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(@DFunLike.coe.{u_1 + 1, u_1 + 1, u_1 + 1}\n (@AddMonoidHom.{u_1, u_1} R R\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst))))\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst)))))\n R (fun x => R)\n (@AddMonoidHom.instFunLike.{u_1, u_1} R R\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst))))\n (@AddZeroClass.toAddZero.{u_1} R\n (@AddMonoid.toAddZeroClass.{u_1} R\n (@AddCommMonoid.toAddMonoid.{u_1} R (@NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R inst)))))\n (@AddMonoidHom.mulRight.{u_1} R inst r))\n fun x =>\n @HMul.hMul.{u_1, u_1, u_1} R R R\n (@instHMul.{u_1} R (@Distrib.toMul.{u_1} R (@NonUnitalNonAssocSemiring.toDistrib.{u_1} 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(Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t) x)\n (@Subtype.val.{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 x)\n ⋯)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-1eb6aaca90766173325ebf2b03d1d5b1e356a109f4c4807087b9ea2040157486","qualified_name":"Equiv.Finset.union_inr","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) (y : ↥t),\n @Eq.{u_1 + 1} (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t))\n (@DFunLike.coe.{u_1 + 1, u_1 + 1, u_1 + 1}\n (Equiv.{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 (Sum.{u_1, u_1} ↥s ↥t) (fun x => ↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t))\n (@EquivLike.toFunLike.{u_1 + 1, u_1 + 1, u_1 + 1}\n (Equiv.{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 (Sum.{u_1, u_1} ↥s ↥t) (↥(@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t))\n (@Equiv.instEquivLike.{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) (@Sum.inr.{u_1, u_1} (↥s) (↥t) y))\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 (@Subtype.val.{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 y)\n ⋯)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-21a58e1fc4671754152cc1dea0e19cc64f0aadf9c7802defd5154f71106c289e","qualified_name":"Equiv.Finset.union_symm_left","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)) 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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-3f17c31cd125ab2fb782cab771ac45c3fbbabe87581343fb8e8b5484ec0f0841","qualified_name":"Equiv.Finset.union_symm_right","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))"},{"kind":"definition","module":"Mathlib.Data.Finset.Basic","object_id":"object-caebe61b574ecec837bcb3cb4b3d31f918db225f35f3c60408327a199ebbb044","qualified_name":"Equiv.piFinsetUnion","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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-9ee6db5f467cfc8d1a8a77b441820d04e403380dd8c2341b90cade2253b1cb26","qualified_name":"Equiv.piFinsetUnion_left","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} 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) →\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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-fc335c7feddcbde8cdce3627eb9f22e4151387449124df6678805ab04f03cbfb","qualified_name":"Equiv.piFinsetUnion_right","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) →\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 (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))"},{"kind":"definition","module":"Mathlib.Data.Set.Basic","object_id":"object-60e916d8bfbc154e921847c89dca6bf41a4560ef5eb4651be6820847e50a9b4f","qualified_name":"Equiv.setSubtypeComm","statement":"{α : Type u_1} →\n (p : α → Prop) →\n Equiv.{u_1 + 1, max 1 (u_1 + 1)} (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)"},{"kind":"theorem","module":"Mathlib.Data.Set.Basic","object_id":"object-a43aa2e84c9a1340e5c77f1aac4db701cd48aaf686405caba45bffcd6fcbb612","qualified_name":"Equiv.setSubtypeComm_apply","statement":"∀ {α : Type u_1} (p : α → Prop) (s : Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a)),\n @Eq.{u_1 + 1}\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (@DFunLike.coe.{u_1 + 1, u_1 + 1, u_1 + 1}\n (Equiv.{u_1 + 1, u_1 + 1} (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a))\n (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (fun x =>\n @Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (@EquivLike.toFunLike.{u_1 + 1, u_1 + 1, u_1 + 1}\n (Equiv.{u_1 + 1, u_1 + 1} (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a))\n (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (@Equiv.instEquivLike.{u_1 + 1, u_1 + 1} (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)))\n (@Equiv.setSubtypeComm.{u_1} α p) s)\n (@Subtype.mk.{u_1 + 1} (Set.{u_1} α)\n (fun s => ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (@Set.ofPred.{u_1} α fun a =>\n ∃ (h : p a),\n @Membership.mem.{u_1, u_1} (@Subtype.{u_1 + 1} α fun a => p a) (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@Set.instMembership.{u_1} (@Subtype.{u_1 + 1} α fun a => p a)) s\n (@Subtype.mk.{u_1 + 1} α (fun a => p a) a h))\n ⋯)"},{"kind":"theorem","module":"Mathlib.Data.Set.Basic","object_id":"object-e25b907b44f020384015442297580511f80acfa45b24d183992a74bc65e6943d","qualified_name":"Equiv.setSubtypeComm_symm_apply","statement":"∀ {α : Type u_1} (p : α → Prop)\n (s :\n @Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a),\n @Eq.{u_1 + 1} (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@DFunLike.coe.{u_1 + 1, u_1 + 1, u_1 + 1}\n (Equiv.{u_1 + 1, u_1 + 1}\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a)))\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (fun x => Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@EquivLike.toFunLike.{u_1 + 1, u_1 + 1, u_1 + 1}\n (Equiv.{u_1 + 1, u_1 + 1}\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a)))\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@Equiv.instEquivLike.{u_1 + 1, u_1 + 1}\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))))\n (@Equiv.symm.{u_1 + 1, u_1 + 1} (Set.{u_1} (@Subtype.{u_1 + 1} α fun a => p a))\n (@Subtype.{u_1 + 1} (Set.{u_1} α) fun s =>\n ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a)\n (@Equiv.setSubtypeComm.{u_1} α p))\n s)\n (@Set.ofPred.{u_1} (@Subtype.{u_1 + 1} α fun a => p a) fun a =>\n @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α)\n (@Subtype.val.{u_1 + 1} (Set.{u_1} α)\n (fun s => ∀ (a : α), @Membership.mem.{u_1, u_1} α (Set.{u_1} α) (@Set.instMembership.{u_1} α) s a → p a) s)\n (@Subtype.val.{u_1 + 1} α (fun a => p a) 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α},\n @Finset.Nontrivial.{u_1} α s → @Finset.Nonempty.{u_1} α (@Finset.erase.{u_1} α inst s a)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-0043f186461afcd86a86c8c9e2050f0fdf091680ef90c26a6422145107adda98","qualified_name":"Finset.Nontrivial.exists_cons_eq","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"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-12f48084bcabd65866a9b6c7ab552d07337e20abd9d89c3a21fe9d1b1d0e71a8","qualified_name":"Finset.attach_empty","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))))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-4c9a3470eac1cca6a14e32e1cc60fe3a499bd7daa96346b92eff40a8737a1904","qualified_name":"Finset.attach_eq_empty_iff","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} 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α"},{"kind":"definition","module":"Mathlib.Data.Finset.Basic","object_id":"object-26c90d022387e31ba33a9feae9cb8c266f8775c29b0f30f0c41d079a47f68c35","qualified_name":"Finset.chooseX","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)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-3edf8adc74054fa085eb8853bead4f182f289c4fb08036ca159267a77bf6cdbb","qualified_name":"Finset.choose_eq_iff","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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-914bf2812a326f0d7b2c97ddb851514329f293782e095d86321f4cd49efeda5f","qualified_name":"Finset.choose_mem","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)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-465fc66b05e6a73b7dc31bcb3f2f7dbd568264563bd5b76b082b75c495eb8344","qualified_name":"Finset.choose_property","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)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-1be1ee31a0298e1d28c644dd3650c1acc02a6bf5d6c76fd434b40d20a66f6783","qualified_name":"Finset.choose_spec","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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-a7d06ba598b8ebeed2fbb4ea46ac98e98b7874b10f583489a4ca872a65b20942","qualified_name":"Finset.disjUnion_eq_union","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),\n @Eq.{u_1 + 1} (Finset.{u_1} α) (@Finset.disjUnion.{u_1} α s t h)\n (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-b59c142f14547a24e913145c20217bfa3421e000b883057585c3a34a9acaabe5","qualified_name":"Finset.disjoint_disjUnion_left","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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-364e3ebf9d0c8bfdca7cbb96b39777ffa675d7353c52ed39d40d0b42b0c7c009","qualified_name":"Finset.disjoint_disjUnion_right","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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-3b250d2182c4dd7307633f1a94aa54fa0233f97ca1bb177d39b10f948327066b","qualified_name":"Finset.disjoint_erase_comm","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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-4931761fb71ae2735d97e0386127b6f2cf9e98e8bafb209484c35ebedf5a95c5","qualified_name":"Finset.disjoint_erase_insert","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)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-9f69918d1da74e543cb142ed42c7150b7ea703232544c65ab99407f3165dbc2e","qualified_name":"Finset.disjoint_filter","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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-94ee2d73c0fc1ab7b6085e3b91fc51f64728eec92902f17222f0d76381c80089","qualified_name":"Finset.disjoint_filter_filter'","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)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-2cf779b1055a8e489fda57eeed4505a7db2eaef82736a7b83d2c3356588722cb","qualified_name":"Finset.disjoint_filter_filter_not","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)})"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-a9329a24cf4ab9aa15109e1c8115ffba608e3fd5165df755f74b2436619951af","qualified_name":"Finset.disjoint_insert_erase","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)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-a8717fe78206ba80751e03cfa442091123165dedfaaf35eaa522fbbfa0761a28","qualified_name":"Finset.disjoint_of_erase_left","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 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(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)))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-781c122396e2924ac078bfa478daf2d2976b4540bd6f208910718f47ac7907b2","qualified_name":"Finset.disjoint_or_nonempty_inter","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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-1860f11d443b70cba0682868d574d8d37983a8a14ab8171003a4e7bcb822b47e","qualified_name":"Finset.disjoint_sdiff","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)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-098ee0fd74e0124fd4525f5c553bdc5f8b5e72c6198a7511bf537dba019c9ed5","qualified_name":"Finset.disjoint_sdiff_inter","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) s t)\n (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) s t)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-709ad6ae3079b595bcadf911d484fc887024db680b3c49f6b26ccdbcc4f22a2f","qualified_name":"Finset.disjoint_union_left","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) (@Finset.instOrderBot.{u_1} α)\n (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t) 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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-9672773524870b11488d77eb03e699d890671784e2cc1332fa410a9f1ea7b116","qualified_name":"Finset.disjoint_union_right","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) (@Finset.instOrderBot.{u_1} α) s\n (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) t u))\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} 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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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-45ee7c01c4366fb51089de23bee8dd8ac82ca00e3a342346bbd28f04baa6d4c4","qualified_name":"Finset.erase_injOn'","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 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a)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-f56ef70659d23748dede7fbd531dd754ed318b138866eb34d5a029e2035cc5cd","qualified_name":"Finset.erase_insert_of_ne","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 b))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-c7193108654593e79b1642f6b294c52620f5865ee71f61fd4df29a2a45d74ffa","qualified_name":"Finset.erase_insert_subset","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (a : α) (s : 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) 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(@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) (@Finset.erase.{u_1} α inst s a) t)\n (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst) s (@Finset.erase.{u_1} α inst t a))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-4999a420522d4f57280c972363f1b2397fdff6650069e5cc98d095e33a1a99b1","qualified_name":"Finset.erase_nonempty","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 Iff (@Finset.Nonempty.{u_1} α (@Finset.erase.{u_1} α inst s a)) (@Finset.Nontrivial.{u_1} α s)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-e2db1d3f45b82be84f1f783ba1b599aebc05ba7836513d97e8cc406e84eec00f","qualified_name":"Finset.erase_sdiff_comm","statement":"∀ {α : Type u_1} [inst : DecidableEq.{u_1 + 1} α] (s t : Finset.{u_1} 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(Finset.{u_1} α)\n (@Finset.erase.{u_1} α inst (@Singleton.singleton.{u_1, u_1} α (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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-c5447bb670f99755bc7a19dc6573f603cfa140dd78cfd11e369f7e52d4069d5b","qualified_name":"Finset.erase_ssubset","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) 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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)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-01862ad539bb2fe2dc89a64120b93d1ab18ea804b9ae7bc9ffec5bf21f0ebf1d","qualified_name":"Finset.erase_union_distrib","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 (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) s t) a)\n (@Union.union.{u_1} (Finset.{u_1} α) (@Finset.instUnion.{u_1} α inst) (@Finset.erase.{u_1} α inst s a)\n (@Finset.erase.{u_1} α inst t 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(@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)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-035f7e50a776fba1af18acceff48360b61f5efaa3a76599225b08bb5ce8348e0","qualified_name":"Finset.filter_and","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} α),\n @Eq.{u_1 + 1} (Finset.{u_1} α) ({a ∈ s | And (p a) (q a)})\n (@Inter.inter.{u_1} (Finset.{u_1} α) (@Finset.instInter.{u_1} α inst_2) (@Finset.filter.{u_1} α p inst s)\n (@Finset.filter.{u_1} α q inst_1 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(@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)))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-1f5dcaa2a7f3f93cab2a24e92fc2087571d7d2aeb0b6d92704386a7516825fa4","qualified_name":"Finset.filter_erase","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} (Finset.{u_1} α) (@Finset.filter.{u_1} α p inst (@Finset.erase.{u_1} α inst_1 s a))\n (@Finset.erase.{u_1} α inst_1 (@Finset.filter.{u_1} α p inst s) a)"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-c50a09c0678ed9234246aa187566fe1fb3c4cd203d0739428e4eecec04fca85b","qualified_name":"Finset.filter_insert","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} (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))"},{"kind":"theorem","module":"Mathlib.Data.Finset.Basic","object_id":"object-645bcbcb6cae3863dba3dab239c76aa854e57f45d8cef386393b0d0a4a959f80","qualified_name":"Finset.filter_inter","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 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(@instHPow.{0, 0} Real Real Real.instPow) a b) nr dr"},{"kind":"theorem","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","object_id":"object-3894eeb122727286137fde8bafb3f6ac6aca19cde028f85292cb0ade1a042e7e","qualified_name":"Mathlib.Meta.NormNum.IsNat.rpow_eq_pow","statement":"∀ {b : Real} {n : Nat},\n @Mathlib.Meta.NormNum.IsNat.{0} Real\n (@AddGroupWithOne.toAddMonoidWithOne.{0} Real (@Ring.toAddGroupWithOne.{0} Real Real.instRing)) b n →\n ∀ (a : Real),\n @Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) a b)\n (@HPow.hPow.{0, 0, 0} Real Nat Real\n (@instHPow.{0, 0} Real Nat (@NPow.toPow.{0} Real (@Monoid.toNPow.{0} Real Real.instMonoid))) a n)"},{"kind":"theorem","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","object_id":"object-9f228f746f82e39a07eba0cd1539d6f5df4a6ac3d155e0d2441de1045ad45285","qualified_name":"Mathlib.Meta.NormNum.IsNat.rpow_isNNRat","statement":"∀ {a b : Real} {m n d r : Nat},\n @Mathlib.Meta.NormNum.IsNat.{0} Real\n (@AddGroupWithOne.toAddMonoidWithOne.{0} Real (@Ring.toAddGroupWithOne.{0} Real Real.instRing)) a m →\n @Mathlib.Meta.NormNum.IsNNRat.{0} Real Real.semiring b n d →\n ∀ (k : Nat),\n @Eq.{1} Nat\n (@HPow.hPow.{0, 0, 0} Nat Nat Nat\n (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid))) r d)\n k →\n ∀ (l : Nat),\n @Eq.{1} Nat\n (@HPow.hPow.{0, 0, 0} Nat Nat Nat\n (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid))) m n)\n l →\n @Eq.{1} Nat k l →\n @Mathlib.Meta.NormNum.IsNat.{0} Real\n (@AddGroupWithOne.toAddMonoidWithOne.{0} Real (@Ring.toAddGroupWithOne.{0} Real Real.instRing))\n (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) a b) 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g)"},{"kind":"theorem","module":"Mathlib.Algebra.Group.Basic","object_id":"object-77f7ca15c0ea5fdd806070a30d6a3dc9ca33e0df608479871e16e5abace40ea0","qualified_name":"zsmul_induction_right","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {g : G} {P : G → Prop},\n P\n (@OfNat.ofNat.{u_3} G (nat_lit 0)\n (@Zero.toOfNat0.{u_3} G\n (@NegZeroClass.toZero.{u_3} G\n (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst)))))) →\n (∀ (a : G),\n P a →\n P\n (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n (@instHAdd.{u_3} G\n (@AddZero.toAdd.{u_3} G\n (@AddZeroClass.toAddZero.{u_3} G\n (@AddMonoid.toAddZeroClass.{u_3} G\n (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n a g)) →\n (∀ (a : G),\n P a →\n P\n (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n (@instHAdd.{u_3} G\n (@AddZero.toAdd.{u_3} G\n (@AddZeroClass.toAddZero.{u_3} G\n (@AddMonoid.toAddZeroClass.{u_3} G\n (@SubNegMonoid.toAddMonoid.{u_3} 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