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{n : Nat},\n  Nat.Coprime (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n → @Odd.{0} Nat Nat.instSemiring n","statement_sha256":"3a2b47f5184a74bb38b2d177aa9641c0cf0088ba85c9d164473b61bc92f37d27","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-f0b1db6dc6e88bf5feb3e51d8a89db50378cab92835d21f849d7ecc4097be3c0","qualified_name":"Nat.Coprime.odd_of_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":7060,"column":93,"line":162},"start":{"byte":6967,"column":0,"line":162}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat}, n.Coprime (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) → @Odd.{0} Nat Nat.instSemiring n","statement_sha256":"a2fa1efbffd5ae98eea15cdf7ceab8bad77a5c2069c6311f8680f836a8604b7e","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-3c5d8c380ebfbd953dd8b9831d280005dfaa7820c94a73f76bad49bc7de1d5d1","qualified_name":"Nat.Prime.coprime_pow_of_not_dvd","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":9010,"column":47,"line":207},"start":{"byte":8858,"column":0,"line":206}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {p m a : Nat},\n  Nat.Prime p →\n    Not (@Dvd.dvd.{0} Nat Nat.instDvd p a) →\n      a.Coprime\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))) p 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(@Eq.{1} Nat p a)","statement_sha256":"91e372c9fc987655dba18214110a0ed3effad34d6a9c4ba0007a43b7203e567d","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-d71ee965aef1812981c4bca250e7ec2f6723c7a9b25f838fd1da6e73e1543772","qualified_name":"Nat.Prime.dvd_iff_not_coprime","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":5496,"column":39,"line":130},"start":{"byte":5365,"column":0,"line":129}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {p n : Nat}, Nat.Prime p → Iff (@Dvd.dvd.{0} Nat Nat.instDvd p n) (Not (p.Coprime 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Nat Nat.instDvd p2 n →\n            @Dvd.dvd.{0} Nat Nat.instDvd (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) p1 p2) n","statement_sha256":"c25d058d0a7a380ce98af6909f4b8329839c4da8048349328a0b74e8ed8729e6","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-46960ee72f42df2474095f781cdfe93f5e4cdf39c362b08533c8860320abefe1","qualified_name":"Nat.Prime.dvd_of_dvd_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":7178,"column":27,"line":165},"start":{"byte":7062,"column":0,"line":164}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {p m n : Nat},\n  Nat.Prime p →\n    @Dvd.dvd.{0} Nat Nat.instDvd p\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      @Dvd.dvd.{0} Nat Nat.instDvd p m","statement_sha256":"ac5a9e789c17accda119df119e80e427a807930246e5db9ee97c17542687f8e6","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-a06b93ecd90f306e4f4a693aec343b4162061f377ec5de418a4cee1bddeff0a0","qualified_name":"Nat.Prime.eq_one_of_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":7512,"column":39,"line":174},"start":{"byte":7402,"column":0,"line":173}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x n : Nat},\n  Nat.Prime\n      (@HPow.hPow.{0, 0, 0} Nat Nat Nat\n        (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat 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(@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))\n      (@Eq.{1} Nat\n        (@HMod.hMod.{0, 0, 0} Nat Nat Nat (@instHMod.{0} Nat Nat.instMod) p\n          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))\n        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 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p)","statement_sha256":"a24b2f8530ea6702787ca901ba806ddb2017e15300eed6a8181854b9ea2b0423","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-78c7a810c54374cce3db30c33f8c8f5fff12bcd9cbfa09e23aec206345ac3154","qualified_name":"Nat.Prime.even_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":3868,"column":69,"line":97},"start":{"byte":3726,"column":0,"line":96}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {p : Nat},\n  Nat.Prime p →\n    Iff (@Even.{0} Nat instAddNat p) (@Eq.{1} Nat p (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))","statement_sha256":"cb47a92a7b1c912efb73c954d099dbd7aa0917a3d96168cf11029d9db43ac1ac","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-03cc5295f9c840215e7281a468f75be28b99ae4234ccd5a3c3e9d27e7b1b3001","qualified_name":"Nat.Prime.even_sub_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":4323,"column":90,"line":107},"start":{"byte":4148,"column":0,"line":106}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {p : Nat},\n  Nat.Prime p →\n    @Ne.{1} Nat p (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) →\n      @Even.{0} Nat instAddNat\n        (@HSub.hSub.{0, 0, 0} Nat Nat Nat (@instHSub.{0} Nat instSubNat) p\n          (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))","statement_sha256":"92c673bfdec9f1d4cda570a7d79e90c1104ac78c79e6032f7ba631f4eaccd110","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-ed85f9e76b9dd48a22e4d500f9b3dab31383d875be7929a879882fb16ac1b634","qualified_name":"Nat.Prime.five_le_of_ne_two_of_ne_three","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":1817,"column":16,"line":54},"start":{"byte":1633,"column":0,"line":50}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {p : Nat},\n  Nat.Prime p →\n    @Ne.{1} Nat p (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) →\n      @Ne.{1} Nat p (@OfNat.ofNat.{0} Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) →\n        @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) p","statement_sha256":"19a50fa55a1793650fbf5bb65c0e224ef7d36dd9737bc8a7499ca9a7f21c075f","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-3234cf4d717ba609c6a5da713481c508166fb60ae2877a35beb99d28e4b4e6d4","qualified_name":"Nat.Prime.mod_two_eq_one_iff_ne_two","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":5153,"column":11,"line":124},"start":{"byte":4904,"column":0,"line":120}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {p : Nat},\n  Nat.Prime p →\n    Iff\n      (@Eq.{1} Nat\n        (@HMod.hMod.{0, 0, 0} Nat Nat Nat (@instHMod.{0} Nat Nat.instMod) p\n          (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))\n        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))\n      (@Ne.{1} Nat p (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))","statement_sha256":"1da9d6ca2eb2e3d3d2b87292abff33af2b125bffbc11c51f7c6412b8e096d966","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-53a2ef25fcf59731af48a80c1183436db9b35bbd928439003ac4878c741ff2b9","qualified_name":"Nat.Prime.mul_eq_prime_sq_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":8856,"column":11,"line":204},"start":{"byte":7764,"column":0,"line":181}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y p : Nat},\n  Nat.Prime p →\n    @Ne.{1} Nat x (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) →\n      @Ne.{1} Nat y (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) →\n        Iff\n          (@Eq.{1} Nat (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) x y)\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))) p\n              (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))\n          (And (@Eq.{1} Nat x p) (@Eq.{1} Nat y p))","statement_sha256":"2927a44dc8f460089fa0a411cd1e65bae61e6f7c7af37994a91be912e910cdbd","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-c8c44bbeec1627f4526b3aa9aa5b0b8782c935ff127a3fe35703b397b20ab2a1","qualified_name":"Nat.Prime.not_coprime_iff_dvd","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":5916,"column":73,"line":140},"start":{"byte":5498,"column":0,"line":132}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {m n : Nat},\n  Iff (Not (m.Coprime n))\n    (∃ p, And (Nat.Prime p) (And (@Dvd.dvd.{0} Nat Nat.instDvd p m) (@Dvd.dvd.{0} Nat Nat.instDvd p 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@Dvd.dvd.{0} Nat Nat.instDvd k (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) →\n    m.Coprime n","statement_sha256":"a43204f550293acbfc1fa5059e79b2bc7f4fa6dcd173b6c11e935d8b5fb9e349","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-236e9ffa5021efeb428c9eb4d87ff7ceebcfc251eed5d78a935d8efdf87f9dc7","qualified_name":"Nat.coprime_of_lt_minFac","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":6267,"column":91,"line":147},"start":{"byte":5918,"column":0,"line":142}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n m : Nat},\n  @Ne.{1} Nat m (@OfNat.ofNat.{0} 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instLENat n p → Nat.Prime p → Or (@Eq.{1} Nat p n) (p.Coprime n)","statement_sha256":"4eb649f0e909c0d71b42d66bcca0249d01223f66b5e2b23c40301dc4e0c88b85","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-20090a42b4453287c441a009e9ead8cce3a5aa297e8247a1676f1fa977d0602a","qualified_name":"Nat.eq_prime_pow_of_dvd_least_prime_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":10892,"column":98,"line":245},"start":{"byte":10471,"column":0,"line":237}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {a p k : Nat},\n  Nat.Prime p →\n    Not\n        (@Dvd.dvd.{0} Nat Nat.instDvd a\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))) p k)) →\n      @Dvd.dvd.{0} Nat Nat.instDvd a\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))) p\n            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k\n              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))) →\n        @Eq.{1} Nat a\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))) p\n            (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k\n              (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))","statement_sha256":"4a77937cbe855b22d23d5fb556edbb847a8b56b377c45ebc11ec6fee8bd07408","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-cb1e31c469cdfdf4a82f9163dad7ce6862843a9d70925eff392284d17ac35f71","qualified_name":"Nat.exists_dvd_of_not_prime","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":2262,"column":52,"line":66},"start":{"byte":2017,"column":0,"line":64}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n →\n    Not (Nat.Prime n) →\n      ∃ m,\n        And (@Dvd.dvd.{0} Nat Nat.instDvd m n)\n          (And (@Ne.{1} Nat m (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@Ne.{1} Nat m n))","statement_sha256":"2642db280c518ce4215b002771cd919f6a44082a7a99bdf5c5ec913ef513d4d3","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-830bee5e5b797389e5560820c26670c982ae7b2869076cfcd46fe00808857859","qualified_name":"Nat.exists_dvd_of_not_prime2","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":2491,"column":40,"line":71},"start":{"byte":2264,"column":0,"line":68}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n →\n    Not (Nat.Prime n) →\n      ∃ m,\n        And (@Dvd.dvd.{0} Nat Nat.instDvd m n)\n          (And (@LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) m)\n            (@LT.lt.{0} Nat instLTNat m n))","statement_sha256":"7fd7a718ab0867f1e308a1c7ffefdaac2e4da69da0027ba94184af39f49d3a3c","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-2f0866ea51518faa12f05767cae2e498d6787935bf657b35c2d361a717b95f08","qualified_name":"Nat.forall_prime_iff_two_and_odd","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":4728,"column":42,"line":115},"start":{"byte":4325,"column":0,"line":109}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {P : Nat → Prop},\n  Iff (∀ (p : Nat), Nat.Prime p → P p)\n    (And (P (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))\n      (∀ (p : Nat), Nat.Prime p → @Odd.{0} Nat Nat.instSemiring p → P p))","statement_sha256":"981567fa5fd56eca9bffcb1a0ada208aafb24802383f5d3330c614458e376473","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-d73fe8a0a1d94e971e6902a9b2ba07b5cef734d5752d492a25ad9a481d01c3ee","qualified_name":"Nat.forall_prime_of_two_of_odd","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":4902,"column":73,"line":118},"start":{"byte":4730,"column":0,"line":117}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {P : Nat → Prop},\n  And (P (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))\n      (∀ (p : Nat), Nat.Prime p → @Odd.{0} Nat Nat.instSemiring p → P p) →\n    ∀ (p : Nat), Nat.Prime p → P p","statement_sha256":"3d6de0eb32e8460bb5ce126ca826a7272d26934a90fbe15fa32c4eccd8d966fa","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-100e5652d74ebc1d0eb77a109eea642c64050fa2e8e3854f98be94f7e40e2f04","qualified_name":"Nat.gcd_eq_one_of_lt_minFac","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":6492,"column":58,"line":151},"start":{"byte":6269,"column":0,"line":149}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n m : Nat},\n  @Ne.{1} Nat m (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n    @LT.lt.{0} Nat instLTNat m n.minFac →\n      @Eq.{1} Nat (n.gcd m) (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))","statement_sha256":"388da98bc3a5ec5ea71d6917062b4026d25ae8f43ce95a9994fdca3a33c30c9d","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-3429afc74c81524594c3da914d0ec816802e29a5a0eccfce679a249c55fb5df4","qualified_name":"Nat.ne_one_iff_exists_prime_dvd","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":11269,"column":59,"line":254},"start":{"byte":10894,"column":0,"line":247}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  Iff (@Ne.{1} Nat n (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))\n    (∃ p, And (Nat.Prime p) (@Dvd.dvd.{0} Nat Nat.instDvd p n))","statement_sha256":"11be381264d82221d308b9750d427725749fdeddc5a9c5bd56fad7856f846148","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-6a94c9a0ef64ded0e7b3105d48f294e0603af27aa86999c2da2a817ebd34d41a","qualified_name":"Nat.not_prime_iff_exists_dvd_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":3238,"column":94,"line":83},"start":{"byte":3026,"column":0,"line":82}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n →\n    Iff (Not (Nat.Prime n))\n      (∃ m,\n        And (@Dvd.dvd.{0} Nat Nat.instDvd m n)\n          (And (@LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) m)\n            (@LT.lt.{0} Nat instLTNat m n)))","statement_sha256":"d786bb499855a9f425c66b43554ddeac75c5eda05b7841bc1f4e78b3684103e6","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-ab58d18060b2e9652a21fbf891e921b15c24fd14c9f0bffc6779194e5f0a7a83","qualified_name":"Nat.not_prime_iff_exists_dvd_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":3024,"column":93,"line":80},"start":{"byte":2811,"column":0,"line":79}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n →\n    Iff (Not (Nat.Prime n))\n      (∃ m,\n        And (@Dvd.dvd.{0} Nat Nat.instDvd m n)\n          (And (@Ne.{1} Nat m (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (@Ne.{1} Nat m n)))","statement_sha256":"e7fc1d1e692831f840a185f0f5017c45ed7174b783ee740775e00b7e343d4b10","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-8ad636de6f8c3efbd6f4e21cc96762e160743edc8fe520b1b3f8ed0ff77c9967","qualified_name":"Nat.not_prime_iff_exists_mul_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":3437,"column":70,"line":87},"start":{"byte":3240,"column":0,"line":85}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n →\n    Iff (Not (Nat.Prime n))\n      (∃ a b,\n        And (@LT.lt.{0} Nat instLTNat a n)\n          (And (@LT.lt.{0} Nat instLTNat b n)\n            (@Eq.{1} Nat (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) a b) n)))","statement_sha256":"c1ae7b01987166288b113c641339a48b83eb54432fb3ac0ef6f07451020aa4f9","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-2732a9c332094154c208d6f3c3096977b03a8a8d930f881cca16d126961127e9","qualified_name":"Nat.not_prime_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":829,"column":25,"line":29},"start":{"byte":716,"column":0,"line":28}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {a b : Nat},\n  @Ne.{1} Nat a (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) →\n    @Ne.{1} Nat b (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) →\n      Not (Nat.Prime (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) a b))","statement_sha256":"eefdd3596fb63ef6b70be1144d9c3f8d43f64b0759b9f75ea0a6df4ee174d615","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-cfc1d5be0746a88601799911f497da406565957258023c328f05bad3654d08a0","qualified_name":"Nat.not_prime_of_dvd_of_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":2809,"column":55,"line":77},"start":{"byte":2653,"column":0,"line":76}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {m n : Nat},\n  @Dvd.dvd.{0} Nat Nat.instDvd m n →\n    @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) m →\n      @LT.lt.{0} Nat instLTNat m n → Not (Nat.Prime n)","statement_sha256":"516f2d61277764dcab41c7dcc1a0edabffd9c5a9e938b9624669346b9b817fa9","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-71149446539e657955dbbc797fe8ca02c6513a17479c6d9149bbc98d56ee8d54","qualified_name":"Nat.not_prime_of_dvd_of_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":2651,"column":55,"line":74},"start":{"byte":2493,"column":0,"line":73}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {m n : Nat},\n  @Dvd.dvd.{0} Nat Nat.instDvd m n →\n    @Ne.{1} Nat m (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) → @Ne.{1} Nat m n → Not (Nat.Prime n)","statement_sha256":"40d4c03e808a96f73c2cc7c216f1f052bd659207359f0daa16d2236cbef6cf21","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-1f53c5c2a6f31affb23cc41be1bcea0df4d725bd2bd0c75503770efe0b64de43","qualified_name":"Nat.not_prime_of_mul_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":969,"column":31,"line":32},"start":{"byte":831,"column":0,"line":31}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {a b n : Nat},\n  @Eq.{1} Nat (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) a b) n →\n    @Ne.{1} Nat a (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) →\n      @Ne.{1} Nat b (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) → Not (Nat.Prime 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(@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid))) q m) →\n        @Eq.{1} Nat p q","statement_sha256":"b260ccd9851224ae2569312676e9c21dedfb680e8bce77570a0a8058065da02e","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-f6a2a0bec618fc17c516b46b2d321ff05c9fcac0d4657f860dc24f50ccb4e04e","qualified_name":"Nat.prime_mul_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":714,"column":80,"line":26},"start":{"byte":530,"column":0,"line":25}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {a b : Nat},\n  Iff (Nat.Prime (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) a b))\n    (Or (And (Nat.Prime a) (@Eq.{1} Nat b (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))\n      (And (Nat.Prime b) (@Eq.{1} Nat a (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 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Nat\n            (@instHPow.{0, 0} Nat Nat (@NPow.toPow.{0} Nat (@Monoid.toNPow.{0} Nat Nat.instMonoid))) p k)\n          m →\n        @Dvd.dvd.{0} Nat Nat.instDvd\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))) p l)\n            n →\n          @Dvd.dvd.{0} Nat Nat.instDvd\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))) p\n                (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat)\n                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k l)\n                  (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))\n              (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) m n) →\n            Or\n              (@Dvd.dvd.{0} Nat Nat.instDvd\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))) p\n                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) k\n                    (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))\n                m)\n              (@Dvd.dvd.{0} Nat Nat.instDvd\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))) p\n                  (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) l\n                    (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))\n                n)","statement_sha256":"209e323384b0bff51192b976549bddab5d2e047573963f4f63b1963581f25ea4","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-b0fb45d6b8c9cd5a3f1304239a5d378a0f292e13727e22fe78e7ff4407dda39d","qualified_name":"Nat.succ_pred_prime","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Prime.Basic","path":"Mathlib/Data/Nat/Prime/Basic.lean","range":{"end":{"byte":2015,"column":28,"line":62},"start":{"byte":1915,"column":0,"line":61}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {p : Nat}, Nat.Prime p → @Eq.{1} Nat p.pred.succ 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