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{m : Int} {n : Nat},\n  Iff (@Dvd.dvd.{0} Int Int.instDvd m (@Nat.cast.{0} Int instNatCastInt n)) (@Dvd.dvd.{0} Nat Nat.instDvd m.natAbs n)","statement_sha256":"739c3a029856530f9612c95f4ce1595dcd6ecb89dfd244658dfeb42206913117","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-8f39f1960a6c322556cd40009e2911724304609c84cb0b3e49fcc40fe47c6477","qualified_name":"Int.eq_of_mod_eq_of_natAbs_sub_lt_natAbs","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":3071,"column":92,"line":83},"start":{"byte":2771,"column":0,"line":80}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {a b c : Int},\n  @Eq.{1} Int (@HMod.hMod.{0, 0, 0} Int Int Int (@instHMod.{0} Int Int.instMod) a b) c →\n    @LT.lt.{0} Nat instLTNat (@HSub.hSub.{0, 0, 0} Int Int Int (@instHSub.{0} Int Int.instSub) a c).natAbs b.natAbs →\n      @Eq.{1} Int a 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@Dvd.dvd.{0} Int Int.instDvd n m → @Eq.{1} Int m (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0)))","statement_sha256":"98b55f28322ef8b89619b776cde41d30e518c873fe8e9adf98d2cb552df32a3f","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-4c7e30a004d92f3f2aac26cd2f26742f6f94ae63fbe4fc8020b96f08f94a78a0","qualified_name":"Int.gcd_emod","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":3352,"column":67,"line":89},"start":{"byte":3224,"column":0,"line":88}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (m n : Int), @Eq.{1} Nat ((@HMod.hMod.{0, 0, 0} Int Int Int (@instHMod.{0} Int Int.instMod) m n).gcd n) (m.gcd n)","statement_sha256":"c327d937ed6114e16e99646d298f7c98cbf260bfaebcf6f2b7c44227ecc4cf26","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-d1628d71b72e501496f520b5e8e85f7caf043ca294f4e89b2c9260d4664225c8","qualified_name":"Int.instNontrivial","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":706,"column":77,"line":31},"start":{"byte":629,"column":0,"line":31}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"Nontrivial.{0} Int","statement_sha256":"8b633b0b7af7c802ebce59ea3e6059e24bc48157d0460799fdeaba7786c844b8","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-07efbb58d539060710139c6b920fdff89c0304df022baeb077796fa0df3e0f09","qualified_name":"Int.natAbs_le_of_dvd_ne_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":3222,"column":60,"line":86},"start":{"byte":3073,"column":0,"line":85}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {m n : Int},\n  @Dvd.dvd.{0} Int Int.instDvd m n →\n    @Ne.{1} Int n (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))) →\n      @LE.le.{0} Nat instLENat m.natAbs n.natAbs","statement_sha256":"9fe3bc6257d638f55be1d3e9afe633ab636696d2843eacab11d6921650d4147d","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-7c15fd55d5e9f2d1a79d1cfaa4c537b0832923eb7ed287e9ab34e0b357661b70","qualified_name":"Int.natAbs_surjective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":1737,"column":81,"line":56},"start":{"byte":1656,"column":0,"line":56}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"@Function.Surjective.{1, 1} Int Nat Int.natAbs","statement_sha256":"20f5094597085aedad17270970f82ea657c9aaf0498735f56f68624329f9ad04","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-9a145602123742addca7daf9f9ec71412f0f9f0556c3072ac059d4ad4da7d2b2","qualified_name":"Int.natCast_dvd","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":2428,"column":91,"line":72},"start":{"byte":2266,"column":0,"line":71}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Int} {m : Nat},\n  Iff (@Dvd.dvd.{0} Int Int.instDvd (@Nat.cast.{0} Int instNatCastInt m) n) (@Dvd.dvd.{0} Nat Nat.instDvd m n.natAbs)","statement_sha256":"be38d8bafff482aa21361e33a34a7bf96c013c0dfde1c5242fff2666fd2fac53","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-c7d86741b5fbe5643b2b516ab439029287176083b6fd86c8779fcbb59411dfde","qualified_name":"Int.natCast_dvd_ofNat","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":2264,"column":21,"line":69},"start":{"byte":2132,"column":0,"line":68}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Nat},\n  Iff (@Dvd.dvd.{0} Int Int.instDvd (@Nat.cast.{0} Int instNatCastInt x) (@OfNat.ofNat.{0} Int y (@instOfNat y)))\n    (@Dvd.dvd.{0} Nat Nat.instDvd x (@OfNat.ofNat.{0} Nat y (instOfNatNat y)))","statement_sha256":"6265cc8dfc158bb113e70b6d333476d1f1b4334789bc6b9dbb2b0fa16bf7033e","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-46ee7a3ec5631df0aeac862824b97c0f5c1c905e9f00629210494cfaf62507ce","qualified_name":"Int.ofNat_dvd_natCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":2130,"column":21,"line":66},"start":{"byte":1998,"column":0,"line":65}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Nat},\n  Iff (@Dvd.dvd.{0} Int Int.instDvd (@OfNat.ofNat.{0} Int x (@instOfNat x)) (@Nat.cast.{0} Int instNatCastInt y))\n    (@Dvd.dvd.{0} Nat Nat.instDvd (@OfNat.ofNat.{0} Nat x (instOfNatNat x)) y)","statement_sha256":"3c3b21df5182982f3708f565112d157699359c16886fae733ca0a38d2408430a","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-53ccb9eecb87374b68d1cc9adac115348b1568cb9cff09261429c2649bd05b35","qualified_name":"Int.ofNat_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":782,"column":74,"line":33},"start":{"byte":708,"column":0,"line":33}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"@Function.Injective.{1, 1} Nat Int Int.ofNat","statement_sha256":"04b38ad2e280ed48da7df4a83eadb6eb9dabb790fe914ff514229c65ebc08807","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-d311cee0dd0ab17f1b0cc4cde508a11b59a58ada81b74a3a61bba6b6758fbb3c","qualified_name":"Int.pow_right_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":1980,"column":38,"line":61},"start":{"byte":1739,"column":0,"line":58}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {a : Int},\n  @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) a.natAbs →\n    @Function.Injective.{1, 1} Nat Int fun x =>\n      @HPow.hPow.{0, 0, 0} Int Nat Int (@instHPow.{0, 0} Int Nat (@instPowNat.{0} Int Int.instNatPow)) a x","statement_sha256":"0b3415b3c728195c13468f776166ade0860ccd88fb503cf2debcf25b6994be99","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-580580901779ce8b9df2a92a63f2853afb30476430533766e65f61a4aa91716f","qualified_name":"Int.strongRec_of_ge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Data.Int.Basic","path":"Mathlib/Data/Int/Basic.lean","range":{"end":{"byte":1619,"column":83,"line":50},"start":{"byte":900,"column":0,"line":39}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {m n : Int} {P : Int → Sort u_1} {lt : (n : Int) → @LT.lt.{0} Int 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