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Complex.instZero)) →\n      @Eq.{1} Real w.re (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        @Eq.{1} Complex w (@OfNat.ofNat.{0} Complex (nat_lit 0) (@Zero.toOfNat0.{0} Complex Complex.instZero))) →\n    @Eq.{1} Real\n      (@Norm.norm.{0} Complex Complex.instNorm\n        (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow) z w))\n      (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@Norm.norm.{0} Complex Complex.instNorm z) w.re)\n        (Real.exp (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) z.arg w.im)))","statement_sha256":"fc8cb92bc425ae3e8ccad349f76f6b93a5662b93ccc29b6ba279b7d41b7e3f29","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-59370d5e6b6cdd0f4eed413548d4ac77428ba54644b4602202eca41a7bf22837","qualified_name":"Complex.norm_cpow_of_ne_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":12945,"column":47,"line":315},"start":{"byte":12689,"column":0,"line":312}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {z : Complex},\n  @Ne.{1} Complex z (@OfNat.ofNat.{0} Complex (nat_lit 0) (@Zero.toOfNat0.{0} Complex Complex.instZero)) →\n    ∀ (w : Complex),\n      @Eq.{1} Real\n        (@Norm.norm.{0} Complex Complex.instNorm\n          (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow) z w))\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n            (@Norm.norm.{0} Complex Complex.instNorm z) w.re)\n          (Real.exp (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) z.arg w.im)))","statement_sha256":"be90e522f7b9ed1be0e78505c0b01876832041c7de79d28f8272b55622b24fe6","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-096c4d5618c222c28284a4d7fca4db1e7f9172f63790e7143d2de229900542d0","qualified_name":"Complex.norm_cpow_real","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":13647,"column":32,"line":332},"start":{"byte":13522,"column":0,"line":330}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Complex) (y : Real),\n  @Eq.{1} Real\n    (@Norm.norm.{0} Complex Complex.instNorm\n      (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow) x ↑y))\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n      (@Norm.norm.{0} Complex Complex.instNorm x) y)","statement_sha256":"9afda40266df2c7c21d6d0e1d92b9f6ef76ce132778a485c33c12f740a2d19dc","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-d17239bbfbd149849f74aa14a59afd09cf9ec96b4c5272898c1ecf94d9f831cf","qualified_name":"Complex.norm_log_natCast_le_rpow_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":43157,"column":42,"line":970},"start":{"byte":42759,"column":0,"line":965}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (n : Nat) {ε : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) ε →\n    @LE.le.{0} Real Real.instLE\n      (@Norm.norm.{0} Complex Complex.instNorm (Complex.log (@Nat.cast.{0} Complex Complex.instNatCast n)))\n      (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@Nat.cast.{0} Real Real.instNatCast n) ε)\n        ε)","statement_sha256":"eedc93f62d2e20881d4c919d9806e5c017f7b4db29228d62d0dad51edeced870","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-f2a99e3ee4b83c35645b754347867c63affa16edaeb6daca90b9693a55aa25da","qualified_name":"Complex.norm_natCast_cpow_le_norm_natCast_cpow_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":42757,"column":56,"line":963},"start":{"byte":42483,"column":0,"line":960}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) n →\n    ∀ {w z : Complex},\n      Iff\n        (@LE.le.{0} Real Real.instLE\n          (@Norm.norm.{0} Complex Complex.instNorm\n            (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n              (@Nat.cast.{0} Complex Complex.instNatCast n) w))\n          (@Norm.norm.{0} Complex Complex.instNorm\n            (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n              (@Nat.cast.{0} Complex Complex.instNatCast n) z)))\n        (@LE.le.{0} Real Real.instLE w.re z.re)","statement_sha256":"7325c0f47c7a0ae278fceed70551cd55e731167687317932fa08312cf5740263","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-5d0a054a89f775aefc747abf2c6df491e367b66640a35b2c62ed98fa2a908434","qualified_name":"Complex.norm_natCast_cpow_le_norm_natCast_cpow_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":42481,"column":65,"line":958},"start":{"byte":42209,"column":0,"line":954}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) n →\n    ∀ {w z : Complex},\n      @LE.le.{0} Real Real.instLE w.re z.re →\n        @LE.le.{0} Real Real.instLE\n          (@Norm.norm.{0} Complex Complex.instNorm\n            (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n              (@Nat.cast.{0} Complex Complex.instNatCast n) w))\n          (@Norm.norm.{0} Complex Complex.instNorm\n            (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n              (@Nat.cast.{0} Complex Complex.instNatCast n) z))","statement_sha256":"b9c2e6f184e1edae0429827b9e114661567931446d108307039c17285ad225b3","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-576e8d76cc87a3347899dd3ea40101cba4b905d74f2322a09e142d0fe1a23a6e","qualified_name":"Complex.norm_natCast_cpow_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":14906,"column":78,"line":358},"start":{"byte":14711,"column":0,"line":356}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) n →\n    ∀ (s : Complex),\n      @Eq.{1} Real\n        (@Norm.norm.{0} Complex Complex.instNorm\n          (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n            (@Nat.cast.{0} Complex Complex.instNatCast n) s))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@Nat.cast.{0} Real Real.instNatCast n) s.re)","statement_sha256":"67a475190f98df9dfdbbeb343ea3a4d5641f70ec5bb53abd3fba790f111e4677","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-a9c94a99ee30b6653120b45cb79258bedc197178ae6497c7d6d3811b62971870","qualified_name":"Complex.norm_natCast_cpow_of_re_ne_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":14709,"column":74,"line":354},"start":{"byte":14506,"column":0,"line":352}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (n : Nat) {s : Complex},\n  @Ne.{1} Real s.re (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    @Eq.{1} Real\n      (@Norm.norm.{0} Complex Complex.instNorm\n        (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n          (@Nat.cast.{0} Complex Complex.instNatCast n) s))\n      (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n        (@Nat.cast.{0} Real Real.instNatCast n) s.re)","statement_sha256":"8b5fdf91f6e06ad2c779427d6ec8083fc99800b7b93e10a88b74b7cd417a7592","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-c805a92ca919efcf9e60f8b5924841ccd5abb3b6e986578775a62c0c8609057e","qualified_name":"Complex.norm_natCast_cpow_pos_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":15092,"column":87,"line":361},"start":{"byte":14908,"column":0,"line":360}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {n : Nat},\n  @LT.lt.{0} Nat instLTNat (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) n →\n    ∀ (s : Complex),\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))\n        (@Norm.norm.{0} Complex Complex.instNorm\n          (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n            (@Nat.cast.{0} Complex Complex.instNatCast n) s))","statement_sha256":"035c973ac0150bce183974b60664011e2473493952a27b7b2a8e284d66241b29","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-66509a7396b0b368d46afee28e6c44a7d15cbe7ee2de1bfe4d567ecb31aa0cbf","qualified_name":"Complex.norm_ofReal_cpow_eventually_eq_atTop","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":14504,"column":37,"line":350},"start":{"byte":14282,"column":0,"line":347}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (c : Complex),\n  @Filter.EventuallyEq.{0, 0} Real Real (@Filter.atTop.{0} Real Real.instPreorder)\n    (fun t =>\n      @Norm.norm.{0} Complex Complex.instNorm\n        (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow) (↑t) c))\n    fun t => @HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) t c.re","statement_sha256":"539e1bd35dc2b14d4ce73436aff4d42c8d72c95b069cca56c78b18baf7cd09e4","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-502d9483f37e35aa158d2b20768fa9d4d4da10200031d897a62151c9a618a3c6","qualified_name":"Complex.norm_prime_cpow_le_one_half","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":41920,"column":73,"line":944},"start":{"byte":41474,"column":0,"line":938}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (p : Nat.Primes) {s : Complex},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) s.re →\n    @LE.le.{0} Real Real.instLE\n      (@Norm.norm.{0} Complex Complex.instNorm\n        (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n          (@Nat.cast.{0} Complex Complex.instNatCast (@Subtype.val.{1} Nat (fun p => Nat.Prime p) p))\n          (@Neg.neg.{0} Complex Complex.instNeg s)))\n      (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n        (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n        (@OfNat.ofNat.{0} Real (nat_lit 2) (@instOfNatAtLeastTwo.{0} Real (nat_lit 2) Real.instNatCast ⋯)))","statement_sha256":"7c044a17f14d48bdc64bd974ae1a118af00ed9eddfc10d84c3c1098605f2ab91","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-484ef8edc10febc958ed98bd332cc2335fda76e875f25611c70874769f352656","qualified_name":"Complex.ofReal_cpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":11401,"column":32,"line":285},"start":{"byte":11170,"column":0,"line":283}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real),\n      @Eq.{1} Complex (↑(@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))\n        (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow) ↑x ↑y)","statement_sha256":"00815a09a818f696efc5e653b00326e73e00d2c45960e7e18c8e9a14f4eddf67","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-328cd590016b29c67549696b9c93f5af4f91cd973daae8442935ef33224e9f7c","qualified_name":"Complex.ofReal_cpow_of_nonpos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":11917,"column":36,"line":294},"start":{"byte":11403,"column":0,"line":287}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    ∀ (y : Complex),\n      @Eq.{1} Complex\n        (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow) (↑x) y)\n        (@HMul.hMul.{0, 0, 0} Complex Complex Complex (@instHMul.{0} Complex Complex.instMul)\n          (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n            (@Neg.neg.{0} Complex Complex.instNeg ↑x) y)\n          (Complex.exp\n            (@HMul.hMul.{0, 0, 0} Complex Complex Complex (@instHMul.{0} Complex Complex.instMul)\n              (@HMul.hMul.{0, 0, 0} Complex Complex Complex (@instHMul.{0} Complex Complex.instMul) (↑Real.pi)\n                Complex.I)\n              y)))","statement_sha256":"76e5d0fcc15fa1dc9e5f3a72d8088f4c446b0b2e8020f2f7c6d75d0086342a42","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-9d09afc5b08e918556dd2adabccb7f12c3d822cb6a087eb471cd6c8c32de0186","qualified_name":"Complex.one_sub_prime_cpow_ne_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":42207,"column":18,"line":952},"start":{"byte":41922,"column":0,"line":946}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {p : Nat},\n  Nat.Prime p →\n    ∀ {s : Complex},\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) s.re →\n        @Ne.{1} Complex\n          (@HSub.hSub.{0, 0, 0} Complex Complex Complex (@instHSub.{0} Complex Complex.instSub)\n            (@OfNat.ofNat.{0} Complex (nat_lit 1) (@One.toOfNat1.{0} Complex Complex.instOne))\n            (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow)\n              (@Nat.cast.{0} Complex Complex.instNatCast p) (@Neg.neg.{0} Complex Complex.instNeg s)))\n          (@OfNat.ofNat.{0} Complex (nat_lit 0) (@Zero.toOfNat0.{0} Complex Complex.instZero))","statement_sha256":"053939b35f8890a9c257dd6474fa477bd81d9c325498d0c91c9351613ef2963a","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-c83c95c95a915d122c7bc6eea709ffd25652ca98b07db1f2b5c7efcceab39c96","qualified_name":"HasCompactSupport.rpow_const","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":11093,"column":50,"line":272},"start":{"byte":10838,"column":0,"line":270}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : TopologicalSpace.{u_1} α] {f : α → Real},\n  @HasCompactSupport.{u_1, 0} α Real inst Real.instZero f →\n    ∀ {r : Real},\n      @Ne.{1} Real r (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        @HasCompactSupport.{u_1, 0} α Real inst Real.instZero fun x =>\n          @HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) (f x) r","statement_sha256":"368f07233501fffafa1c6bafbd29dcdd5b72df0af5ac3133da25f46aded119c2","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-da12b1e14843228001a70c6fec8a72f58e9c12c2c8b715031875febcf43997a8","qualified_name":"Mathlib.Meta.NormNum.IsInt.rpow_eq_inv_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":46135,"column":92,"line":1054},"start":{"byte":45922,"column":0,"line":1052}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {b : Real} {n : Nat},\n  @Mathlib.Meta.NormNum.IsInt.{0} Real Real.instRing b (Int.negOfNat 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        (@Inv.inv.{0} Real Real.instInv\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))","statement_sha256":"49c799656f44e5fffb7c2e14aff976096a0bfe504bf9b907d0c6a2843cf832fd","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-22850819a0085d8f88583f5d8dafeb5d43bc903d2ef72312b2ae9b6f359a58c4","qualified_name":"Mathlib.Meta.NormNum.IsNNRat.rpow_isNNRat","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":47195,"column":48,"line":1079},"start":{"byte":46771,"column":0,"line":1071}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (a b : Real) (na da : Nat),\n  @Mathlib.Meta.NormNum.IsNNRat.{0} Real Real.semiring a na da →\n    ∀ (nr dr : Nat),\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)\n            (@Nat.cast.{0} Real Real.instNatCast na) b)\n          nr →\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)\n              (@Nat.cast.{0} Real Real.instNatCast da) b)\n            dr →\n          @Mathlib.Meta.NormNum.IsNNRat.{0} Real Real.semiring\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) a b) nr dr","statement_sha256":"471aa057cbec967c45131c3fa451a5920a910f8f1a47c3c890aa2845381febc5","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-3894eeb122727286137fde8bafb3f6ac6aca19cde028f85292cb0ade1a042e7e","qualified_name":"Mathlib.Meta.NormNum.IsNat.rpow_eq_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":45920,"column":29,"line":1050},"start":{"byte":45797,"column":0,"line":1049}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","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)","statement_sha256":"359fb38b6d9a8932a55732f2bb36304c3e05a5520d1da0c041427b7e574393c4","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-9f228f746f82e39a07eba0cd1539d6f5df4a6ac3d155e0d2441de1045ad45285","qualified_name":"Mathlib.Meta.NormNum.IsNat.rpow_isNNRat","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":46769,"column":59,"line":1069},"start":{"byte":46137,"column":0,"line":1056}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","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) r","statement_sha256":"7d8291e5f8b91372b6b9ab9685fa63ff7e4a380b431ab172426b1e99a60e77ca","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-77604a68bb4ad787e632d3591d9f125f16c37d3408bffaf43ed0576d327fc5a0","qualified_name":"Mathlib.Meta.NormNum.evalRPow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":50094,"column":16,"line":1147},"start":{"byte":48353,"column":0,"line":1110}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"Mathlib.Meta.NormNum.NormNumExt","statement_sha256":"15be327462bc2bcb4150c260071c751c9de606cf428d8215ba2b7842eebd9a3f","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-6f2385fa853f89a5d1aeb4060b725081744ca8af0251c5ca8cadf0f2099ec717","qualified_name":"Mathlib.Meta.NormNum.proveIsNatRPowIsNNRat","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":48351,"column":72,"line":1108},"start":{"byte":47404,"column":0,"line":1086}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"(a : Qq.Quoted (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))) →\n  (na : Qq.Quoted (Lean.Expr.const `Nat (@List.nil.{0} Lean.Level))) →\n    Qq.Quoted\n        (((((Lean.Expr.const `Mathlib.Meta.NormNum.IsNat\n                          (@List.cons.{0} Lean.Level Lean.Level.zero (@List.nil.{0} Lean.Level))).app\n                      (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                  (((Lean.Expr.const `AddGroupWithOne.toAddMonoidWithOne\n                            (@List.cons.{0} Lean.Level Lean.Level.zero (@List.nil.{0} Lean.Level))).app\n                        (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                    (((Lean.Expr.const `Ring.toAddGroupWithOne\n                              (@List.cons.{0} Lean.Level Lean.Level.zero (@List.nil.{0} Lean.Level))).app\n                          (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                      (Lean.Expr.const `Real.instRing (@List.nil.{0} Lean.Level))))).app\n              a).app\n          na) →\n      (b : Qq.Quoted (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))) →\n        (nb db : Qq.Quoted (Lean.Expr.const `Nat (@List.nil.{0} Lean.Level))) →\n          Qq.Quoted\n              ((((((Lean.Expr.const `Mathlib.Meta.NormNum.IsNNRat\n                                    (@List.cons.{0} Lean.Level Lean.Level.zero (@List.nil.{0} Lean.Level))).app\n                                (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                            (Lean.Expr.const `Real.semiring (@List.nil.{0} Lean.Level))).app\n                        b).app\n                    nb).app\n                db) →\n            Lean.Meta.MetaM\n              (Prod.{0, 0} Nat\n                (@Sigma.{0, 0} (Qq.Quoted (Lean.Expr.const `Nat (@List.nil.{0} Lean.Level))) fun r =>\n                  Qq.Quoted\n                    (((((Lean.Expr.const `Mathlib.Meta.NormNum.IsNat\n                                      (@List.cons.{0} Lean.Level Lean.Level.zero (@List.nil.{0} Lean.Level))).app\n                                  (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                              (((Lean.Expr.const `AddGroupWithOne.toAddMonoidWithOne\n                                        (@List.cons.{0} Lean.Level Lean.Level.zero (@List.nil.{0} Lean.Level))).app\n                                    (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                                (((Lean.Expr.const `Ring.toAddGroupWithOne\n                                          (@List.cons.{0} Lean.Level Lean.Level.zero (@List.nil.{0} Lean.Level))).app\n                                      (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                                  (Lean.Expr.const `Real.instRing (@List.nil.{0} Lean.Level))))).app\n                          (((((((Lean.Expr.const `HPow.hPow\n                                                    (@List.cons.{0} Lean.Level Lean.Level.zero\n                                                      (@List.cons.{0} Lean.Level Lean.Level.zero\n                                                        (@List.cons.{0} Lean.Level Lean.Level.zero\n                                                          (@List.nil.{0} Lean.Level))))).app\n                                                (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                                            (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                                        (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                                    ((((Lean.Expr.const `instHPow\n                                                  (@List.cons.{0} Lean.Level Lean.Level.zero\n                                                    (@List.cons.{0} Lean.Level Lean.Level.zero\n                                                      (@List.nil.{0} Lean.Level)))).app\n                                              (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                                          (Lean.Expr.const `Real (@List.nil.{0} Lean.Level))).app\n                                      (Lean.Expr.const `Real.instPow (@List.nil.{0} Lean.Level)))).app\n                                a).app\n                            b)).app\n                      r)))","statement_sha256":"3ec4bd4a11d78746c1d25376f0966236149a144c66ab6ec234566d16812fcd34","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-dcdd055f470219564995b4b9c8b04c21c58a2dc7e2c9d54c40c542a81bbd002c","qualified_name":"Mathlib.Meta.NormNum.rpow_isRat_eq_inv_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":47389,"column":58,"line":1083},"start":{"byte":47197,"column":0,"line":1081}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (a b : Real) (n d : Nat),\n  @Mathlib.Meta.NormNum.IsRat.{0} Real Real.instRing b (Int.negOfNat n) d →\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 Real Real (@instHPow.{0, 0} Real Real Real.instPow) (@Inv.inv.{0} Real Real.instInv a)\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n          (@Nat.cast.{0} Real Real.instNatCast n) (@Nat.cast.{0} Real Real.instNatCast d)))","statement_sha256":"170c43e5c36a94cd69ca86bd2c9682ddb29e678946442c6b420e967edd14ade4","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-fb55df63614781a2e75853517a5d906c158ad4aa92ada26f578e3d5084a186e6","qualified_name":"Mathlib.Meta.Positivity.evalRpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":16728,"column":41,"line":402},"start":{"byte":16023,"column":0,"line":387}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"Mathlib.Meta.Positivity.PositivityExt","statement_sha256":"2dfaf4795fcaaf9a3a15001a85aaa191024ff62e1af209f4c80c32c2fad46de7","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-8bf16dc9ee7c8b71de2c46718ea24318351fddf231ef9f79d027676d2677c905","qualified_name":"Mathlib.Meta.Positivity.evalRpowZero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":16021,"column":41,"line":385},"start":{"byte":15521,"column":0,"line":376}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"Mathlib.Meta.Positivity.PositivityExt","statement_sha256":"2dfaf4795fcaaf9a3a15001a85aaa191024ff62e1af209f4c80c32c2fad46de7","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-8b8f8d70ac019c138008468ea290825fa54fa601695d3d647fb9c435d4268ad0","qualified_name":"Real.abs_log_mul_self_rpow_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":38995,"column":72,"line":882},"start":{"byte":38603,"column":0,"line":877}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x t : Real),\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) t →\n        @LT.lt.{0} Real Real.instLT\n          (@abs.{0} Real Real.lattice Real.instAddGroup\n            (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (Real.log x)\n              (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x t)))\n          (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n            (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) t)","statement_sha256":"6e7bfb10f0be0b48a4a57f8abe5ad9aab8ad9c43f4a7525fba77d7c54a6c7373","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-c94cdea1fed497c8dc2ba74415d6061cf38013b98ffbeae1e36a359dad538f1b","qualified_name":"Real.abs_rpow_le_abs_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":6636,"column":66,"line":178},"start":{"byte":6294,"column":0,"line":172}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x y : Real),\n  @LE.le.{0} Real Real.instLE\n    (@abs.{0} Real Real.lattice Real.instAddGroup\n      (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n      (@abs.{0} Real Real.lattice Real.instAddGroup x) 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Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))\n    (Real.exp (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (Real.log x) 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@Eq.{1} Real\n      (@abs.{0} Real Real.lattice Real.instAddGroup\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))\n      (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n        (@abs.{0} Real Real.lattice Real.instAddGroup x) y)","statement_sha256":"b94e4be71cf25bb71bb43d945d9dd6ab695025e3ec1e0cd680a22d0dc199072f","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-902dd2c2fec23da485345ef904028f6e719bf18be75993de0446c6c3c2a0575c","qualified_name":"Real.antitoneOn_rpow_Ioi_of_exponent_nonpos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":26953,"column":54,"line":627},"start":{"byte":26770,"column":0,"line":625}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {r : Real},\n  @LE.le.{0} Real Real.instLE r (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    @AntitoneOn.{0, 0} Real Real Real.instPreorder Real.instPreorder\n      (fun x => @HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x r)\n      (@Set.Ioi.{0} Real Real.instPreorder (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real 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Real.instZero)) b →\n    @LE.le.{0} Real Real.instLE b (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @Antitone.{0, 0} Real Real Real.instPreorder Real.instPreorder fun x =>\n        @HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) b x","statement_sha256":"6189f523239ea3c77ef2967751435ce4be577f1a09b799a12a47be06efc4bbb0","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-d957662d307ac038106de71aba2eea97d950fb57094bb5f78ebaa892b426574d","qualified_name":"Real.div_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":20406,"column":72,"line":489},"start":{"byte":20241,"column":0,"line":488}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      ∀ (z : Real),\n        @Eq.{1} Real\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n            (@HDiv.hDiv.{0, 0, 0} Real Real Real\n              (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid)) x y)\n            z)\n          (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))","statement_sha256":"7a17bbf7662453137a3ef6264f07acb023611fd4ea460977f0497784d3675060","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-3455e6f4d28fc376a24014c9a8fa36a29314b11005531661e09f705946c31e96","qualified_name":"Real.eq_rpow_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":33951,"column":65,"line":774},"start":{"byte":33786,"column":0,"line":773}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @Ne.{1} Real z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        Iff\n          (@Eq.{1} Real x\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y\n              (@Inv.inv.{0} Real Real.instInv z)))\n          (@Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z) y)","statement_sha256":"019606743f846012459a73148c9d886eae17002d91b9b67f8f5eb614938f26ae","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-921c29e5e736195fe49724205aec50d4ca4a6edd41e4264c431a322a57b633ad","qualified_name":"Real.eq_zero_rpow_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":5412,"column":36,"line":146},"start":{"byte":5271,"column":0,"line":145}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x a : Real},\n  Iff\n    (@Eq.{1} Real a\n      (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n        (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x))\n    (Or\n      (And (@Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n        (@Eq.{1} Real a (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))\n      (And (@Eq.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n        (@Eq.{1} Real a (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))))","statement_sha256":"f04d72ddea940d3b72fbae52ff55b0fd4356c760d57f97f2115e9367a930192a","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-370a0c447fe5d48e1b8b388d10e2b9beea52a0560cba53853a4e38c0553dbf32","qualified_name":"Real.exp_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":1887,"column":101,"line":54},"start":{"byte":1786,"column":0,"line":54}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x y : Real),\n  @Eq.{1} Real (Real.exp (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) x y))\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) (Real.exp x) y)","statement_sha256":"c0b20e87b99c05a055bf3ced0398fcc8d15195fabfb624c912d111a592740527","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-e75eda6e06a5c2802f30f8133a6351bf68ddd5835ef9444f1df45188d1cf6478","qualified_name":"Real.exp_one_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":2841,"column":97,"line":80},"start":{"byte":2744,"column":0,"line":80}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (n : Nat),\n  @Eq.{1} Real\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)))\n      (Real.exp (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))) n)\n    (Real.exp (@Nat.cast.{0} Real Real.instNatCast n))","statement_sha256":"5f1bc131c594bbfdf05001419e70551ff850c34e6c77c9cb9db0a470b500a4ff","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-6b004e583d65785ff76e9928ca083f471838c50b5713886f376648a4730eeca1","qualified_name":"Real.exp_one_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":2742,"column":82,"line":78},"start":{"byte":2652,"column":0,"line":77}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real),\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n      (Real.exp (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))) x)\n    (Real.exp x)","statement_sha256":"8f0efce0737f2cef4e2b13235d85c60d2901d766c2781cfd6633edd1c3ca5224","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-e67e4c4b2ca9b7aff760aaafccedb0f6a85eab4ab58477bcd01cedb13d2ca1dd","qualified_name":"Real.instPow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":1141,"column":50,"line":38},"start":{"byte":1091,"column":0,"line":38}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"Pow.{0, 0} Real Real","statement_sha256":"5253b6282b27a3178d11180abed682780bee5d80a114d2ab3a1032fd09bfa92c","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-b36f1ffac3745f0ac082e25490e4bd59daba1d6496ecb74b31c5d00416e9db24","qualified_name":"Real.inv_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":20239,"column":44,"line":486},"start":{"byte":20119,"column":0,"line":485}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@Inv.inv.{0} Real Real.instInv x) y)\n        (@Inv.inv.{0} Real Real.instInv\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))","statement_sha256":"2133542b97675ab6b0bac8918d1f08096eb95b0986e0fdf132eb4b5d2e6442f8","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-e294e6ef96e4869a81d8d34759fa018b29ad66545404cf28c58e0e42faf07692","qualified_name":"Real.le_log_of_pow_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":36491,"column":47,"line":829},"start":{"byte":36365,"column":0,"line":828}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Nat},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE\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))) x n)\n        y →\n      @LE.le.{0} Real Real.instLE\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n          (Real.log x))\n        (Real.log y)","statement_sha256":"e7b69145c6d05ed9a50003bacba501c141bdb5dbbdc5d5dc1f1127964f6d52d9","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-081e965fd4d34e4aad73214b6d6b9b7c369462814ac6aa491528d1a6e5b09b4f","qualified_name":"Real.le_log_of_rpow_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":36363,"column":48,"line":826},"start":{"byte":36235,"column":0,"line":825}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z) y →\n      @LE.le.{0} Real Real.instLE (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) z (Real.log x))\n        (Real.log 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Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE\n        (@HPow.hPow.{0, 0, 0} Real Int Real\n          (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) x n)\n        y →\n      @LE.le.{0} Real Real.instLE\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Int.cast.{0} Real Real.instIntCast n)\n          (Real.log x))\n        (Real.log 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Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      Iff\n        (@LE.le.{0} Real Real.instLE x\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))) y n))\n        (@LE.le.{0} Real Real.instLE (Real.log x)\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n            (Real.log y)))","statement_sha256":"eed452f276af9ef5bdf22adf1ba3ad2a87ccd504a953d807472e8cff14355a23","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-e8488860a34ec6842cb0bbdda94fbf7618ec62a8f2433ffada616908df5ce11b","qualified_name":"Real.le_pow_of_log_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":34736,"column":45,"line":791},"start":{"byte":34612,"column":0,"line":790}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Nat},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n    @LE.le.{0} Real Real.instLE (Real.log x)\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n          (Real.log y)) →\n      @LE.le.{0} Real Real.instLE x\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))) y n)","statement_sha256":"f3033a17315830519b1e9a0ce0fbfaf348e8c44513595211b2500672e981b518","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-2afa540b94be223f76842ab21eec622a142269b518f9985749db4e79affdea17","qualified_name":"Real.le_rpow_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":9314,"column":24,"line":241},"start":{"byte":8611,"column":0,"line":228}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y z : Real),\n      @LE.le.{0} Real Real.instLE\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y z))","statement_sha256":"77cb666f972e8b26a0e0bf009c2b44001c81a14e407d40f78bfdb3cda8625928","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-9c96550d71f3ac57b0b36f9a5d3ec915b18e8ffcde8a2219c8ad8a6c1ab7396d","qualified_name":"Real.le_rpow_iff_log_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":34114,"column":64,"line":777},"start":{"byte":33953,"column":0,"line":776}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      Iff\n        (@LE.le.{0} Real Real.instLE x\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))\n        (@LE.le.{0} Real Real.instLE (Real.log x)\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) z (Real.log y)))","statement_sha256":"9329a182bc22eff6c0a798bf1602a78b1bacc6d159f98a55b87d6caf04812f74","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-9466af277a940bf46819978ef7bb272ffe2f1353aca6bf02186abd37aa319c32","qualified_name":"Real.le_rpow_inv_iff_of_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":25584,"column":80,"line":598},"start":{"byte":25389,"column":0,"line":596}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        Iff\n          (@LE.le.{0} Real Real.instLE x\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y\n              (@Inv.inv.{0} Real Real.instInv z)))\n          (@LE.le.{0} Real Real.instLE y\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))","statement_sha256":"4e9cb209c1a7d0b5d9b976a9dedf15ede39b92032cc7df08d29be9751aecfa5c","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-06b802e404fb2130337e75ac5b20850bf362c388274f927d306b1151a0e439a1","qualified_name":"Real.le_rpow_inv_iff_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":24873,"column":65,"line":585},"start":{"byte":24695,"column":0,"line":584}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        Iff\n          (@LE.le.{0} Real Real.instLE x\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y\n              (@Inv.inv.{0} Real Real.instInv z)))\n          (@LE.le.{0} Real Real.instLE\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z) y)","statement_sha256":"cd37043a8be69eaee404c4910a5557797e601b180fb2dabd02258d7b863cd32c","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-e8790c439727ec314fe63c430e1e37b47e5cf6e51febeee7b081e0118aa91469","qualified_name":"Real.le_rpow_of_log_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":34610,"column":41,"line":788},"start":{"byte":34407,"column":0,"line":785}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n    @LE.le.{0} Real Real.instLE (Real.log x)\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) z (Real.log y)) →\n      @LE.le.{0} Real Real.instLE x (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z)","statement_sha256":"1a6b6cc106a7f6fcac769797d022b278bb5ad443e89c612d35eb693b8be5102c","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-6304d7b2035bdf316116279aad8fa1df19e02b6c551a5085b8eec94f3a4b2b34","qualified_name":"Real.le_zpow_iff_log_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":34405,"column":47,"line":783},"start":{"byte":34256,"column":0,"line":782}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Int},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      Iff\n        (@LE.le.{0} Real Real.instLE x\n          (@HPow.hPow.{0, 0, 0} Real Int Real\n            (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) y\n            n))\n        (@LE.le.{0} Real Real.instLE (Real.log x)\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Int.cast.{0} Real Real.instIntCast n)\n            (Real.log y)))","statement_sha256":"c51608c0e1a44b58d888ba732ef103566886bca1a2dadaf74919758d3e253a4e","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-3cbef3b685e076055979c57e0726c2f47513d89d8fc0e9d34bc3f1bb49f4933c","qualified_name":"Real.le_zpow_of_log_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":34873,"column":45,"line":794},"start":{"byte":34738,"column":0,"line":793}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Int},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n    @LE.le.{0} Real Real.instLE (Real.log x)\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Int.cast.{0} Real Real.instIntCast n)\n          (Real.log y)) →\n      @LE.le.{0} Real Real.instLE x\n        (@HPow.hPow.{0, 0, 0} Real Int Real\n          (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) y n)","statement_sha256":"bf228eed4c55009e1b3a5df85e5f2d5b93f696256b045e248746ea69fd6a9ca3","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-b8af591035138a9b81ae31eacbceebf55fff59b1b6368e20080428a3afe72874","qualified_name":"Real.log_le_rpow_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":39408,"column":21,"line":890},"start":{"byte":38997,"column":0,"line":884}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x ε : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) ε →\n      @LE.le.{0} Real Real.instLE (Real.log x)\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x ε) ε)","statement_sha256":"dad377dbaad814542e3ed6600777417a4bd964147b8cdd2823e759bf2b7faf08","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-f66f0eac4a27a983acd33cf5e71d1f184cc4d793ae6c10ce4cba8c3a3408e552","qualified_name":"Real.log_natCast_le_rpow_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":39661,"column":35,"line":895},"start":{"byte":39410,"column":0,"line":892}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (n : Nat) {ε : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) ε →\n    @LE.le.{0} Real Real.instLE (Real.log (@Nat.cast.{0} Real Real.instNatCast n))\n      (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@Nat.cast.{0} Real Real.instNatCast n) ε)\n        ε)","statement_sha256":"bfdb38cbc1d27e481ef728d0674e31f73ea45d90bc60f2a101eb5a3c0bc7d787","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-8d7d18f07e0f6262a3b65bbb71105c16c69ab67a6d96a78c18e97cfa7afc2322","qualified_name":"Real.log_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":20795,"column":100,"line":496},"start":{"byte":20408,"column":0,"line":491}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real),\n      @Eq.{1} Real (Real.log (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) y (Real.log 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Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT\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))) x n)\n        y →\n      @LT.lt.{0} Real Real.instLT\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n          (Real.log x))\n        (Real.log 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Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      Iff\n        (@LT.lt.{0} Real Real.instLT x\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))) y n))\n        (@LT.lt.{0} Real Real.instLT (Real.log x)\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n            (Real.log 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  @LT.lt.{0} Real Real.instLT (Real.log x)\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) z (Real.log y)) →\n      @LT.lt.{0} Real Real.instLT x (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z)","statement_sha256":"ef27aff55fba30a732399479a2890e4f017cb6fb6290cbff57a8750518d39b12","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-8eedc6da11ab1a2e3ce0c71baea30f60c28a5c8d26bdf508e7aad94cf046eeaa","qualified_name":"Real.lt_zpow_iff_log_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":35315,"column":47,"line":803},"start":{"byte":35170,"column":0,"line":802}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Int},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real 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y)))","statement_sha256":"c5357991a10988175b920fb83b0431d5a69caceded64abd390bc7c0da88e62de","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-8d184d66d3b6e44c713f0461dac13177b85c37cc39f37da505941437be178dc4","qualified_name":"Real.lt_zpow_of_log_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":35771,"column":45,"line":814},"start":{"byte":35640,"column":0,"line":813}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Int},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real 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n)","statement_sha256":"56a60976fdd45c603e8a09da6e714f1b78ec4d6c6a16be4803f70b25b7a4ea36","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-66f77146f76f56d8a97f0486463f96bf4566fa82fe999b0905b010be50b744ba","qualified_name":"Real.monotoneOn_rpow_Ici_of_exponent_nonneg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":23475,"column":44,"line":557},"start":{"byte":23302,"column":0,"line":555}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {r : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real 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Real.instOne)) b →\n    @Monotone.{0, 0} Real Real Real.instPreorder Real.instPreorder fun x =>\n      @HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) b 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  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n      Iff\n        (@Eq.{1} Real (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) y (Real.log x))\n          (Real.log z))\n        (@Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) 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@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) x y) z)\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))","statement_sha256":"2eef29c8636bf58eb87f051d6adc1a8975c6624fd3b6615f84c6395cd389e2ef","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-43877a15ce879d549ab6a753d09717988fdf767a622ab1d3f295e51d40ff59a9","qualified_name":"Real.norm_rpow_of_nonneg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":7647,"column":36,"line":204},"start":{"byte":7486,"column":0,"line":202}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Eq.{1} Real\n      (@Norm.norm.{0} Real Real.norm\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))\n      (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) (@Norm.norm.{0} Real Real.norm x)\n        y)","statement_sha256":"47da16ae63b7b64fdc9a30544e9f5bd31e7164c88d36b23aed1e669360e445e4","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-1c39da42673340e907e30764c0c9b3472732bba5f8fd4938add4f30628bf5969","qualified_name":"Real.one_le_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":29164,"column":8,"line":680},"start":{"byte":29052,"column":0,"line":678}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x →\n    @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n      @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x 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Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LE.le.{0} Real Real.instLE z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)","statement_sha256":"64472e2c7c73bd5b96a59fd0b39b37eb1a4b2fe6c452603a70488d40a2ae417d","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-302ffd694ee8b9aa5da7ecb62354b6b64213a579531f28760fc118aded8ad01e","qualified_name":"Real.one_lt_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":29050,"column":38,"line":676},"start":{"byte":28914,"column":0,"line":674}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)","statement_sha256":"9471c62a7cc9a306ca8d3877944cf9f0a454ad01412c130e76ea1da3c975831b","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-de321ecf67c93134f27a4c19729d60fd696013ebb441207980912acd6a837a87","qualified_name":"Real.one_lt_rpow_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":30649,"column":41,"line":711},"start":{"byte":30378,"column":0,"line":708}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    Iff\n      (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))\n      (Or\n        (And (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x)\n          (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y))\n        (And (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x)\n          (And (@LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n            (@LT.lt.{0} Real Real.instLT y\n              (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))))","statement_sha256":"2a5a02f5a17eb446b0bc7c0784dcb6af30d52991ea833d8fdfb6bacf76b746c9","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-8bc5a34ca724321551c13ab6a1d6186c2b959facf210d772893d6e7ce3925efe","qualified_name":"Real.one_lt_rpow_iff_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":30376,"column":89,"line":706},"start":{"byte":30185,"column":0,"line":705}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    Iff\n      (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))\n      (Or\n        (And (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x)\n          (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y))\n        (And (@LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n          (@LT.lt.{0} Real Real.instLT y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))))","statement_sha256":"f6ef48fc20ddb4a2561dab15bca0f9ec3fb4012daf2e5fa1aadd05c8b6f415b6","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-860e84d99d63051f4716edf44e51b0c2ff616e07c5ceb8ff8e165de034b0ff70","qualified_name":"Real.one_lt_rpow_of_pos_of_lt_one_of_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":29349,"column":26,"line":685},"start":{"byte":29166,"column":0,"line":682}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LT.lt.{0} Real Real.instLT z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)","statement_sha256":"daa625135067e5722655df379cf98fa938411eb1926509f74426cb58b00f84fc","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-ebddc058a1e4d57b8946d15c5d965503425baf63b2e0f79bb5ed965d2b531125","qualified_name":"Real.one_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":5663,"column":68,"line":155},"start":{"byte":5587,"column":0,"line":154}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real),\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n      (@OfNat.ofNat.{0} 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: Type u_1} (f : α → Real),\n  @Eq.{u_1 + 1} (α → Real)\n    (@HPow.hPow.{u_1, 0, u_1} (α → Real) Real (α → Real)\n      (@instHPow.{u_1, 0} (α → Real) Real (@Pi.instPow.{u_1, 0, 0} α Real (fun a => Real) fun i => Real.instPow)) f\n      (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n    f","statement_sha256":"019ca6fda9ac61d4c65410604e8d9b78bd3fd01001181efd53166395a73fe0dc","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-d790dc4a2019f1ac8838348f4317941b3c9eb617d4bfb0ee65067f460b391bc0","qualified_name":"Real.pi_rpow_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":4629,"column":86,"line":126},"start":{"byte":4535,"column":0,"line":125}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} (f : α → Real),\n  @Eq.{u_1 + 1} (α → Real)\n    (@HPow.hPow.{u_1, 0, u_1} (α → Real) Real (α → Real)\n      (@instHPow.{u_1, 0} (α → Real) Real (@Pi.instPow.{u_1, 0, 0} α Real (fun a => Real) fun i => Real.instPow)) f\n      (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n    (@OfNat.ofNat.{u_1} (α → Real) (nat_lit 1)\n      (@One.toOfNat1.{u_1} (α → Real) (@Pi.instOne.{u_1, 0} α (fun a => Real) fun i => Real.instOne)))","statement_sha256":"8749a35184997fa17c0d08070168aca506603256317ee6ac68628bbe8dfb015a","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-06c815a845bc784188f12fdd767b393ec2624ef9f0489cc9c44b0f7ff127a46c","qualified_name":"Real.pow_le_iff_le_log","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":36077,"column":49,"line":820},"start":{"byte":35934,"column":0,"line":819}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Nat},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      Iff\n        (@LE.le.{0} Real Real.instLE\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))) x n)\n          y)\n        (@LE.le.{0} Real Real.instLE\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n            (Real.log x))\n          (Real.log y))","statement_sha256":"5b7f3e50782273aa4db9e0e64e84b7ab356843fd8c14c3ade6ef36b19eefc925","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-b0a5ee05bed706ff6db01d41ff8f79095195926469b414972787e787e15fb992","qualified_name":"Real.pow_le_of_le_log","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":36961,"column":45,"line":840},"start":{"byte":36837,"column":0,"line":839}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Nat},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n    @LE.le.{0} Real Real.instLE (Real.log x)\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n          (Real.log y)) →\n      @LE.le.{0} Real Real.instLE x\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))) y n)","statement_sha256":"f3033a17315830519b1e9a0ce0fbfaf348e8c44513595211b2500672e981b518","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-c338b43c337b10b1638be8178f49cecf9491bdabac1056eb48982bdeca84d9ec","qualified_name":"Real.pow_lt_iff_lt_log","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":37396,"column":49,"line":849},"start":{"byte":37257,"column":0,"line":848}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Nat},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      Iff\n        (@LT.lt.{0} Real Real.instLT\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))) x n)\n          y)\n        (@LT.lt.{0} Real Real.instLT\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n            (Real.log x))\n          (Real.log y))","statement_sha256":"f6d15d6a41f6ad186ac03b22f7b6215e94beb474ca7ddfc7a3b7edca6728f81e","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-13427ca3d058e5b73846a8560781f389a396ba7eb5593e0bec28aa30e75af71a","qualified_name":"Real.pow_lt_of_lt_log","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":38256,"column":45,"line":869},"start":{"byte":38136,"column":0,"line":868}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Nat},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n    @LT.lt.{0} Real Real.instLT (Real.log x)\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n          (Real.log y)) →\n      @LT.lt.{0} Real Real.instLT x\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))) y n)","statement_sha256":"a32aa2c650728bfb6789167ec5c443b4478af3b6316b3504a3a21ceac03f4cd2","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-1a39e1c574c30f3fc78100f8e50ae00d544b748b4af4077d7f84b88f567e763d","qualified_name":"Real.pow_rpow_inv_natCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":21536,"column":73,"line":511},"start":{"byte":21313,"column":0,"line":509}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real} {n : Nat},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Nat n (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\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))) x n)\n          (@Inv.inv.{0} Real Real.instInv (@Nat.cast.{0} Real Real.instNatCast n)))\n        x","statement_sha256":"6aca45203f3c4ad5894313ed76a0f2aa3cebeb5d29ff59dcb000672990825973","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-d4f47a456aef565016039d955ab630e0ea7fe7ebc9497be27b66858572fad84a","qualified_name":"Real.rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":1089,"column":28,"line":36},"start":{"byte":633,"column":0,"line":31}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"Real → Real → Real","statement_sha256":"0d4222242bb169ad0bbda1f344df403b005d93fb54c8c3e5d10e4395ea38fb1d","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-760ce2cb78e80d975964cbd0371ce915d59a26fc42b265d77ea7e834a9c21282","qualified_name":"Real.rpow_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":7803,"column":50,"line":209},"start":{"byte":7675,"column":0,"line":208}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y z : Real),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y z))\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))","statement_sha256":"c621fde360bd43a776254b3aae9a5b264560ea4e6354731d67f6c7d57a6640d4","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-dd3272c80d133ae1deb05b68170d1fd164adb99c1c77cda5db5ac993fd0e2fb0","qualified_name":"Real.rpow_add'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":8138,"column":27,"line":216},"start":{"byte":7805,"column":0,"line":211}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y z)\n        (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y z))\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))","statement_sha256":"c9a4ec175c638a70e7cacc3d16bb11604fa9726e97d834cff5f7b6df3bff23c3","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-1d4717ec89f065a17171e3fb106b5f0877325f5bd94b0b3a6c8bc510c316ac00","qualified_name":"Real.rpow_add_intCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":17896,"column":93,"line":430},"start":{"byte":17571,"column":0,"line":427}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    ∀ (y : Real) (n : Int),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y\n            (@Int.cast.{0} Real Real.instIntCast n)))\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Int Real\n            (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) x\n            n))","statement_sha256":"6ff1465b7e569e0311f31fc19a5ca2235cf9332af6ec2a9713d4a5d449115ca4","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-bb44ec68a7a7ad2855d52aa1749e6c5622163ad3026923c47679cd13324e9f1b","qualified_name":"Real.rpow_add_intCast'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":18469,"column":35,"line":442},"start":{"byte":18331,"column":0,"line":441}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ {n : Int},\n      @Ne.{1} Real\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y\n            (@Int.cast.{0} Real Real.instIntCast n))\n          (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        @Eq.{1} Real\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n            (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y\n              (@Int.cast.{0} Real Real.instIntCast n)))\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n            (@HPow.hPow.{0, 0, 0} Real Int Real\n              (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) x\n              n))","statement_sha256":"5778dfc7b61269e5d4c35b0554615243a593b270d1b25e4514306c275286574d","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-bc3e3031a1a76dace5a5feb01b82f42c067adf141b5ccc8340547f387cdfef6b","qualified_name":"Real.rpow_add_natCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":18039,"column":37,"line":433},"start":{"byte":17898,"column":0,"line":432}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    ∀ (y : Real) (n : Nat),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y\n            (@Nat.cast.{0} Real Real.instNatCast n)))\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\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))) x n))","statement_sha256":"b012424b6de409f9c685ce5b6d9588926a8b66a0cfbf449cb05b534648937022","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-175eb8297428a4aa5a41c2ec5c9ab3f4b659899aa860b9823508a7aa5feb12c3","qualified_name":"Real.rpow_add_natCast'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":18599,"column":35,"line":445},"start":{"byte":18471,"column":0,"line":444}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Nat},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real\n        (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y\n          (@Nat.cast.{0} Real Real.instNatCast n))\n        (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y\n            (@Nat.cast.{0} Real Real.instNatCast n)))\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\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))) x n))","statement_sha256":"c9141cf6e86786ac899525b9973a00a9336f6315a9e0ffa15eac32779027b5be","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-9dda899c15def4dc0651db4d14e0358b7f6f2b3baf5fbf04f1977abd7357efed","qualified_name":"Real.rpow_add_of_nonneg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":8609,"column":65,"line":226},"start":{"byte":8356,"column":0,"line":222}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        @Eq.{1} Real\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n            (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y z))\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))","statement_sha256":"8ea18257bf98a1d43b4e723eef04cfa381dc049ecbd749eb1312fbfa93cfb4b4","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-a2fbad9877762e3a326889cb1f6bdd270cec881c77e3657c20ec6c6ec3fdaf56","qualified_name":"Real.rpow_add_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":18996,"column":37,"line":454},"start":{"byte":18871,"column":0,"line":453}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    ∀ (y : Real),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y\n            (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))))\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) x)","statement_sha256":"111482a3d6f9990d4b109b3c4adb3dbbc4d35d65da8c2690d6ac886f6dbc9f38","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-8edc42c298c14a78f9afd37b3b3c189694bbe3969b390fba7f940c8bb62a06e1","qualified_name":"Real.rpow_add_one'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":19241,"column":31,"line":460},"start":{"byte":19125,"column":0,"line":459}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real\n        (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y\n          (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n        (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y\n            (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))))\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) x)","statement_sha256":"eb8f85d82297e479e6970cb8adf52912dc0c2013cc186e3d31edffe38902a45f","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-d2eee135fcc8827335461251f9294bff238ba2c5bd8b9c629544a243d250a547","qualified_name":"Real.rpow_def","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":1282,"column":72,"line":43},"start":{"byte":1210,"column":0,"line":43}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x y : Real),\n  @Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n    (@HPow.hPow.{0, 0, 0} Complex Complex Complex (@instHPow.{0, 0} Complex Complex Complex.instPow) ↑x ↑y).re","statement_sha256":"bec67e98839a61e2a476654b1801a88c52abcd54ea55ff8d9e9f649f4ae7a19b","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-3096d67f596cfe5949fea3dda1b85e225e6d4363feaf2c5e4a288685275aad90","qualified_name":"Real.rpow_def_of_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":4005,"column":21,"line":107},"start":{"byte":3242,"column":0,"line":95}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    ∀ (y : Real),\n      @Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n          (Real.exp (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (Real.log x) y))\n          (Real.cos (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) y Real.pi)))","statement_sha256":"e96ef3fd2b1d6978d41a08ce39b6818df75b948d26e719951f716709076f9ad3","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-2e1ddc148444def4930a4ab6bd757fe1f60b6e51e57fee18355311367b3dab4c","qualified_name":"Real.rpow_def_of_nonneg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":1626,"column":83,"line":49},"start":{"byte":1284,"column":0,"line":45}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real),\n      @Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@ite.{1} Real (@Eq.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n          (x.decidableEq (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n          (@ite.{1} Real (@Eq.{1} Real y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n            (y.decidableEq (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n            (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n            (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n          (Real.exp (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (Real.log x) y)))","statement_sha256":"99f663e87355251dc0f6bca743bde0c3f08499cece456efed6caea350fceac8e","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-905c8e4868b54d471688f579016a9837d417a3c20941c7cbcae69f92aafd7cd3","qualified_name":"Real.rpow_def_of_nonpos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":4264,"column":44,"line":114},"start":{"byte":4007,"column":0,"line":109}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    ∀ (y : Real),\n      @Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@ite.{1} Real (@Eq.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n          (x.decidableEq (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n          (@ite.{1} Real (@Eq.{1} Real y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n            (y.decidableEq (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n            (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n            (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n            (Real.exp (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (Real.log x) y))\n            (Real.cos (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) y Real.pi))))","statement_sha256":"345f362b69bfdbcd4e1ce4f748f80ddd64bc27d16efcba79a22afedc1b123d9d","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-de7bdaee05d962caacca9628f758b97afc5a1f90f540f2133bcba1bf02ce42b7","qualified_name":"Real.rpow_def_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":1784,"column":67,"line":52},"start":{"byte":1628,"column":0,"line":51}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real),\n      @Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (Real.exp (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (Real.log x) y))","statement_sha256":"417e987a6da65abfc8e99a02eeb38d0d90b5d4ea9f49f4ac3070e13ebe21ac52","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-3d57a336185e76b1bceae6ad66b366c54455873177bb2ca4bbd1381dc043a66d","qualified_name":"Real.rpow_div_two_eq_sqrt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":43793,"column":6,"line":997},"start":{"byte":43647,"column":0,"line":994}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real} (r : Real),\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Eq.{1} Real\n      (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid)) r\n          (@OfNat.ofNat.{0} Real (nat_lit 2) (@instOfNatAtLeastTwo.{0} Real (nat_lit 2) Real.instNatCast ⋯))))\n      (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x.sqrt r)","statement_sha256":"94fb8c99a1d0d9d7f219b126c857e26a3d6848e3f4fdacc20b70ec7ab121f6e0","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-5947259d142fa9c9caa016139d07b9d3303ffe54fb776cac193e689edff8cc25","qualified_name":"Real.rpow_eq_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":1208,"column":57,"line":41},"start":{"byte":1143,"column":0,"line":40}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x y : Real),\n  @Eq.{1} Real (x.rpow y) (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)","statement_sha256":"423ecccc51d641ee2b9954238de6f7b3befe410885cc32a1d2e8356d135e78cc","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-9eabaf92dc005ea3d85dd46cd3e8270768b1046727669a6b6231fceffad031ed","qualified_name":"Real.rpow_eq_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":3131,"column":38,"line":88},"start":{"byte":3008,"column":0,"line":86}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      Iff\n        (@Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n        (@Eq.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))","statement_sha256":"1c20a64c7bf198e2e58a1c6408f34fe7eaeec6437cc1807a309446ddeb50401a","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-ddb7f38330dcf5a146276c209af48c0f1f537da2f1b74dbbaa0acf031fde1b27","qualified_name":"Real.rpow_eq_zero_iff_of_nonneg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":3006,"column":37,"line":84},"start":{"byte":2843,"column":0,"line":82}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    Iff\n      (@Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n      (And (@Eq.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n        (@Ne.{1} Real y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))","statement_sha256":"1d43ca83fe20f6ae8c97d1a7dfcb47f34354c9263cfd008ceef16126959bfaa7","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-db23e9bc7223f8ebae182953fa198c30243d7872ddb15379f1f8bc105b51aab0","qualified_name":"Real.rpow_intCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":2095,"column":22,"line":59},"start":{"byte":1889,"column":0,"line":56}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real) (n : Int),\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@Int.cast.{0} Real Real.instIntCast n))\n    (@HPow.hPow.{0, 0, 0} Real Int Real\n      (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) x n)","statement_sha256":"73baf6129a588e416b0be45be6f6338ed7270504507867f5fb4a4bbeea8560ca","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-6a8007b8b5b802e7ec43f882a82f2116f10872f92610ab441ee7592c6f78cfdc","qualified_name":"Real.rpow_intCast_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":22139,"column":32,"line":524},"start":{"byte":22015,"column":0,"line":523}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (n : Int) (z : Real),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Int.cast.{0} Real Real.instIntCast n)\n            z))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@HPow.hPow.{0, 0, 0} Real Int Real\n            (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) x\n            n)\n          z)","statement_sha256":"5c721a888a0ee43d3b6d93d673722d47dc5aa6afe67674100f74ef08c17e5520","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-980698f3967c305ba1abd60d4845b8b1769e5dc38fb6d8fe0a95b24cb3942041","qualified_name":"Real.rpow_inv_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":33784,"column":65,"line":771},"start":{"byte":33619,"column":0,"line":770}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @Ne.{1} Real z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        Iff\n          (@Eq.{1} Real\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n              (@Inv.inv.{0} Real Real.instInv z))\n            y)\n          (@Eq.{1} Real x (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))","statement_sha256":"7985a49998a6017b12d30d97cd07ae4aff5c35e02e500e30933d9af53ddfdb00","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-4886f5a1c478fc12c7a1c6130bdd10a04ef33ab776dd3d89c3cb1bf8ecdcfbf7","qualified_name":"Real.rpow_inv_le_iff_of_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":26167,"column":80,"line":610},"start":{"byte":25972,"column":0,"line":608}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        Iff\n          (@LE.le.{0} Real Real.instLE\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n              (@Inv.inv.{0} Real Real.instInv z))\n            y)\n          (@LE.le.{0} Real Real.instLE\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z) x)","statement_sha256":"2be86e41f08600cc8149cf37241f5de4528586cee450eb93920721d1d6082b47","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-e1f6f72f2eb87e95b952859b959c3339d057c5c380eae46816ab73db00d2c207","qualified_name":"Real.rpow_inv_le_iff_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":25053,"column":65,"line":588},"start":{"byte":24875,"column":0,"line":587}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        Iff\n          (@LE.le.{0} Real Real.instLE\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n              (@Inv.inv.{0} Real Real.instInv z))\n            y)\n          (@LE.le.{0} Real Real.instLE x\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))","statement_sha256":"0378cce609f73add13d1c98b2bc1150edae68fb0e558acb3485e8330ed50e204","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-6edf2f369a14d0be952d33e8a1cac228cae7d2ab96c619057ab74700da0f0636","qualified_name":"Real.rpow_inv_log","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":7080,"column":54,"line":188},"start":{"byte":6892,"column":0,"line":186}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@Inv.inv.{0} Real Real.instInv (Real.log x)))\n        (Real.exp (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))","statement_sha256":"999c330b37fe0ea164a59ee97b2f358ec91e598c7f4a94623626b133ee34b3e0","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-f2a3871d030fbbecd0f8740a657ecee1166bfe2d224e4afbc98885512bdf1a77","qualified_name":"Real.rpow_inv_log_le_exp_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":7484,"column":24,"line":200},"start":{"byte":7082,"column":0,"line":190}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@Inv.inv.{0} Real Real.instInv (Real.log x)))\n    (Real.exp (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))","statement_sha256":"8e8d4837743093f16d950059b865a8a970166bee66f10925e807ed2b6c88a7dd","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-94fc1714f833f4756c5fefcb4e547ac81dc448f27d1f6ff23b228828a2025b89","qualified_name":"Real.rpow_inv_lt_iff_of_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":25970,"column":80,"line":606},"start":{"byte":25779,"column":0,"line":604}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        Iff\n          (@LT.lt.{0} Real Real.instLT\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n              (@Inv.inv.{0} Real Real.instInv z))\n            y)\n          (@LT.lt.{0} Real Real.instLT\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z) x)","statement_sha256":"0729190dfab98b7e850682f1fd72d524ca20cb450ef020c81bc7f05f87b5c510","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-07bf27c0d2069fe73f2866aec0ca3f763ed68c0f6ed2e8f6914383cfb9df7772","qualified_name":"Real.rpow_inv_lt_iff_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":25387,"column":59,"line":594},"start":{"byte":25222,"column":0,"line":593}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        Iff\n          (@LT.lt.{0} Real Real.instLT\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n              (@Inv.inv.{0} Real Real.instInv z))\n            y)\n          (@LT.lt.{0} Real Real.instLT x\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))","statement_sha256":"e4f7409b0eba58f8494d321839b2a3f00837997e8f6167732c3dc0e1954c4e8f","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-59d1c27016d58e8ded9c39bb89932a94572f3e3bb1ee087de0485742f273d8c0","qualified_name":"Real.rpow_inv_natCast_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":21761,"column":73,"line":515},"start":{"byte":21538,"column":0,"line":513}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real} {n : Nat},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Nat n (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n      @Eq.{1} Real\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)))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n            (@Inv.inv.{0} Real Real.instInv (@Nat.cast.{0} Real Real.instNatCast n)))\n          n)\n        x","statement_sha256":"bfba648fa37b2825352d420a2254d84689e13b637939ed5620823fd16554e56a","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-df77ff5362cee901b2eb894fc0c4b1eeca0824a4b702fac597a997a48e86ddbf","qualified_name":"Real.rpow_inv_rpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":21311,"column":54,"line":507},"start":{"byte":21170,"column":0,"line":506}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n            (@Inv.inv.{0} Real Real.instInv y))\n          y)\n        x","statement_sha256":"0d2eafb1901129c0d0ce920c5d2a88762221a7f4735c050fa6937bcc2b840e9a","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-54b376b28e9225bd8f9c378729a94f95fde883f8398615809bdee8cd0c9ac5e3","qualified_name":"Real.rpow_le_iff_le_log","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":35932,"column":64,"line":817},"start":{"byte":35773,"column":0,"line":816}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      Iff\n        (@LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n          y)\n        (@LE.le.{0} Real Real.instLE\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) z (Real.log x)) (Real.log y))","statement_sha256":"d49661aabf9c84a02ed44847d093b50f09bda6114a39418bedb03f5c256a0c40","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-b57009dc5b1df1e02f22e3c5094df5894a0ed857f0840bf237ab709620b49ce4","qualified_name":"Real.rpow_le_of_le_log","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":36835,"column":41,"line":837},"start":{"byte":36632,"column":0,"line":834}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n    @LE.le.{0} Real Real.instLE (Real.log x)\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) z (Real.log y)) →\n      @LE.le.{0} Real Real.instLE x (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z)","statement_sha256":"1a6b6cc106a7f6fcac769797d022b278bb5ad443e89c612d35eb693b8be5102c","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-c917aea9f520e7254f3a397fbeae5b8790d85d236ae3228f151fca2202e0295f","qualified_name":"Real.rpow_le_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":28571,"column":8,"line":664},"start":{"byte":28442,"column":0,"line":662}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n          (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real 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Real.instZero)) x →\n    Iff\n      (@LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n      (Or\n        (And (@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x)\n          (@LE.le.{0} Real Real.instLE y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))\n        (And (@LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n          (@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y)))","statement_sha256":"f2a98c4c8a5e209e46097f687fd59ff5670847ee9492924836429e7cac50bc50","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-cf04844230916a41e2d11fe2d094d2a2b1173fcd931e444dbcd9c6ef31d9de73","qualified_name":"Real.rpow_le_one_of_one_le_of_nonpos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":28912,"column":26,"line":672},"start":{"byte":28739,"column":0,"line":670}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x →\n    @LE.le.{0} Real Real.instLE z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n        (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real 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x →\n    @LE.le.{0} Real Real.instLE x y →\n      @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y 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 @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        Iff\n          (@LE.le.{0} Real Real.instLE\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))\n          (@LE.le.{0} Real Real.instLE x 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x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        Iff\n          (@LE.le.{0} Real Real.instLE\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))\n          (@LE.le.{0} Real Real.instLE y x)","statement_sha256":"47c67b53bfaf24448e0f00e42e42897c314854dc952eb4d725b12a3a3e9592e2","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-65cf34d661f4cf176f880893a108bc908138587a7c62bb0774ffd234a479bf4a","qualified_name":"Real.rpow_le_rpow_left_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":27240,"column":56,"line":633},"start":{"byte":26955,"column":0,"line":629}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x →\n    Iff\n      (@LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))\n      (@LE.le.{0} Real Real.instLE y z)","statement_sha256":"ac1c259afc8a4b08172ef3f4106cf40c7e4ccbefeeec186361f2feefc39a61c9","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-135b3de77292d47e760cd89e348afaf17825566a2baa541db9185f1bcc03619d","qualified_name":"Real.rpow_le_rpow_left_iff_of_base_lt_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":28092,"column":46,"line":651},"start":{"byte":27823,"column":0,"line":647}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      Iff\n        (@LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))\n        (@LE.le.{0} Real Real.instLE z y)","statement_sha256":"80a565be2602ceae12897651e7d1fc2a1e66bd4ce8c58710fc97155b9850296a","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-36b5a35aef04e92ceaeb76004768a6a74e070e0843adee280c4d01ca9c92be2c","qualified_name":"Real.rpow_le_rpow_of_exponent_ge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":27821,"column":86,"line":645},"start":{"byte":27594,"column":0,"line":643}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LE.le.{0} Real Real.instLE z y →\n        @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)","statement_sha256":"22b195df793e5a926201e75f4ff79aa952c88eaf562fd6eb5f54ee9a6a715bee","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-ad49de81184f1e28f1c27cc057f1d7b7b3fec9d4ebde189b9c4d86f9c40cd5fb","qualified_name":"Real.rpow_le_rpow_of_exponent_ge'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":31579,"column":94,"line":729},"start":{"byte":31192,"column":0,"line":725}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        @LE.le.{0} Real Real.instLE z y →\n          @LE.le.{0} Real Real.instLE\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)","statement_sha256":"f1550eef439a6ecdd6bbae372ad1400cab4a8c733b9d62283fc03de4df729779","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-36d3f4602643a5c917ae3b7c7f06c476304a94edeef4e555d301b7a12531889c","qualified_name":"Real.rpow_le_rpow_of_exponent_ge_of_imp","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":31190,"column":51,"line":723},"start":{"byte":30651,"column":0,"line":713}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LE.le.{0} Real Real.instLE z y →\n        (@Eq.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n            @Eq.{1} Real y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n              @Eq.{1} Real z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))) →\n          @LE.le.{0} Real Real.instLE\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)","statement_sha256":"de6f19ea75a9c1b08f351667535d0fd1156b73f7d31615af5c9b10efa2989201","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-e143d803dee30ea4cb8270522bd9ba3613100f73acf5f77494b29b183f1a2e4d","qualified_name":"Real.rpow_le_rpow_of_exponent_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":26587,"column":46,"line":619},"start":{"byte":26377,"column":0,"line":616}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x →\n    @LE.le.{0} Real Real.instLE y z →\n      @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)","statement_sha256":"dd9beef7d74e2b7b392de58edd7125f263bbac91392d244f5dcab73d71009d77","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-df04abc17998fed9990518f469cfa3eeed26911a916553e1a1dad90536207d0b","qualified_name":"Real.rpow_le_rpow_of_exponent_le_or_ge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":41443,"column":52,"line":932},"start":{"byte":41008,"column":0,"line":926}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  Or\n      (And (@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x)\n        (@LE.le.{0} Real Real.instLE y z))\n      (And (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x)\n        (And (@LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n          (@LE.le.{0} Real Real.instLE z y))) →\n    @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n      (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)","statement_sha256":"d513ac5fedf38c2b837197f3c4c2e7811012bb14a804f4d7f480a466d489dc9c","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-95491f3101f5cb0dd1db41159fbe724b69757f7eeca2e4c5839a858cc8f567cd","qualified_name":"Real.rpow_le_rpow_of_nonpos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":23980,"column":22,"line":569},"start":{"byte":23722,"column":0,"line":565}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE x y →\n      @LE.le.{0} Real Real.instLE z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)","statement_sha256":"98718f4e371d23d4dbd9d91e4ce268a221d8b890f1a3b0c387d35b6ef0ca6bfc","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-2ee10886eca6af211342d017aa338ea0b5a53fd73b673b58f74e1019a5eb2a69","qualified_name":"Real.rpow_le_self_of_le_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":32455,"column":109,"line":746},"start":{"byte":32220,"column":0,"line":744}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) y →\n        @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          x","statement_sha256":"18fd71569429779944ec97c1142a794c215491066573fcef1d0370bc8d2ea43e","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-41b56a444cc822e7d06faedfc70a2766887754f6255aab893cbcd9ae77f7e1ac","qualified_name":"Real.rpow_le_self_of_one_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":32609,"column":67,"line":749},"start":{"byte":32457,"column":0,"line":748}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x →\n    @LE.le.{0} Real Real.instLE y (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LE.le.{0} Real Real.instLE (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) 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@LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @Ne.{1} Real z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        Iff\n          (@Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))\n          (@Eq.{1} Real x 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@LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        @LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n          (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real 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Real.instZero)) x →\n    Iff\n      (@LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n      (Or\n        (And (@Eq.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n          (@Ne.{1} Real y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))\n        (Or\n          (And (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x)\n            (@LT.lt.{0} Real Real.instLT y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))\n          (And (@LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n            (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))\n              y))))","statement_sha256":"e75cb313432c173f94fd0a1b0a4cef9945118c476cf4ebbd5166bd11a9f5cb11","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-a95fd12538d53969769c635cf43198fb23dde8a8cd24eaed8558fe60db5f92da","qualified_name":"Real.rpow_lt_one_iff'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":30183,"column":65,"line":703},"start":{"byte":30021,"column":0,"line":701}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      Iff\n        (@LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n        (@LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real 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Real.instZero)) x →\n    Iff\n      (@LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n      (Or\n        (And (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x)\n          (@LT.lt.{0} Real Real.instLT y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))))\n        (And (@LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n          (@LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y)))","statement_sha256":"4886f10315083210a5a69a04baf4457abdd66f58a0f5fbc1bc0a7c4148fbacf4","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-801e8fa3dd7ad4081ddd1421e60677f84973b33bbacefd54d3e4dfd9698883a4","qualified_name":"Real.rpow_lt_one_of_one_lt_of_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":28737,"column":26,"line":668},"start":{"byte":28573,"column":0,"line":666}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real 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 @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) z →\n        Iff\n          (@LT.lt.{0} Real Real.instLT\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))\n          (@LT.lt.{0} Real Real.instLT x 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x →\n    @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LT.lt.{0} Real Real.instLT z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        Iff\n          (@LT.lt.{0} Real Real.instLT\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z))\n          (@LT.lt.{0} Real Real.instLT y x)","statement_sha256":"b72b42dd5e83fb577e5467cd2657f2a4146120c961aad98bb86fcb88cc4a9985","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-4ae58ec7cf73c9210db97a9531303aeec3f2193d46a662a84175b53a2bcd5b8f","qualified_name":"Real.rpow_lt_rpow_left_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":27386,"column":61,"line":637},"start":{"byte":27242,"column":0,"line":635}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x →\n    Iff\n      (@LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))\n      (@LT.lt.{0} Real Real.instLT y 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Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      Iff\n        (@LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))\n        (@LT.lt.{0} Real Real.instLT z 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Real.instZero)) x →\n    @LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LT.lt.{0} Real Real.instLT z y →\n        @LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x 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Real.instOne)) x →\n    @LT.lt.{0} Real Real.instLT y z →\n      @LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x 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→\n    @LT.lt.{0} Real Real.instLT x y →\n      @LT.lt.{0} Real Real.instLT z (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        @LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y z)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x 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→\n    @LT.lt.{0} Real Real.instLT x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) y →\n        @LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          x","statement_sha256":"69ab53820ab374b6f811c49375065f43d7234b2fb934affbbad0f7aac60feca8","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-8aed9f3272bd8fc0cd9ef97f3a53ab8237c14eb0a0a86ebee5f458ae497f2800","qualified_name":"Real.rpow_lt_self_of_one_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":33241,"column":67,"line":761},"start":{"byte":33095,"column":0,"line":760}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) x →\n    @LT.lt.{0} Real Real.instLT y (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      @LT.lt.{0} Real Real.instLT (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) x","statement_sha256":"2084eb0f546355fd7c2fbcda086fd46f3d54c09bfbadd7882e0a118e0225bbb1","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-1a7356fcba5cb3bcda9e31ed03976962a89bc4aa962ba2475bc253c618c6ac3f","qualified_name":"Real.rpow_max","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":31843,"column":50,"line":735},"start":{"byte":31581,"column":0,"line":731}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y p : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) y →\n      @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) p →\n        @Eq.{1} Real\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n            (@Max.max.{0} Real Real.instMax x y) p)\n          (@Max.max.{0} Real Real.instMax\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x p)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) y p))","statement_sha256":"7e36adec4a72ddae9efa9f68b41bc9aac1c93360503c5b92b33082cb32deec52","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-7a72619642622cedf8eb6cebd6cff6cd16670483cdc4df4deef03835bdb60d4d","qualified_name":"Real.rpow_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":17252,"column":43,"line":419},"start":{"byte":16860,"column":0,"line":415}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y z : Real),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) y z))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) z)","statement_sha256":"afde72a261ad3c2ec7d6a46cc0f834e33bfe48551906c85e60619864f339d44e","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-631c9a9b88879511d8cec9b60839df5c48fd596e5201c5406705bfbbc70a01a5","qualified_name":"Real.rpow_mul_intCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":22265,"column":32,"line":527},"start":{"byte":22141,"column":0,"line":526}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real) (n : Int),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) y\n            (@Int.cast.{0} Real Real.instIntCast n)))\n        (@HPow.hPow.{0, 0, 0} Real Int Real\n          (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid)))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) n)","statement_sha256":"4309ec7cb53da6bf438163db193e30fce8f0c9540545cd6b3a2510541f9ad0e3","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-d80ca1cd51b4e56020ee23c419449bf348de0cfa965478489f91d087656c0aec","qualified_name":"Real.rpow_mul_natCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":22013,"column":32,"line":521},"start":{"byte":21889,"column":0,"line":520}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real) (n : Nat),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) y\n            (@Nat.cast.{0} Real Real.instNatCast n)))\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)))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) n)","statement_sha256":"f4b4a7ab2e9ff7803c3feeea919f036234bab1c8d4b49a918077b514cafe8830","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-2b1af0f888dea2d4548fe5e5f9dd574826c5bb92fb63c19a67818d5b6defd770","qualified_name":"Real.rpow_natCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":2215,"column":99,"line":62},"start":{"byte":2097,"column":0,"line":61}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real) (n : Nat),\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@Nat.cast.{0} Real Real.instNatCast n))\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))) x n)","statement_sha256":"81afa7504891820e5e27e575eda4e7c23cbac927baab41c6a91f76daf3958c28","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-d0e62a4d278fca5495e966ce47438584c4c146762d120d231620dd16db13074a","qualified_name":"Real.rpow_natCast_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":21887,"column":32,"line":518},"start":{"byte":21763,"column":0,"line":517}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (n : Nat) (z : Real),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) (@Nat.cast.{0} Real Real.instNatCast n)\n            z))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\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))) x n)\n          z)","statement_sha256":"281125e09f9c71f16fe4243129090e39b3d16968dcf6405350d99ccf2fb8d955","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-f453e1efe50928f4cc3f8760ee10ba2b8a0d92fca87fb9dd7edcafd1ed29c38c","qualified_name":"Real.rpow_ne_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":3229,"column":15,"line":91},"start":{"byte":3133,"column":0,"line":90}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      Iff\n        (@Ne.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n        (@Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))","statement_sha256":"feaa30b8c442679eed84cd36cdc47d7eeebfd35dec4e4348ba6223eb793ee4ab","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-6e2d20548bb6866bd99a60f2485981f04945df9aa8c4cb54f8a0d8ec5afc37ce","qualified_name":"Real.rpow_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":10459,"column":69,"line":261},"start":{"byte":10223,"column":0,"line":259}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@Neg.neg.{0} Real Real.instNeg y))\n        (@Inv.inv.{0} Real Real.instInv\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x 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(@Neg.neg.{0} Real Real.instNeg y))\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) (@Inv.inv.{0} Real Real.instInv x) y)","statement_sha256":"bbea0b2414fa3a371b9613fa755b07ad1ef5da191490e23041019d835948c88b","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-1125442553a83cbdaf111d723e3d663d020071a5756a22f96d58ac967845cf8c","qualified_name":"Real.rpow_neg_natCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":2376,"column":55,"line":66},"start":{"byte":2217,"column":0,"line":64}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real) (n : Nat),\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@Neg.neg.{0} Real Real.instNeg (@Nat.cast.{0} Real Real.instNatCast n)))\n    (@HPow.hPow.{0, 0, 0} Real Int Real\n      (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) x\n      (@Neg.neg.{0} Int Int.instNegInt (@Nat.cast.{0} Int instNatCastInt n)))","statement_sha256":"b7e7516c1593af72e1042e3167547342d3d9b3e395d82b101c7d97d014ec5f89","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-8a9688724e0c93adcc8643ac65c5e81c39032921252f4d6a3afbb406bcb8835a","qualified_name":"Real.rpow_neg_ofNat","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":2650,"column":22,"line":75},"start":{"byte":2511,"column":0,"line":73}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real) (n : Nat) [inst : n.AtLeastTwo],\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@Neg.neg.{0} Real Real.instNeg\n        (@OfNat.ofNat.{0} Real n (@instOfNatAtLeastTwo.{0} Real n Real.instNatCast inst))))\n    (@HPow.hPow.{0, 0, 0} Real Int Real\n      (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) x\n      (@Neg.neg.{0} Int Int.instNegInt (@OfNat.ofNat.{0} Int n (@instOfNat n))))","statement_sha256":"fe3c42baac5b93f97703dfd47dcb5be173981ffbc6e10ecb469c49b63152b6bc","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-c537aaa6c4322b1b7b2c37220be1f4c3cc5e75974ba605b2ca49e34e96598dde","qualified_name":"Real.rpow_neg_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":19762,"column":37,"line":475},"start":{"byte":19662,"column":0,"line":474}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real),\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@Neg.neg.{0} Real Real.instNeg (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))))\n    (@Inv.inv.{0} Real Real.instInv x)","statement_sha256":"298dda4b4cae94ab2b35869485ac5625f31bd0ff0bd24e7a0cbd50b400eed0df","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-f3b9abc804e6be0cfbe9f5807498ecc6e005754c481b0c811e15fa8a8058cb57","qualified_name":"Real.rpow_nonneg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":6073,"column":58,"line":166},"start":{"byte":5887,"column":0,"line":163}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real),\n      @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)","statement_sha256":"c9917df178ae2f79be2e2c077aea08103272d0e70b4f045b3a2805bdb1f5e47a","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-6ddbe39df5e34e4117419b6feb17acc285549da03943e23de7e336db2bc44093","qualified_name":"Real.rpow_ofNat","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":2509,"column":18,"line":71},"start":{"byte":2378,"column":0,"line":68}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real) (n : Nat) [inst : n.AtLeastTwo],\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@OfNat.ofNat.{0} Real n (@instOfNatAtLeastTwo.{0} Real n Real.instNatCast inst)))\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))) x\n      (@OfNat.ofNat.{0} Nat n (instOfNatNat n)))","statement_sha256":"9b52d3c875c836b51711333a10b8749060cbf910d78eba72ad4cab2af9b2f4fc","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-2cf078a0c4aa79033c06172983f1adf0df2e68c9334c7e09bf2f130a99be4829","qualified_name":"Real.rpow_of_add_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":8354,"column":35,"line":220},"start":{"byte":8140,"column":0,"line":218}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {w x y z : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real w (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd) y z) w →\n        @Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x w)\n          (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))","statement_sha256":"73e980742095ae278e0530a6ce47f4c38ce191e686324c8f60abc7d082f9b271","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-d1f37671f7b41720290a65703e3bf116a0858ffabe8d00ab56924c6609d2ab02","qualified_name":"Real.rpow_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":5490,"column":68,"line":149},"start":{"byte":5414,"column":0,"line":148}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real),\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n    x","statement_sha256":"15ca4f5e32d70b6b4b7287df9e5d2872ba53ac94d5949559813f69af9c276573","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-665eaa36eb6ee410f1afb67eba6d4fdf5fd8b8e8695b79739f1bc23cdeff322c","qualified_name":"Real.rpow_one_add'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":19359,"column":31,"line":463},"start":{"byte":19243,"column":0,"line":462}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real\n        (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd)\n          (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) y)\n        (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HAdd.hAdd.{0, 0, 0} Real Real Real (@instHAdd.{0} Real Real.instAdd)\n            (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) y))\n        (@HMul.hMul.{0, 0, 0} Real Real Real (@instHMul.{0} Real Real.instMul) x\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))","statement_sha256":"8876ee106b3d5e5762c341325aa5bbc549d19bedc90e4547ac32020e5ba8042a","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-a2c2375d6ba71dfd2e6eae9dd8448fed151b935318886fef3c56c64e0e51ad66","qualified_name":"Real.rpow_one_sub'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":19595,"column":31,"line":469},"start":{"byte":19479,"column":0,"line":468}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real\n        (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub)\n          (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) y)\n        (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub)\n            (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) y))\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid)) x\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y))","statement_sha256":"590abe6259849dc7888575a0f7f0018eb81e8bb50c2418fba56f4b9b0e1d9992","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-50d29f7fed37cb94d421b75e8e6375513be4d2ddaf42900d6c82a5fb0c702e30","qualified_name":"Real.rpow_pos_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":4390,"column":40,"line":118},"start":{"byte":4266,"column":0,"line":116}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real),\n      @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)","statement_sha256":"a5feba397817ee303f71e0d5cf7d6209e7c70c8eacfc290a1116414fd5511dd7","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-eb74cf4076715df5fddacfda12f12f4592b9249e5306145d6c9ddffff929e077","qualified_name":"Real.rpow_pow_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":17410,"column":57,"line":422},"start":{"byte":17254,"column":0,"line":421}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y : Real) (n : Nat),\n      @Eq.{1} Real\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)))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) n)\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\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))) x n)\n          y)","statement_sha256":"0cdb3a9be50deee88bc8e08c1c667a3d513211db57e685ccfcb82540bb7defb6","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-eee76348f3e76fe92f9c406ab653db41dcb259d99370f429de9aea88c52051cf","qualified_name":"Real.rpow_right_inj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":41006,"column":57,"line":924},"start":{"byte":40714,"column":0,"line":920}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y z : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)) →\n      Iff\n        (@Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))\n        (@Eq.{1} Real y z)","statement_sha256":"fc44f6dc19979e9f8756d8a09cc88c2100057d546f7b58d6377e57cacfada04f","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-d14b7d2d9275c7679e8947bf4226cea30cba55d45207b61ae2200cd3b114308c","qualified_name":"Real.rpow_rpow_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":21168,"column":54,"line":504},"start":{"byte":21027,"column":0,"line":503}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real y (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@Inv.inv.{0} Real Real.instInv y))\n        x","statement_sha256":"f1ecb32e8d0cc2c436de5649f9014f50f64d18a883806a766a612e39b6ee4ac1","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-803cfed90278b6ba34671449b78be307764a79701281a873956845f91b23024e","qualified_name":"Real.rpow_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":10630,"column":81,"line":264},"start":{"byte":10461,"column":0,"line":263}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ (y z : Real),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y z))\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x 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Real},\n      @Ne.{1} Real (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y z)\n          (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        @Eq.{1} Real\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n            (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y z))\n          (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x z))","statement_sha256":"3b542279e0f4686bb5756c8c31dcb6ef0aee85f0887df7c26fe24451e522efac","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-395e0599c6566e0543a028d60f3149573805269d3aa6bb75d32af7247309013d","qualified_name":"Real.rpow_sub_intCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":18186,"column":41,"line":436},"start":{"byte":18041,"column":0,"line":435}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    ∀ (y : Real) (n : Int),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y\n            (@Int.cast.{0} Real Real.instIntCast n)))\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n          (@HPow.hPow.{0, 0, 0} Real Int Real\n            (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) x\n            n))","statement_sha256":"54b82b0dd75a46716af6d829187664ea502a07cf33752461234c2786ef9e0e82","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-771e79395aa23bcc1b5179f9b43d1d1abdf7c7471fea612b399f7d4421a2756d","qualified_name":"Real.rpow_sub_intCast'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":18739,"column":35,"line":448},"start":{"byte":18601,"column":0,"line":447}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    ∀ {n : Int},\n      @Ne.{1} Real\n          (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y\n            (@Int.cast.{0} Real Real.instIntCast n))\n          (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n        @Eq.{1} Real\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n            (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y\n              (@Int.cast.{0} Real Real.instIntCast n)))\n          (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n            (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\n            (@HPow.hPow.{0, 0, 0} Real Int Real\n              (@instHPow.{0, 0} Real Int (@ZPow.toPow.{0} Real (@DivInvMonoid.toZPow.{0} Real Real.instDivInvMonoid))) x\n              n))","statement_sha256":"6516a876a7bdbe7f11de2f17be6306e5791cbe4a2512023e718656b7a3d08462","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-96a6ccf5ffec56796d3caa39f488d4ac7f57ce86c617f05b03d1112a2933c294","qualified_name":"Real.rpow_sub_natCast","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":18329,"column":37,"line":439},"start":{"byte":18188,"column":0,"line":438}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    ∀ (y : Real) (n : Nat),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y\n            (@Nat.cast.{0} Real Real.instNatCast n)))\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\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))) x n))","statement_sha256":"92aaddd86c17f2d9308a4a60d491b17e5676634f3db8abc163b25dcad2691021","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-0dca4908ce3db496fc7126c914c14bbbaae1fd3e3f5e2d4251676e280f4c4320","qualified_name":"Real.rpow_sub_natCast'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":18869,"column":35,"line":451},"start":{"byte":18741,"column":0,"line":450}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real} {n : Nat},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real\n        (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y\n          (@Nat.cast.{0} Real Real.instNatCast n))\n        (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y\n            (@Nat.cast.{0} Real Real.instNatCast n)))\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y)\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))) x n))","statement_sha256":"f03baa6f544f509ae02d98859c42e282ea3f36b7cbf85fbe81b6220ec913d95b","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-291c96546657724a2e3e6ad65c1bfcc1aa6ccdd130d07c53bb14112f5e190f23","qualified_name":"Real.rpow_sub_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":19123,"column":37,"line":457},"start":{"byte":18998,"column":0,"line":456}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x : Real},\n  @Ne.{1} Real x (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n    ∀ (y : Real),\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y\n            (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))))\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) x)","statement_sha256":"21075799e9ddcff1a4a1676f89897a923e7e9ab92d6fa3d659c5d65c6643fae5","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-69de0fa6989e537c9428ce1a20f57f459b637390c95c13460497f623ee1c2996","qualified_name":"Real.rpow_sub_one'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":19477,"column":31,"line":466},"start":{"byte":19361,"column":0,"line":465}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {x y : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) x →\n    @Ne.{1} Real\n        (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y\n          (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne)))\n        (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) →\n      @Eq.{1} Real\n        (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n          (@HSub.hSub.{0, 0, 0} Real Real Real (@instHSub.{0} Real Real.instSub) y\n            (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))))\n        (@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n          (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x y) x)","statement_sha256":"867b2b51031f79f6e59d4ac833bfbaa18e332c79ca232ca46698b30d8659234a","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-ae2ff60599b7713b7323dbe750ba2f483ff998f8e9f9106ab86687a1b4bad58a","qualified_name":"Real.rpow_sum_of_nonneg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":10004,"column":85,"line":253},"start":{"byte":9576,"column":0,"line":247}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {a : Real},\n  @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) a →\n    ∀ {s : Finset.{u_1} ι} {f : ι → Real},\n      (∀ (x : ι),\n          @Membership.mem.{u_1, u_1} ι (Finset.{u_1} ι)\n              (@SetLike.instMembership.{u_1, u_1} (Finset.{u_1} ι) ι (@Finset.instSetLike.{u_1} ι)) s x →\n            @LE.le.{0} Real Real.instLE (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero))\n              (f x)) →\n        @Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) a (∑ x ∈ s, f x))\n          (∏ x ∈ s, @HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) a (f x))","statement_sha256":"67306b58103a1f3dc305bb0c198a417588fc93dcdcbbbcf608ad4301b4d600d9","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-f336a4f47750af73077f9d19306307302381885101eb01f2d359853d94dad624","qualified_name":"Real.rpow_sum_of_pos","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":9574,"column":110,"line":245},"start":{"byte":9316,"column":0,"line":243}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {a : Real},\n  @LT.lt.{0} Real Real.instLT (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)) a →\n    ∀ (f : ι → Real) (s : Finset.{u_1} ι),\n      @Eq.{1} Real (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) a (∑ x ∈ s, f x))\n        (∏ x ∈ s, @HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) a (f x))","statement_sha256":"d1a11aeea15989d9f7e6fb40e46a57cafcdc405dc1c5aec9f503146e01e2e9b5","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-461087b655bb18fbb2eca784b8b77e27d90df7495627d7bd4e563f2a92934e21","qualified_name":"Real.rpow_two","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":19660,"column":6,"line":472},"start":{"byte":19597,"column":0,"line":471}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real),\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@OfNat.ofNat.{0} 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2))))","statement_sha256":"d4578cb6f2046ba7441bcda33dfbca88c962ba8c700a5af372138725bebdb5a5","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-fdb613e0a5b28aa91f67a0dfc9384343ebe0c15f0c6afc88caf6693703863ce9","qualified_name":"Real.rpow_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Analysis.SpecialFunctions.Pow.Real","path":"Mathlib/Analysis/SpecialFunctions/Pow/Real.lean","range":{"end":{"byte":4469,"column":69,"line":121},"start":{"byte":4392,"column":0,"line":120}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (x : Real),\n  @Eq.{1} Real\n    (@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real Real.instZero)))\n    (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real 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(@HPow.hPow.{0, 0, 0} Real Real Real (@instHPow.{0, 0} Real Real Real.instPow) x\n      (@OfNat.ofNat.{0} Real (nat_lit 0) (@Zero.toOfNat0.{0} Real 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(@HDiv.hDiv.{0, 0, 0} Real Real Real (@instHDiv.{0} Real (@DivInvMonoid.toDiv.{0} Real Real.instDivInvMonoid))\n        (@OfNat.ofNat.{0} Real (nat_lit 1) (@One.toOfNat1.{0} Real Real.instOne))\n        (@OfNat.ofNat.{0} Real (nat_lit 2) (@instOfNatAtLeastTwo.{0} Real (nat_lit 2) Real.instNatCast 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