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Nat instSubNat) m n))\n    (Or (@Dvd.dvd.{0} Nat Nat.instDvd n m) (@LE.le.{0} Nat instLENat m n))","statement_sha256":"175b997231c93e93d97b2e337457ce402a8996423f73ef4cd2aac3a5bd552c96","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-f6e412cbe8af6d476b64adc9d7d4a527ba470b227a8692facb51e5aaa3e95f9c","qualified_name":"Nat.instLinearOrder","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Basic","path":"Mathlib/Data/Nat/Basic.lean","range":{"end":{"byte":1305,"column":32,"line":44},"start":{"byte":964,"column":0,"line":34}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"LinearOrder.{0} 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(@instHAdd.{0} Nat instAddNat) k\n            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))),\n  (∀ (n : Nat),\n      @Function.Injective.{u_1, u_1} (C n)\n        (C\n          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) n\n            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))\n        (@next n)) →\n    @Function.Injective.{u_1, u_1} (C n) (C m) (@Nat.leRecOn.{u_1} C n m hnm fun {k} => @next k)","statement_sha256":"090c5ba7d3cade59f361cf08fb5a90d2adab03bd63f4c55f1965e69bfbbe14d6","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-20ea9f9a1c8ebcdd32219589745030a5e939226e01479ff4fb7c0914eafd3d32","qualified_name":"Nat.leRecOn_surjective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Basic","path":"Mathlib/Data/Nat/Basic.lean","range":{"end":{"byte":3303,"column":21,"line":105},"start":{"byte":2879,"column":0,"line":93}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {C : Nat → Sort u_1} {n m : Nat} (hnm : @LE.le.{0} Nat instLENat n m)\n  (next :\n    {k : Nat} →\n      C k →\n        C\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  (∀ (n : Nat),\n      @Function.Surjective.{u_1, u_1} (C n)\n        (C\n          (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) n\n            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))\n        (@next n)) →\n    @Function.Surjective.{u_1, u_1} (C n) (C m) (@Nat.leRecOn.{u_1} C n m hnm fun {k} => @next k)","statement_sha256":"1ae0e3c305935d25451d37b79d2fafc3128471c2bd963b0b4de80deb59e0966a","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-e09f0328f76961c8b775c3e8923ac77d804225efc258221c099a67654a6e3990","qualified_name":"Nat.mul_le_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Basic","path":"Mathlib/Data/Nat/Basic.lean","range":{"end":{"byte":5230,"column":77,"line":156},"start":{"byte":4843,"column":0,"line":148}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {a : Nat},\n  @Ne.{1} Nat a (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) →\n    ∀ (b : Nat),\n      @LE.le.{0} Nat instLENat (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat) a b)\n        (@HPow.hPow.{0, 0, 0} Nat Nat Nat (@instHPow.{0, 0} Nat Nat (@instPowNat.{0} Nat instNatPowNat)) a b)","statement_sha256":"cd9261d542b51f0d79595c471b539cf96ab1da4d2bf327703df330be0379c866","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-3560be489e7f1b47f568d356bd2792c760a49d0d5343f73e5f02b5705580f636","qualified_name":"Nat.pow_left_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Basic","path":"Mathlib/Data/Nat/Basic.lean","range":{"end":{"byte":1909,"column":63,"line":66},"start":{"byte":1764,"column":0,"line":65}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @Ne.{1} Nat n (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n    @Function.Injective.{1, 1} Nat Nat fun a =>\n      @HPow.hPow.{0, 0, 0} Nat Nat Nat (@instHPow.{0, 0} Nat Nat (@instPowNat.{0} Nat instNatPowNat)) a n","statement_sha256":"3dd48eb372fd5dbbf0fdc5bbb3024e5392575c1918a8339db69ca77af1755878","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-dd8c23648f56238cdf07fd1599cad8169538af99be3d87ba12f1f65da239a80e","qualified_name":"Nat.pow_right_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Basic","path":"Mathlib/Data/Nat/Basic.lean","range":{"end":{"byte":2053,"column":64,"line":69},"start":{"byte":1911,"column":0,"line":68}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {a : Nat},\n  @LE.le.{0} Nat instLENat (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) a →\n    @Function.Injective.{1, 1} Nat Nat fun x =>\n      @HPow.hPow.{0, 0, 0} Nat Nat Nat (@instHPow.{0, 0} Nat Nat (@instPowNat.{0} Nat instNatPowNat)) a x","statement_sha256":"46b00820c4b2fc34c2c90e1fe9e5677229703767a94e704335cbf6b32457855b","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-c6d75d935e7d31c8a823005380093a0724304d13278e4c8aa92ee2b1b96f8515","qualified_name":"Nat.pow_sub_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Basic","path":"Mathlib/Data/Nat/Basic.lean","range":{"end":{"byte":2255,"column":92,"line":73},"start":{"byte":2055,"column":0,"line":71}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x a : Nat},\n  @Ne.{1} Nat x (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n    @Ne.{1} Nat a (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n      @Eq.{1} Nat\n        (@HPow.hPow.{0, 0, 0} Nat Nat Nat (@instHPow.{0, 0} Nat Nat (@instPowNat.{0} Nat instNatPowNat)) x\n          (@HSub.hSub.{0, 0, 0} Nat Nat Nat (@instHSub.{0} Nat instSubNat) a\n            (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))\n        (@HDiv.hDiv.{0, 0, 0} Nat Nat Nat (@instHDiv.{0} Nat Nat.instDiv)\n          (@HPow.hPow.{0, 0, 0} Nat Nat Nat (@instHPow.{0, 0} Nat Nat (@instPowNat.{0} Nat instNatPowNat)) x a) x)","statement_sha256":"1fb5353830a6feb7049d558d304e55669edaec55a10a5a10e4e59b547448bf66","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-c86cafdc6e5c7d5ae2daadecf78eeb0bdd94e60bd89658cc77e1b241166452cb","qualified_name":"Nat.set_induction","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Basic","path":"Mathlib/Data/Nat/Basic.lean","range":{"end":{"byte":3851,"column":44,"line":116},"start":{"byte":3592,"column":0,"line":113}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {S : Set.{0} Nat},\n  @Membership.mem.{0, 0} Nat (Set.{0} Nat) (@Set.instMembership.{0} Nat) S\n      (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n    (∀ (k : Nat),\n        @Membership.mem.{0, 0} Nat (Set.{0} Nat) (@Set.instMembership.{0} Nat) S k →\n          @Membership.mem.{0, 0} Nat (Set.{0} Nat) (@Set.instMembership.{0} Nat) S\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      ∀ (n : Nat), @Membership.mem.{0, 0} Nat (Set.{0} Nat) (@Set.instMembership.{0} Nat) S n","statement_sha256":"823c8d15412cb207dd61cb096cd937ef81aa8f028f29c835f191febebcb4fa72","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-4b0b396a08bb76a3eeaf0b28381175bf34b24dc96c932b8d6af93f3031c44573","qualified_name":"Nat.set_induction_bounded","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Basic","path":"Mathlib/Data/Nat/Basic.lean","range":{"end":{"byte":3590,"column":47,"line":111},"start":{"byte":3306,"column":0,"line":108}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n k : Nat} {S : Set.{0} Nat},\n  @Membership.mem.{0, 0} Nat (Set.{0} Nat) (@Set.instMembership.{0} Nat) S k →\n    (∀ (k : Nat),\n        @Membership.mem.{0, 0} Nat (Set.{0} Nat) (@Set.instMembership.{0} Nat) S k →\n          @Membership.mem.{0, 0} Nat (Set.{0} Nat) (@Set.instMembership.{0} Nat) S\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      @LE.le.{0} Nat instLENat k n → @Membership.mem.{0, 0} Nat (Set.{0} Nat) (@Set.instMembership.{0} Nat) S n","statement_sha256":"3701287d12dd39979f30fadf3b973675f5f13b7aa3e2033692a86e2c6efa94d2","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-965ee422158fc53f469a898072a555cc013bc3d5dd63c4cecf3a0a55bcddd76b","qualified_name":"Nat.succ_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Nat.Basic","path":"Mathlib/Data/Nat/Basic.lean","range":{"end":{"byte":1576,"column":54,"line":54},"start":{"byte":1522,"column":0,"line":54}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"@Function.Injective.{1, 1} Nat Nat 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Nat (@instHMul.{0} Nat instMulNat)\n        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))\n        (@HPow.hPow.{0, 0, 0} Nat Nat Nat (@instHPow.{0, 0} Nat Nat (@instPowNat.{0} Nat instNatPowNat)) k\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    (@HPow.hPow.{0, 0, 0} Nat Nat Nat (@instHPow.{0, 0} Nat Nat (@instPowNat.{0} Nat instNatPowNat))\n      (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))\n      (@HMul.hMul.{0, 0, 0} Nat Nat Nat (@instHMul.{0} Nat instMulNat)\n        (@OfNat.ofNat.{0} Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) 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