte":16702,"column":0,"line":447}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (m n : Nat),\n  @Eq.{1} (Finset.{0} Nat)\n    (@Inter.inter.{0} (Finset.{0} Nat) (@Finset.instInter.{0} Nat instDecidableEqNat) (Finset.range m) 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(@Finset.instUnion.{0} Nat instDecidableEqNat) (Finset.range m) (Finset.range n))\n    (Finset.range (@Max.max.{0} Nat Nat.instMax m n))","statement_sha256":"ff775b3b9a63c5b9787d2fef8b0ad18211b82a3873a6be47563797ef2ac4a670","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-c277ec2a32fed60693064394548e7e6ec2e5693168c4977ae354ab023915ee2e","qualified_name":"Finset.sdiff_disjoint","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":10079,"column":49,"line":271},"start":{"byte":9983,"column":0,"line":270}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α},\n  @Disjoint.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α) (@Finset.instOrderBot.{u_1} α)\n    (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) t s) s","statement_sha256":"8d05a346fb2eee13b04873a06f859c7bc483ba5189d67e322907b2152b617c95","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-39bdecaf6fc8a869b79f068e8a01a109138fe0ce7ef9f9f8895092df0498cf50","qualified_name":"Finset.sdiff_eq_filter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":14617,"column":101,"line":388},"start":{"byte":14516,"column":0,"line":388}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s₁ s₂ : Finset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α) (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s₁ s₂)\n    ({x ∈ s₁ | x ∉ s₂})","statement_sha256":"f7ea0b97bfbe7cf4996ccc502c226fabd39257d22f3b13e3c2301209e1331610","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-19070fb4afdf72c8562aa918322713b6ddd944e177e0e42980da4b18230c8be1","qualified_name":"Finset.sdiff_erase","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":9660,"column":7,"line":261},"start":{"byte":9578,"column":0,"line":260}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s a →\n    @Eq.{u_1 + 1} (Finset.{u_1} α)\n      (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s (@Finset.erase.{u_1} α inst t a))\n      (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a\n        (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s t))","statement_sha256":"2cee605e971bdefeb54121775057e24d35c12dad2a417ad67e038c44673a7387","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-880ef6a304c8635073f00863be0a344fec5cb334bb4c434f7df66fdd8b8dfcb7","qualified_name":"Finset.sdiff_erase_self","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":9737,"column":7,"line":264},"start":{"byte":9662,"column":0,"line":263}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Finset.{u_1} α} {a : α},\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s a →\n    @Eq.{u_1 + 1} (Finset.{u_1} α)\n      (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s (@Finset.erase.{u_1} α inst s a))\n      (@Singleton.singleton.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instSingleton.{u_1} α) a)","statement_sha256":"fffa5cffd3dd58d1f57c0d4d28042474dcfa911aa73d79b5efd2fbb1714fc75e","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-3003d59670ee0c4751e80080da8caa7c57efb30423d37e3a038c66a2e1c2dd1f","qualified_name":"Finset.sdiff_insert","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":9417,"column":7,"line":254},"start":{"byte":9321,"column":0,"line":253}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s t : Finset.{u_1} α) (x : α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s\n      (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) x t))\n    (@Finset.erase.{u_1} α inst (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s t) x)","statement_sha256":"f27e88e6f1275517fd7d01d70aac6d02f4580cfddcf9eabaa04d988930f36e5f","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-14a0bfba272135c7bdbbd70cf209c4a302fc5fa7e37d09b0aba3bd751010ecec","qualified_name":"Finset.sdiff_insert_insert_of_mem_of_notMem","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":9576,"column":7,"line":258},"start":{"byte":9419,"column":0,"line":256}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {x : α},\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s x →\n    x ∉ t →\n      @Eq.{u_1 + 1} (Finset.{u_1} α)\n        (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) x\n          (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s\n            (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) x t)))\n        (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s t)","statement_sha256":"e1d7269a7539a1408f6adacbc2a55247d508151dce195257ba8d4edefeffa157","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-a733a4c350ecc61ca3a7aa9f967e4f0118537fc5a2bac9a113faf0d07a108c0e","qualified_name":"Finset.sdiff_singleton_eq_erase","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":7242,"column":91,"line":203},"start":{"byte":7151,"column":0,"line":203}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (a : α) (s : Finset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s\n      (@Singleton.singleton.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instSingleton.{u_1} α) a))\n    (@Finset.erase.{u_1} α inst s a)","statement_sha256":"9938e0d1527a07a2364ad87a924ef81276b591ebbaacf8cda8ecc7507ac6cb56","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-0dd7d32aaa51321babb3e24e089ad79dbe250ed9b7299c8b4f60ab3aef38848b","qualified_name":"Finset.sdiff_union_erase_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":9319,"column":7,"line":251},"start":{"byte":9208,"column":0,"line":250}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  @LE.le.{u_1} (Finset.{u_1} α)\n      (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n        (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n      t s →\n    @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n        (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) t a →\n      @Eq.{u_1 + 1} (Finset.{u_1} α)\n        (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst)\n          (@SDiff.sdiff.{u_1} (Finset.{u_1} α) (@Finset.instSDiff.{u_1} α inst) s t) (@Finset.erase.{u_1} α inst t a))\n        (@Finset.erase.{u_1} α inst s a)","statement_sha256":"cbf09e0103074c0e4d8958aae3ed849833cfc987d464de6f50338d69b72d4559","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-ab7975ba00222ad6517da3aedc0f832a04eb51ca01a6cdbde5fbede3f26a3ac2","qualified_name":"Finset.ssubset_iff_exists_subset_erase","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":5369,"column":7,"line":158},"start":{"byte":5255,"column":0,"line":157}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α},\n  Iff\n    (@LT.lt.{u_1} (Finset.{u_1} α)\n      (@Preorder.toLT.{u_1} (Finset.{u_1} α)\n        (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n      s t)\n    (∃ a,\n      And\n        (@Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n          (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) t a)\n        (@LE.le.{u_1} (Finset.{u_1} α)\n          (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n            (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n          s (@Finset.erase.{u_1} α inst t a)))","statement_sha256":"bfadd0ad65d29072fa0253332ef9a784ed87523edb8ecd88fe6405b4e131ac17","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-2f2f0ada326d227c499e58320495c205de8315062abc899065359a3ad16ef7ee","qualified_name":"Finset.subset_insert_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":5750,"column":103,"line":165},"start":{"byte":5647,"column":0,"line":165}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {a : α} {s t : Finset.{u_1} α},\n  Iff\n    (@LE.le.{u_1} (Finset.{u_1} α)\n      (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n        (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n      s (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a t))\n    (@LE.le.{u_1} (Finset.{u_1} α)\n      (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n        (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n      (@Finset.erase.{u_1} α inst s a) t)","statement_sha256":"93c021b1a1882f4c8221307fcee44d1dfbe588d56e3b67d95b58f213ef67828e","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-33562f9399f3e52696c63b9b03da1159ad6d43f0d8da1939a859194dcd9f7a74","qualified_name":"Finset.subset_insert_iff_of_notMem","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":6123,"column":46,"line":174},"start":{"byte":5990,"column":0,"line":173}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s t : Finset.{u_1} α} {a : α},\n  a ∉ s →\n    Iff\n      (@LE.le.{u_1} (Finset.{u_1} α)\n        (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n          (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n        s (@Insert.insert.{u_1, u_1} α (Finset.{u_1} α) (@Finset.instInsert.{u_1} α inst) a t))\n      (@LE.le.{u_1} (Finset.{u_1} α)\n        (@Preorder.toLE.{u_1} (Finset.{u_1} α)\n          (@PartialOrder.toPreorder.{u_1} (Finset.{u_1} α) (@Finset.instPartialOrder.{u_1} α)))\n        s t)","statement_sha256":"3474b709b561ae61897f8cbe89edeea03189f05a2c4b086453344752c1e45759","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-7ea63d707c4126fd6dd6982105ca6a48f33bf082612b7919cc48f15fc06ad5e8","qualified_name":"Finset.subset_union_elim","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":15077,"column":53,"line":399},"start":{"byte":14619,"column":0,"line":390}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Finset.{u_1} α} {t₁ t₂ : Set.{u_1} α},\n  @LE.le.{u_1} (Set.{u_1} α) (@Set.instLE.{u_1} α)\n      (@SetLike.coe.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α) s)\n      (@Union.union.{u_1} (Set.{u_1} α) (@Set.instUnion.{u_1} α) t₁ t₂) →\n    ∃ s₁ s₂,\n      And\n        (@Eq.{u_1 + 1} (Finset.{u_1} α) (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s₁ s₂) s)\n        (And\n          (@LE.le.{u_1} (Set.{u_1} α) (@Set.instLE.{u_1} α)\n            (@SetLike.coe.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α) s₁) t₁)\n          (@LE.le.{u_1} (Set.{u_1} α) (@Set.instLE.{u_1} α)\n            (@SetLike.coe.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α) s₂)\n            (@SDiff.sdiff.{u_1} (Set.{u_1} α) (@Set.instSDiff.{u_1} α) t₂ t₁)))","statement_sha256":"b0a1eb1e5c1e80733fd63bbfb653af352feaf07ec1f84ea7096e24be6700a1fc","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-d97293238ca89a8ba5fdaf3bf24c2b03657b7cd4e106b3e607c3389dd278c510","qualified_name":"Finset.toList_empty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":19053,"column":23,"line":532},"start":{"byte":18966,"column":0,"line":530}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1},\n  @Eq.{u_1 + 1} (List.{u_1} α)\n    (@Finset.toList.{u_1} α\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))\n    (@List.nil.{u_1} α)","statement_sha256":"604faedd73bbc42bf98897c4242e1591de804f311a66b2ea25dc41e1cbd63e52","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-fdfc9d20c14659bb8b553e0e7f9be0ec586caea7a1679945305ef5238a02f35c","qualified_name":"Finset.toList_eq_nil","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":18884,"column":42,"line":526},"start":{"byte":18765,"column":0,"line":524}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {s : Finset.{u_1} α},\n  Iff (@Eq.{u_1 + 1} (List.{u_1} α) (@Finset.toList.{u_1} α s) (@List.nil.{u_1} α))\n    (@Eq.{u_1 + 1} (Finset.{u_1} α) s\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))","statement_sha256":"a4b44cc608dd8189b593f8d26363a77860f33a77b9ea8661f77a39e5ba503fcd","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-a9fe62ac5d2fffdcaf8485735aa3188bc9f0db2dff80d0baf2b2cedea4743329","qualified_name":"Finset.union_erase_of_mem","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":9206,"column":7,"line":248},"start":{"byte":9107,"column":0,"line":247}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Finset.{u_1} α} {a : α},\n  @Membership.mem.{u_1, u_1} α (Finset.{u_1} α)\n      (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (@Finset.instSetLike.{u_1} α)) s a →\n    ∀ (t : Finset.{u_1} α),\n      @Eq.{u_1 + 1} (Finset.{u_1} α)\n        (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s (@Finset.erase.{u_1} α inst t a))\n        (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t)","statement_sha256":"4095baf1d3a361c55edd17943d74123cae299b847901429c367f5406548d4cf3","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-6199613255e1b6762766f44488de0ebe078c8a1bfebfbdb65d838518dd9e960e","qualified_name":"Finset.univ_finset_eq_singleton_empty_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":25351,"column":69,"line":700},"start":{"byte":25104,"column":0,"line":697}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_3},\n  Iff\n    (@Eq.{u_3 + 1} (Set.{u_3} (Finset.{u_3} α)) (@Set.univ.{u_3} (Finset.{u_3} α))\n      (@Singleton.singleton.{u_3, u_3} (Finset.{u_3} α) (Set.{u_3} (Finset.{u_3} α))\n        (@Set.instSingletonSet.{u_3} (Finset.{u_3} α))\n        (@EmptyCollection.emptyCollection.{u_3} (Finset.{u_3} α) (@Finset.instEmptyCollection.{u_3} α))))\n    (IsEmpty.{u_3 + 1} 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@Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.filter.{u_1} α p inst_1 (@List.toFinset.{u_1} α inst s))\n    (@List.toFinset.{u_1} α inst (@List.filter.{u_1} α (fun b => @Decidable.decide (p b) (inst_1 b)) 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α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@List.toFinset.{u_1} α inst\n      (@Inter.inter.{u_1} (List.{u_1} α) (@List.instInterOfBEq_batteries.{u_1} α (@instBEqOfDecidableEq.{u_1} α inst)) l\n        l'))\n    (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) (@List.toFinset.{u_1} α inst l)\n      (@List.toFinset.{u_1} α inst l'))","statement_sha256":"dae70d0da2bb26f25f46911a4ae4cc4f22912fa1da7428e0e48464ae8bed6ddb","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-2e979e226af5c77ce3d843f031c9281b6b7d8f557d20a1e9696965ad25f20b4d","qualified_name":"List.toFinset_union","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":18185,"column":6,"line":500},"start":{"byte":18068,"column":0,"line":497}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (l l' : List.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@List.toFinset.{u_1} α inst\n      (@Union.union.{u_1} (List.{u_1} α) (@List.instUnionOfBEq_batteries.{u_1} α (@instBEqOfDecidableEq.{u_1} α inst)) l\n        l'))\n    (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) (@List.toFinset.{u_1} α inst l)\n      (@List.toFinset.{u_1} α inst l'))","statement_sha256":"f7ce2b5c7acb53a69f5f7392561d4ca09d14b4ee5aab53096b3c328c87f42794","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-7ee0c765e07b07cca7bb5138a25d3ec1b4d69b28180b02ea839d29e187dca278","qualified_name":"Multiset.Aesop.toFinset_nonempty_of_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":17812,"column":75,"line":484},"start":{"byte":17685,"column":0,"line":483}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Multiset.{u_1} α},\n  @Ne.{u_1 + 1} (Multiset.{u_1} α) s\n      (@OfNat.ofNat.{u_1} (Multiset.{u_1} α) (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} (Multiset.{u_1} α) (@Multiset.instZero.{u_1} α))) →\n    @Finset.Nonempty.{u_1} α (@Multiset.toFinset.{u_1} α inst s)","statement_sha256":"25c9d5d710a00f4d557337bd4c42fcc1ec90c1c926d88258cb852ddd15e6e17d","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-0bc2be466704762b80daa10742e836e4acd02c56457c0072a6fb339db6da31f4","qualified_name":"Multiset.toFinset_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":17159,"column":23,"line":465},"start":{"byte":17037,"column":0,"line":463}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s t : Multiset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Multiset.toFinset.{u_1} α inst\n      (@HAdd.hAdd.{u_1, u_1, u_1} (Multiset.{u_1} α) (Multiset.{u_1} α) (Multiset.{u_1} α)\n        (@instHAdd.{u_1} (Multiset.{u_1} α) (@Multiset.instAdd.{u_1} α)) s t))\n    (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) (@Multiset.toFinset.{u_1} α inst s)\n      (@Multiset.toFinset.{u_1} α inst t))","statement_sha256":"5dd440d3758636f91fdb0a1c7f6a1b509d8530934430923d08cd46ea07e205e9","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-752dd93529aff0e387a273fb032e2efd589a49b420fd089e6a856d441b4b5ee5","qualified_name":"Multiset.toFinset_eq_empty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":17542,"column":50,"line":477},"start":{"byte":17408,"column":0,"line":475}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {m : Multiset.{u_1} α},\n  Iff\n    (@Eq.{u_1 + 1} (Finset.{u_1} α) (@Multiset.toFinset.{u_1} α inst m)\n      (@EmptyCollection.emptyCollection.{u_1} (Finset.{u_1} α) (@Finset.instEmptyCollection.{u_1} α)))\n    (@Eq.{u_1 + 1} (Multiset.{u_1} α) m\n      (@OfNat.ofNat.{u_1} (Multiset.{u_1} α) (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} (Multiset.{u_1} α) (@Multiset.instZero.{u_1} α))))","statement_sha256":"200a7d112b114d64a64d848d6ef81b0a6b9b4fa487abda00bd8eccdb8048c22c","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-722da7a29146c752623337bd380a7eca430c9c2eff1b8b1076c3f346916ed768","qualified_name":"Multiset.toFinset_filter","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":17967,"column":11,"line":489},"start":{"byte":17814,"column":0,"line":486}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s : Multiset.{u_1} α) (p : α → Prop)\n  [inst_1 : @DecidablePred.{u_1 + 1} α p],\n  @Eq.{u_1 + 1} 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{α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s t : Multiset.{u_1} α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α)\n    (@Multiset.toFinset.{u_1} α inst (@Inter.inter.{u_1} (Multiset.{u_1} α) (@Multiset.instInter.{u_1} α inst) s t))\n    (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) (@Multiset.toFinset.{u_1} α inst s)\n      (@Multiset.toFinset.{u_1} α inst t))","statement_sha256":"2eb1e1fa462a80c2e12b5103ad8122e197e81eca50079f4c6e5b51069431e53f","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-c3bcb05f91480384dda7805150d97ec7b0fe8a79fe63f4b4caacfe0fc0014d73","qualified_name":"Multiset.toFinset_nonempty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":17683,"column":65,"line":481},"start":{"byte":17544,"column":0,"line":479}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] {s : Multiset.{u_1} α},\n  Iff (@Finset.Nonempty.{u_1} α (@Multiset.toFinset.{u_1} α inst s))\n    (@Ne.{u_1 + 1} (Multiset.{u_1} α) s\n      (@OfNat.ofNat.{u_1} (Multiset.{u_1} α) (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} (Multiset.{u_1} α) (@Multiset.instZero.{u_1} α))))","statement_sha256":"77557d233035d7700d4493c102ad34193150a4395ed1cc5504f7a985cba06947","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-5092ad982966d7e6948b426601d83271214545d50631c244da0170bc494e672e","qualified_name":"Multiset.toFinset_replicate","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Finset.Basic","path":"Mathlib/Data/Finset/Basic.lean","range":{"end":{"byte":24509,"column":33,"line":675},"start":{"byte":24308,"column":0,"line":670}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (n : Nat) (a : α),\n  @Eq.{u_1 + 1} (Finset.{u_1} α) (@Multiset.toFinset.{u_1} α inst 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