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d)","statement_sha256":"2fd1d2db2be77e2bd5fab9dd4b7fb7effd801becb60073a0b48f13cf0f30e980","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-9d0d7f0c5f7e55ce0e583dd93f7b60f7c09bd63db5d389bcadc9740980b5cfad","qualified_name":"add_add_neg_add_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23712,"column":28,"line":742},"start":{"byte":23686,"column":2,"line":742}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a b)\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          b)\n        c))\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      a c)","statement_sha256":"30b233208c4c26592393dd989c388b3ce16fc86ac4484187b5a83889a53e7f43","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-dfd05790db7863c8c7524acd0b69b280bf407948f4dd791ac42b84a00d7a16a7","qualified_name":"add_add_neg_cancel'_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":32870,"column":13,"line":971},"start":{"byte":32859,"column":2,"line":971}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G\n                (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n      a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        b\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G\n                (@SubtractionCommMonoid.toSubtractionMonoid.{u_3} G\n                  (@AddCommGroup.toDivisionAddCommMonoid.{u_3} G inst)))))\n          a)))\n    b","statement_sha256":"c273452a0cdf2b0eb5a7dcaa1538fd2f66423f8db971dbaaa17418d9a7fff8ee","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-124fc0f06235c24eba62869892b2ace78a45f83f6b6e298a40b8bd4afb8c820f","qualified_name":"add_add_sub_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":33020,"column":28,"line":975},"start":{"byte":32994,"column":2,"line":975}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G\n                (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        a c)\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        b c))\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G\n                (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n      a b)","statement_sha256":"77f28ae89699f1ec1f067da1d3d19e76fc54ff44ca446d39772ab76bf4744ad6","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-788e45287284bd36db490e3444f6ee16082cd8d82e180c47f5eaa7177ad52b36","qualified_name":"add_comm_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17878,"column":13,"line":572},"start":{"byte":17867,"column":2,"line":572}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n      (@instHAdd.{u_1} α\n        (@AddZero.toAdd.{u_1} α\n          (@AddZeroClass.toAddZero.{u_1} α\n            (@AddMonoid.toAddZeroClass.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α\n                (@SubtractionMonoid.toSubNegMonoid.{u_1} α\n                  (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))))\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α\n          (@SubNegMonoid.toSub.{u_1} α\n            (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a b)\n      c)\n    (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n      (@instHAdd.{u_1} α\n        (@AddZero.toAdd.{u_1} α\n          (@AddZeroClass.toAddZero.{u_1} α\n            (@AddMonoid.toAddZeroClass.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α\n                (@SubtractionMonoid.toSubNegMonoid.{u_1} α\n                  (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))))\n      a\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α\n          (@SubNegMonoid.toSub.{u_1} α\n            (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        c b))","statement_sha256":"01ac75344745688616d03ebca9284ef37211224ac0b7ef08aa373dcd3a9a3e67","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-35dd294b9c32f476eb32c828dbe6a48ae3d4869ddf82a366050e3a0668e99323","qualified_name":"add_eq_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":8027,"column":28,"line":238},"start":{"byte":8001,"column":2,"line":238}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M]\n  [@IsLeftCancelAdd.{u_4} M\n      (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst)))]\n  {a b : M},\n  Iff\n    (@Eq.{u_4 + 1} M\n      (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M\n          (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n        a b)\n      a)\n    (@Eq.{u_4 + 1} M b\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M\n          (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))))","statement_sha256":"563e424558aa6ce0d4ef886853a9bd30158b74feaad30983314171f74fa2bc79","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-a2429e42b9d72b92ab57686fa9621473ed58b8c8740898614fed0792864cd987","qualified_name":"add_eq_of_eq_add_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20166,"column":13,"line":643},"start":{"byte":20155,"column":2,"line":643}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        c\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          b)) →\n    @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a b)\n      c","statement_sha256":"464d5098e73c8df7f723e09c0f4febaebacf91fcb9243d74df15b64efb9228ed","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-972a503bd1eb3409b800f1fff5793e7c72fb85b36220c46502f66e747f2f4742","qualified_name":"add_eq_of_eq_neg_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20054,"column":13,"line":640},"start":{"byte":20043,"column":2,"line":640}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G b\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          a)\n        c) →\n    @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a b)\n      c","statement_sha256":"bc81a3e2380e493b8cfc2c86a2ea5c863331b3cefe3ada6eac72e1eb7b2936a7","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-4562114b805df1a4266e2ce2e5e0a7705cfae84c3a7a805176eec47193f17c98","qualified_name":"add_eq_of_eq_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23214,"column":13,"line":726},"start":{"byte":23203,"column":2,"line":726}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) c b) →\n    @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a b)\n      c","statement_sha256":"97ca0adcb428dab8ddef3f1715827ef7d30f3c8b8164eec526b17b3f694e0308","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-1a2d98eeca02f4bb1e29a9a61ff2ab9d927f3c24e7ec419570fa56f75cead67c","qualified_name":"add_eq_of_eq_sub'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31395,"column":13,"line":932},"start":{"byte":31384,"column":2,"line":932}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G b\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        c a) →\n    @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        a b)\n      c","statement_sha256":"0c19e4ea75b183176b31488e888cfff572478a0d4f39b480f4faad076df8a93c","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-47e40c6cf669ea5f1fe08611d4103263dca7f68900acd4a53d5123a20c62bb9b","qualified_name":"add_eq_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":8561,"column":28,"line":259},"start":{"byte":8535,"column":2,"line":259}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M]\n  [@IsRightCancelAdd.{u_4} M\n      (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M 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inst))))))","statement_sha256":"46d6d0e6b1313999525ac5bccbb2adac9ed5e66a07925d4051b7a077c2abde57","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-9b43c33fe91c3a54916a54b53dedb12f8a87488080f028258f0a34b686bd4691","qualified_name":"add_eq_zero_iff_eq_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20259,"column":13,"line":646},"start":{"byte":20248,"column":2,"line":646}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n 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b))","statement_sha256":"85a7f311e2458432dd2875068efde3833ff8080bbefa96b6e74273f669121f6b","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-644cfea00dbf736fb3cd0bc3a5ce67a0dd30a11919f679e157264e70f6af6222","qualified_name":"add_eq_zero_iff_eq_neg'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20598,"column":13,"line":655},"start":{"byte":20587,"column":2,"line":655}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n       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a))","statement_sha256":"a8e44fb6bd3ebd4ba1a4d2af143916cd7117f1532a348d65dac80c48becdcc5a","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-185249a80a4df63cb0525e55cc19afdccccd975e0f690d91cf78640ffb80a6be","qualified_name":"add_eq_zero_iff_neg_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20406,"column":13,"line":650},"start":{"byte":20395,"column":2,"line":650}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n        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b)","statement_sha256":"abdc23a82122f009d82163cf8b077805b9402d3934bde0806095499f579993c3","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-8615c371ac9849fd2026f975a37c73a42a088dc079becc59d1efb7bd707cf179","qualified_name":"add_eq_zero_iff_neg_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20781,"column":13,"line":660},"start":{"byte":20770,"column":2,"line":660}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n        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a)","statement_sha256":"61accc08a5c771ddc646d1c5d43aedd07f78945cb3a2f975ad4dff03a424d0f7","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-ad80a6a890672ae97daa96d860a9184e96027b6a846c3753b0c350f9c53d665c","qualified_name":"add_left_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3556,"column":13,"line":116},"start":{"byte":3545,"column":2,"line":116}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommSemigroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G 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c))","statement_sha256":"fdb5352a174327433ce21250bd098da2b3b5f360a46b891c3d637e82792881dc","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-82f45067ee2cea2c486a35b8edae41abdeb524ea4c07230258f6a11ab54eb1fc","qualified_name":"add_left_iterate","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":5982,"column":28,"line":182},"start":{"byte":5956,"column":2,"line":182}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] (a : M) (n : Nat),\n  @Eq.{u_4 + 1} (M → M)\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, u_4, u_4} M M M\n          (@instHAdd.{u_4} M\n            (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n          a x)\n      n)\n    fun x =>\n    @HAdd.hAdd.{u_4, u_4, u_4} M M M\n      (@instHAdd.{u_4} M\n        (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n      (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n        (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst))) n a)\n      x","statement_sha256":"ad22515551fe6d01dd6ed4f1827f11bbf0da743fc3b182c82960497af49e3b9e","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-f844658c0059b2cc852d0e72dc6c61982eae655b74c3d94c7c666020097be3e4","qualified_name":"add_left_iterate._f","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":5982,"column":28,"line":182},"start":{"byte":5956,"column":2,"line":182}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] (a : M) (x : Nat)\n  (f :\n    @Nat.below.{0}\n      (fun x =>\n        @Eq.{u_4 + 1} (M → M)\n          (@Nat.iterate.{u_4 + 1} M\n            (fun x =>\n              @HAdd.hAdd.{u_4, u_4, u_4} M M M\n                (@instHAdd.{u_4} M\n                  (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n                a x)\n            x)\n          fun x_1 =>\n          @HAdd.hAdd.{u_4, u_4, u_4} M M M\n            (@instHAdd.{u_4} M\n              (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n            (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n              (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst))) x a)\n            x_1)\n      x),\n  @Eq.{u_4 + 1} (M → M)\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, u_4, u_4} M M M\n          (@instHAdd.{u_4} M\n            (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n          a x)\n      x)\n    fun x_1 =>\n    @HAdd.hAdd.{u_4, u_4, u_4} M M M\n      (@instHAdd.{u_4} M\n        (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n      (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n        (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst))) x a)\n      x_1","statement_sha256":"745eaf475a728b901ad7037164a440e7a0bfc8058f79d5c66b0e9575784db5bd","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-5ecf6c1755f8f7cb36b13e3395c77586a14f12ff450b6d5663e04c85043d4d74","qualified_name":"add_left_iterate_apply","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6464,"column":83,"line":193},"start":{"byte":6383,"column":2,"line":193}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] {n : Nat} (a b : M),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, u_4, 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b)","statement_sha256":"d5f4682c917d6bdc8a53edfbd49548cc7f0b089b3abf411d9bfb15d7489ebf0c","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-4aa4bb540daf511940d5bf08813c5b94291af416ea2eca11fa63d2e5fffc14fe","qualified_name":"add_left_iterate_apply_self","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6798,"column":13,"line":200},"start":{"byte":6787,"column":2,"line":200}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] (a : M) (n : Nat),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, u_4, u_4} M M M\n          (@instHAdd.{u_4} M\n            (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n          a x)\n      n a)\n    (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n      (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst)))\n      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) n\n        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))\n      a)","statement_sha256":"54eac439a45fc5c1c605c9e377d7ddea6705488aedd51b99956911d9b53fd8f6","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-b774c958f5e965bf38699bd6c49ef16384e78afc670cbb6f6c17f72d6256b827","qualified_name":"add_left_iterate_apply_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":7091,"column":13,"line":208},"start":{"byte":7080,"column":2,"line":208}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] {n : Nat} (a : M),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, u_4, u_4} M M M\n          (@instHAdd.{u_4} M\n            (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n          a x)\n      n\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M\n          (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))))\n    (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n      (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst))) n a)","statement_sha256":"e16a5ccd04a041a75e8b8f982fa4feff5917c1a775d3a0a52561f9d6ab94156a","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-ddfd58d053634f3fd70fb3c812164f19f2e7971b16d6d950a76d8000500eee01","qualified_name":"add_left_surjective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19175,"column":13,"line":614},"start":{"byte":19164,"column":2,"line":614}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G),\n  @Function.Surjective.{u_3 + 1, u_3 + 1} G G fun x =>\n    @HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      a x","statement_sha256":"0490b9129627df2d81c040aadaade2a9aa4f2a21f3a09ea558db7b733c279ccc","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-77c7ecde54b0435f621afff74743abe6bef6c5c2dc26416c90a7815a87731460","qualified_name":"add_ne_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":8283,"column":13,"line":247},"start":{"byte":8272,"column":2,"line":247}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M]\n  [@IsLeftCancelAdd.{u_4} M\n      (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst)))]\n  {a b : M},\n  Iff\n    (@Ne.{u_4 + 1} M\n      (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M\n          (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n        a b)\n      a)\n    (@Ne.{u_4 + 1} M b\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M\n          (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))))","statement_sha256":"07a6bb46226aca0c0bd392204af6c9da1140f1216e96b915e3193995338d6f8e","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-149af20496688fb25ed7c91c55f5d3244ea95ccfb2141c5df7bb7c532a42fd84","qualified_name":"add_ne_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":8821,"column":13,"line":268},"start":{"byte":8810,"column":2,"line":268}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M]\n  [@IsRightCancelAdd.{u_4} M\n      (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst)))]\n  {a b : M},\n  Iff\n    (@Ne.{u_4 + 1} M\n      (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M\n          (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n        a b)\n      b)\n    (@Ne.{u_4 + 1} M a\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M\n          (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))))","statement_sha256":"1acbd51021c7165591e9c05159f88f658b43b879438518779cc248b110d8e3c2","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-6ad627d953d6ce30896d304ecfe2322bcd1dc67518c3ddc595b7e79f55db3438","qualified_name":"add_neg_add_add_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23880,"column":28,"line":746},"start":{"byte":23854,"column":2,"line":746}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          b))\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        b c))\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      a c)","statement_sha256":"63943dc0ec6f049fc444345f8d5ae1b08894f72e3d624e416fb81451a32999c1","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-38f05c8096c7fff3a3a7599cc18cb1654812267161532604cdef3c17c122cca1","qualified_name":"add_neg_eq_add_neg_iff_add_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":33948,"column":13,"line":1000},"start":{"byte":33937,"column":2,"line":1000}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c d : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        a\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G\n                (@SubtractionCommMonoid.toSubtractionMonoid.{u_3} G\n                  (@AddCommGroup.toDivisionAddCommMonoid.{u_3} G inst)))))\n          b))\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        c\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G\n                (@SubtractionCommMonoid.toSubtractionMonoid.{u_3} G\n                  (@AddCommGroup.toDivisionAddCommMonoid.{u_3} G inst)))))\n          d)))\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        a d)\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        c b))","statement_sha256":"c8819b8aba9680df88d72eca3599fbe206a92827fe60051af0d2a516a69e0b65","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-0d5f2042e0e0bf9ca968719e33e5c0abde5837e00c97339651bf750ef4d247cf","qualified_name":"add_neg_eq_iff_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21636,"column":13,"line":684},"start":{"byte":21625,"column":2,"line":684}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          b))\n      c)\n    (@Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        c b))","statement_sha256":"f5a6b71149ec7b2a65a47264d6e1c781b9eea7f4d1bed794c2b2d0f507ee5c41","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-d144e9d19b2860264f600746d2d3bf0118a2c03767a83d779e7acdb394f8a861","qualified_name":"add_neg_eq_of_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19744,"column":13,"line":631},"start":{"byte":19733,"column":2,"line":631}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        c b) →\n    @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          b))\n      c","statement_sha256":"7a1ccd1224683f196198bdf994989a04f7665086fc2fec5616b007c0bfc227cc","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-cfb057144ed4a94ee50a2e872094bd08b5fcaf785e65927ea196e97282c0b786","qualified_name":"add_neg_eq_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21813,"column":13,"line":688},"start":{"byte":21802,"column":2,"line":688}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          b))\n      (@OfNat.ofNat.{u_3} G (nat_lit 0)\n        (@Zero.toOfNat0.{u_3} G\n          (@NegZeroClass.toZero.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G a b)","statement_sha256":"dd158f02c6682c2a1196dba5d961f4f58a53813ffbb28bf4bbce8f721f905816","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-85142ceb493d497ebff9a8ae8fb360b51ba33e16622aa3dc509de55c56dfb1c1","qualified_name":"add_one_zsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":26158,"column":27,"line":808},"start":{"byte":26133,"column":2,"line":808}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G) (n : Int),\n  @Eq.{u_3 + 1} G\n    (@HSMul.hSMul.{0, u_3, u_3} Int G G\n      (@instHSMul.{0, u_3} Int G\n        (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n      (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd) n\n        (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))\n      a)\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        n a)\n      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b)","statement_sha256":"c94bed1021547717eec32e4fd7014e502ae652edb9b4babd42e53e8aa44a3898","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-c7100567b8e0590bb5fdafa5ba50364e84d7ec689859c434c6f0f960f7275674","qualified_name":"add_right_iterate","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6159,"column":28,"line":187},"start":{"byte":6133,"column":2,"line":187}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] (a : M) (n : Nat),\n  @Eq.{u_4 + 1} (M → M)\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, u_4, 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a)","statement_sha256":"3284c578059eb1dc722ea8cab1e6731a43f737e6a8297492ad94ad4556291403","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-59c1a586343611d1e7cb2b73f1830d5b22716c2f21f1c9380bb929a3353c7b97","qualified_name":"add_right_iterate._f","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6159,"column":28,"line":187},"start":{"byte":6133,"column":2,"line":187}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] (a : M) (x : Nat)\n  (f :\n    @Nat.below.{0}\n      (fun x =>\n        @Eq.{u_4 + 1} (M → M)\n          (@Nat.iterate.{u_4 + 1} M\n            (fun x =>\n              @HAdd.hAdd.{u_4, u_4, u_4} M M M\n                (@instHAdd.{u_4} M\n                  (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n                x a)\n            x)\n          fun x_1 =>\n          @HAdd.hAdd.{u_4, u_4, u_4} M M M\n            (@instHAdd.{u_4} M\n              (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n            x_1\n            (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n              (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst))) x a))\n      x),\n  @Eq.{u_4 + 1} (M → M)\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, u_4, u_4} M M M\n          (@instHAdd.{u_4} M\n            (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n          x a)\n      x)\n    fun x_1 =>\n    @HAdd.hAdd.{u_4, u_4, u_4} M M M\n      (@instHAdd.{u_4} M\n        (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n      x_1\n      (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n        (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst))) x a)","statement_sha256":"b4671e931ff85349f059e84fcd0ff0f043dfaa9d9e53c99979f4e0034be566a3","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-8183483996228c16bf2840e3c6521c6f02282cc60e92cbf6398dad561e55e396","qualified_name":"add_right_iterate_apply","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6701,"column":84,"line":197},"start":{"byte":6619,"column":2,"line":197}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] {n : Nat} (a b : M),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, u_4, u_4} M M M\n          (@instHAdd.{u_4} M\n            (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n          x a)\n      n b)\n    (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n      (@instHAdd.{u_4} M\n        (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n      b\n      (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n        (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst))) n a))","statement_sha256":"482bb08c9c83de6096c62b3cb74b83f4258df7648a2b579ebc72384ed0d26653","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-2afe95944c1f234f4f77ed220b8cae7bca49ff352a3f2d16b7a6b9db1b8d67a2","qualified_name":"add_right_iterate_apply_self","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6943,"column":13,"line":204},"start":{"byte":6932,"column":2,"line":204}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] (a : M) (n : Nat),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, 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a)","statement_sha256":"634b2c6055cab65c4cdf9f4e58671c0cdebf68185a417766bf4ae2fa3700d619","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-ff16f3fc2449f06be9d38e32d7cd581ebc22ed405cfa5d403ff1cfec253b1319","qualified_name":"add_right_iterate_apply_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":7184,"column":13,"line":211},"start":{"byte":7173,"column":2,"line":211}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] {n : Nat} (a : M),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HAdd.hAdd.{u_4, u_4, u_4} M M M\n          (@instHAdd.{u_4} M\n            (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n          x a)\n      n\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M\n          (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))))\n    (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n      (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst))) n a)","statement_sha256":"2a0f19c2682bc716a27525ca4718dbaa561cd471204564ece4c6b917f93d208e","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-44ca8a4919cb3477e7ba7ab998715477f2ae6f0f4bf87c29d039c61a54419c1d","qualified_name":"add_right_surjective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19306,"column":13,"line":618},"start":{"byte":19295,"column":2,"line":618}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G),\n  @Function.Surjective.{u_3 + 1, u_3 + 1} G G fun x =>\n    @HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      x a","statement_sha256":"f9d14d0d0516661749a09c1136156a74e8a65728563bfa140cf5363757638b81","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-da41df0408f69fecc1c65755be980936cb93fad1de6d53f1cb06ae2cd8d1b103","qualified_name":"add_rotate","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":4043,"column":13,"line":132},"start":{"byte":4032,"column":2,"line":132}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommSemigroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G (@AddCommMagma.toAdd.{u_3} G 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a)","statement_sha256":"1365477e5bef936e08af9c79fcdf32cbc23c6b07b3e458b9ac3ebc556061e835","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-4e1c782e03bf08cc3ee116df77dfaee09d3b2c74ffbdbd20ab3c59475a4f05f6","qualified_name":"add_rotate'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":4158,"column":13,"line":136},"start":{"byte":4147,"column":2,"line":136}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommSemigroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G (@AddCommMagma.toAdd.{u_3} 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c)","statement_sha256":"8960cc35406afc53baad8bb84e68e03b41d7dd1c176bdc9b8c3ec5e1d5746d8f","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-73309cb564c13ff00706de3ae287e84cae9a7a68c5a3842521c33ab4e5e824db","qualified_name":"add_sub_add_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18380,"column":13,"line":590},"start":{"byte":18369,"column":2,"line":590}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b c d : α),\n  @Eq.{u_1 + 1} α\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        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(@instHSub.{u_1} α\n          (@SubNegMonoid.toSub.{u_1} α\n            (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a c))","statement_sha256":"8e066d996f9755d7abcee2bc4398ece3b1747afefd2915a6be93b92bb47ec032","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-b0d9dccb30f73bdc79cb343fb773e43d5b0d2fb202abf2eb2a3f8637682d7dc3","qualified_name":"add_sub_one_nsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":5107,"column":31,"line":163},"start":{"byte":5078,"column":2,"line":163}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] {n : Nat},\n  @Ne.{1} Nat n (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n    ∀ (a : M),\n      @Eq.{u_4 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a)","statement_sha256":"e69910bd15fa840472d5b513c946f49fd76dfdf9537b950b884bac5392d40fca","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-85ac837095553d04c7240ca3fc7bfc72a6e19e5ef59205bf7154fc08a4102f9b","qualified_name":"add_sub_right_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17568,"column":13,"line":560},"start":{"byte":17557,"column":2,"line":560}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        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(@instHSub.{u_1} α\n          (@SubNegMonoid.toSub.{u_1} α\n            (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a c)\n      b)","statement_sha256":"a98ec6d5f60f0f6b9279583269a5c783363a280ddaa7eafe07e859a0aa4f2c2f","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-17f6f5ad2d3fa3cde5ec183effc6bd07079addc38bb3842f1a99150418b0212e","qualified_name":"add_sub_sub_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":33449,"column":28,"line":987},"start":{"byte":33423,"column":2,"line":987}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G\n        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inst)))))))\n      b c)","statement_sha256":"a80b86709ec731b0fa1c01fa7f26e57e303b4ed6e8a9adc9bb3620b070589572","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-f898ba70fd417e1b19c54126123549a430ff89d5a92e6b476bc21b79e9379f6e","qualified_name":"add_zero_eq_id","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3408,"column":13,"line":106},"start":{"byte":3397,"column":2,"line":106}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddZeroClass.{u_4} M],\n  @Eq.{u_4 + 1} (M → M)\n    (fun x =>\n      @HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst))) x\n        (@OfNat.ofNat.{u_4} M (nat_lit 0)\n          (@Zero.toOfNat0.{u_4} M (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst)))))\n    (@id.{u_4 + 1} M)","statement_sha256":"b65ea1252c22b39a3e2eecb1f48b57c14b907865d1ff9fad07f4d3c59b91337d","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-d5e61b41c6945b28fc7b5b20994cefdf08fa7e8a46d61c3ad189fbf87b2a2236","qualified_name":"add_zero_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":10407,"column":13,"line":332},"start":{"byte":10396,"column":2,"line":332}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : SubNegMonoid.{u_3} G] (x y : G),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        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y)","statement_sha256":"23b5f16c4bca8cf709e9edb9faa9effbc46dc7f0b12dc3097bf1707ae8b5583f","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-835c6a7d5daf111b9dda42d5c7d5aab0158cb9f0bed28551db56d39fffa38bfe","qualified_name":"add_zsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":26806,"column":23,"line":823},"start":{"byte":26785,"column":2,"line":823}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G) (m n : Int),\n  @Eq.{u_3 + 1} G\n    (@HSMul.hSMul.{0, u_3, u_3} Int G G\n      (@instHSMul.{0, u_3} Int G\n        (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n      (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd) m n) a)\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        m a)\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        n a))","statement_sha256":"3d59b26aeece86ba340bcf15cc412d01908b81a28e096326e97fb6025ba5e4fc","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-c4bacdacee23be4ffae3b68dd84b75971410968fcb9d24ee0416c9789daed93a","qualified_name":"add_zsmul_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":28292,"column":13,"line":858},"start":{"byte":28281,"column":2,"line":858}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a b : G) (n : Int),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        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(@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      a\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        n\n        (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n          (@instHAdd.{u_3} G\n            (@AddZero.toAdd.{u_3} G\n              (@AddZeroClass.toAddZero.{u_3} G\n                (@AddMonoid.toAddZeroClass.{u_3} G\n                  (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n          b a)))","statement_sha256":"77159f0d3f3b84fde39f1707d3f748f3636f6a3c6843a410b24eb68b13c713b5","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-70d687df75d432e2fecd1687b2d1dcdf1ecee8692ecadfe94bee1e7fb1cca54d","qualified_name":"add_zsmul_self","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":27242,"column":28,"line":833},"start":{"byte":27216,"column":2,"line":833}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G) (n : Int),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      a\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        n a))\n    (@HSMul.hSMul.{0, u_3, u_3} Int G G\n      (@instHSMul.{0, u_3} Int G\n        (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n      (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd) n\n        (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))\n      a)","statement_sha256":"0bc242a3b5f78e679b5c5177c88ffecb572496ef3cda66e875b6d2b44a18cc97","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-75eb069bc7c8b652d113e270aab64528a548075320cae23678b4237697b2fcc8","qualified_name":"additive_of_symm_of_total","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":34761,"column":39,"line":1022},"start":{"byte":34724,"column":2,"line":1022}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : AddMonoid.{u_2} β] (p r : α → α → Prop) [@Std.Total.{u_1 + 1} α r]\n  (f : α → α → β) [@Std.Symm.{u_1 + 1} α p],\n  (∀ {a b : α},\n      p a b →\n        @Eq.{u_2 + 1} β\n          (@HAdd.hAdd.{u_2, u_2, u_2} β β β\n            (@instHAdd.{u_2} β\n              (@AddZero.toAdd.{u_2} β (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst))))\n            (f a b) (f b a))\n          (@OfNat.ofNat.{u_2} β (nat_lit 0)\n            (@Zero.toOfNat0.{u_2} β\n              (@AddZero.toZero.{u_2} β (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst)))))) →\n    (∀ {a b c : α},\n        r a b →\n          r b c →\n            p a b →\n              p b c →\n                p a c →\n                  @Eq.{u_2 + 1} β (f a c)\n                    (@HAdd.hAdd.{u_2, u_2, u_2} β β β\n                      (@instHAdd.{u_2} β\n                        (@AddZero.toAdd.{u_2} β\n                          (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst))))\n                      (f a b) (f b c))) →\n      ∀ {a b c : α},\n        p a b →\n          p b c →\n            p a c →\n              @Eq.{u_2 + 1} β (f a c)\n                (@HAdd.hAdd.{u_2, u_2, u_2} β β β\n                  (@instHAdd.{u_2} β\n                    (@AddZero.toAdd.{u_2} β (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst))))\n                  (f a b) (f b c))","statement_sha256":"ed277b3f9b7c5dfbcdbb6f7218bb1b927da3607bdf7b4c9543dd57981ec093b7","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-24e1bfd4317ce2b94508375710acf37afbd9c1bb51c4d7342c56d5ce329ab41c","qualified_name":"additive_of_symmetric_of_total","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":35741,"column":65,"line":1040},"start":{"byte":35638,"column":0,"line":1039}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : AddMonoid.{u_2} β] (p r : α → α → Prop) [@Std.Total.{u_1 + 1} α r]\n  (f : α → α → β) [@Std.Symm.{u_1 + 1} α p],\n  (∀ {a b : α},\n      p a b →\n        @Eq.{u_2 + 1} β\n          (@HAdd.hAdd.{u_2, u_2, u_2} β β β\n            (@instHAdd.{u_2} β\n              (@AddZero.toAdd.{u_2} β (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst))))\n            (f a b) (f b a))\n          (@OfNat.ofNat.{u_2} β (nat_lit 0)\n            (@Zero.toOfNat0.{u_2} β\n              (@AddZero.toZero.{u_2} β (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst)))))) →\n    (∀ {a b c : α},\n        r a b →\n          r b c →\n            p a b →\n              p b c →\n                p a c →\n                  @Eq.{u_2 + 1} β (f a c)\n                    (@HAdd.hAdd.{u_2, u_2, u_2} β β β\n                      (@instHAdd.{u_2} β\n                        (@AddZero.toAdd.{u_2} β\n                          (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst))))\n                      (f a b) (f b c))) →\n      ∀ {a b c : α},\n        p a b →\n          p b c →\n            p a c →\n              @Eq.{u_2 + 1} β (f a c)\n                (@HAdd.hAdd.{u_2, u_2, u_2} β β β\n                  (@instHAdd.{u_2} β\n                    (@AddZero.toAdd.{u_2} β (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst))))\n                  (f a b) (f b c))","statement_sha256":"ed277b3f9b7c5dfbcdbb6f7218bb1b927da3607bdf7b4c9543dd57981ec093b7","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-5dd93dd2b975236997aa81d66af9349f0ef2563342f74aa74040dc6a03ef96cb","qualified_name":"additive_of_total","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":36629,"column":90,"line":1051},"start":{"byte":36255,"column":2,"line":1048}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : AddMonoid.{u_2} β] (r : α → α → Prop) [@Std.Total.{u_1 + 1} α r] (f : α → α → β)\n  (p : α → Prop),\n  (∀ {a b : α},\n      p a →\n        p b →\n          @Eq.{u_2 + 1} β\n            (@HAdd.hAdd.{u_2, u_2, u_2} β β β\n              (@instHAdd.{u_2} β\n                (@AddZero.toAdd.{u_2} β (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst))))\n              (f a b) (f b a))\n            (@OfNat.ofNat.{u_2} β (nat_lit 0)\n              (@Zero.toOfNat0.{u_2} β\n                (@AddZero.toZero.{u_2} β\n                  (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst)))))) →\n    (∀ {a b c : α},\n        r a b →\n          r b c →\n            p a →\n              p b →\n                p c →\n                  @Eq.{u_2 + 1} β (f a c)\n                    (@HAdd.hAdd.{u_2, u_2, u_2} β β β\n                      (@instHAdd.{u_2} β\n                        (@AddZero.toAdd.{u_2} β\n                          (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst))))\n                      (f a b) (f b c))) →\n      ∀ {a b c : α},\n        p a →\n          p b →\n            p c →\n              @Eq.{u_2 + 1} β (f a c)\n                (@HAdd.hAdd.{u_2, u_2, u_2} β β β\n                  (@instHAdd.{u_2} β\n                    (@AddZero.toAdd.{u_2} β (@AddZeroClass.toAddZero.{u_2} β (@AddMonoid.toAddZeroClass.{u_2} β inst))))\n                  (f a b) (f b c))","statement_sha256":"d5112bdd7e9b5ce4776534a6e044b1e5bc5f246e43c595cd30a7efbda36be3e1","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-7befd29495643b14103b95da26e82e0c312d838a747f650e1ca59bcea6b311bd","qualified_name":"boole_nsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":4416,"column":25,"line":147},"start":{"byte":4393,"column":2,"line":147}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] (P : Prop) [inst_1 : Decidable P] (a : M),\n  @Eq.{u_4 + 1} M\n    (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n      (@instHSMul.{0, u_4} Nat M (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M inst)))\n      (@ite.{1} Nat P inst_1 (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))\n        (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))\n      a)\n    (@ite.{u_4 + 1} M P inst_1 a\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M\n          (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))))","statement_sha256":"4aa2cec3d1c5a7ad6b64e8b74b861b999daab2dc1871b8abd9c16cc0c9a558ff","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-53a1e8faaadb8fff7ef3a6fa83aec397ee9741b64316fb16c6c43e902269e1df","qualified_name":"comp_add_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":2174,"column":50,"line":63},"start":{"byte":2041,"column":2,"line":62}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : AddSemigroup.{u_1} α] (x y : α),\n  @Eq.{u_1 + 1} (α → α)\n    (@Function.comp.{u_1 + 1, u_1 + 1, u_1 + 1} α α α\n      (fun x_1 => @HAdd.hAdd.{u_1, u_1, u_1} α α α (@instHAdd.{u_1} α (@AddSemigroup.toAdd.{u_1} α inst)) x x_1)\n      fun x => @HAdd.hAdd.{u_1, u_1, u_1} α α α (@instHAdd.{u_1} α (@AddSemigroup.toAdd.{u_1} α inst)) y x)\n    fun x_1 =>\n    @HAdd.hAdd.{u_1, u_1, u_1} α α α (@instHAdd.{u_1} α (@AddSemigroup.toAdd.{u_1} α inst))\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α (@instHAdd.{u_1} α (@AddSemigroup.toAdd.{u_1} α inst)) x y) x_1","statement_sha256":"46fbcf7f408d2491139532894647a913f4e972254bddb5a55a3a224ab8aa0c78","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-2b6506d8b574dca96d78a5fd59175aa730bb8617f8992fc276c175f4dd20f7f2","qualified_name":"comp_add_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":2537,"column":51,"line":72},"start":{"byte":2403,"column":2,"line":71}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : AddSemigroup.{u_1} α] (x y : α),\n  @Eq.{u_1 + 1} (α → α)\n    (@Function.comp.{u_1 + 1, u_1 + 1, u_1 + 1} α α α\n      (fun x_1 => @HAdd.hAdd.{u_1, u_1, u_1} α α α (@instHAdd.{u_1} α (@AddSemigroup.toAdd.{u_1} α inst)) x_1 x)\n      fun x => @HAdd.hAdd.{u_1, u_1, u_1} α α α (@instHAdd.{u_1} α (@AddSemigroup.toAdd.{u_1} α inst)) x y)\n    fun x_1 =>\n    @HAdd.hAdd.{u_1, u_1, u_1} α α α (@instHAdd.{u_1} α (@AddSemigroup.toAdd.{u_1} α inst)) x_1\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α (@instHAdd.{u_1} α (@AddSemigroup.toAdd.{u_1} α inst)) y x)","statement_sha256":"0ae67594670997583c8145477c46c63515c27a97a2cf3393c905f64201cb885d","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-c8a7c13ba06386056b0efe8959c1114fddb3b6a509af69e60161a129af118ae1","qualified_name":"comp_mul_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":2280,"column":18,"line":66},"start":{"byte":1921,"column":0,"line":59}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Semigroup.{u_1} α] (x y : α),\n  @Eq.{u_1 + 1} (α → α)\n    (@Function.comp.{u_1 + 1, u_1 + 1, u_1 + 1} α α α\n      (fun x_1 => @HMul.hMul.{u_1, u_1, u_1} α α α (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α inst)) x x_1) fun x =>\n      @HMul.hMul.{u_1, u_1, u_1} α α α (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α inst)) y x)\n    fun x_1 =>\n    @HMul.hMul.{u_1, u_1, u_1} α α α (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α inst))\n      (@HMul.hMul.{u_1, u_1, u_1} α α α (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α inst)) x y) x_1","statement_sha256":"100fcdad2b57e9e7d3048e4f8b872e3b278429a9eb5d7a555cd2935a0d6a42d2","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-207a9bb279ad0473fc5e634f917f28d77e2826cdad2727b0bbcf3d36368ae74b","qualified_name":"comp_mul_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":2646,"column":18,"line":75},"start":{"byte":2282,"column":0,"line":68}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Semigroup.{u_1} α] (x y : α),\n  @Eq.{u_1 + 1} (α → α)\n    (@Function.comp.{u_1 + 1, u_1 + 1, u_1 + 1} α α α\n      (fun x_1 => @HMul.hMul.{u_1, u_1, u_1} α α α (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α inst)) x_1 x) fun x =>\n      @HMul.hMul.{u_1, u_1, u_1} α α α (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α inst)) x y)\n    fun x_1 =>\n    @HMul.hMul.{u_1, u_1, u_1} α α α (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α inst)) x_1\n      (@HMul.hMul.{u_1, u_1, u_1} α α α (@instHMul.{u_1} α (@Semigroup.toMul.{u_1} α inst)) y x)","statement_sha256":"0e2728b89410320046a5c425d2a76e644895016bd2531c57573b5f9c614e851c","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-a97f5cffd9d57629dde3e72457f7a680929a78972ab6a79fa9fe2994c77f9c04","qualified_name":"conj_eq_one_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":22141,"column":34,"line":696},"start":{"byte":22016,"column":0,"line":694}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        (@HMul.hMul.{u_3, u_3, u_3} G G G\n          (@instHMul.{u_3} G\n            (@MulOne.toMul.{u_3} G\n              (@MulOneClass.toMulOne.{u_3} G\n                (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n          a b)\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          a))\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G b\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))","statement_sha256":"6968dab4cfc4b44ee5dae798488fd4c26658ddb65abdb20483fa68b850bcb4d0","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-972f027c4ae1ead49b60465969c40a03053f4a10c4a2715759dd411ce5c22fda","qualified_name":"const_div_involutive","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":34595,"column":29,"line":1014},"start":{"byte":34467,"column":0,"line":1012}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a : G),\n  @Function.Involutive.{u_3 + 1} G fun x =>\n    @HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      a x","statement_sha256":"59fee607facc934702b64826403b7df91e040a5e6299d0037f9b5ae5971367c4","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-f893ddd7c7a2db5cfb7f587bf3994902ca8e9d47c6e1bfbdfd1162fc01bab2c5","qualified_name":"const_sub_involutive","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":34495,"column":28,"line":1012},"start":{"byte":34469,"column":2,"line":1012}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] (a : G),\n  @Function.Involutive.{u_3 + 1} G fun x =>\n    @HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G\n        (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n      a x","statement_sha256":"fa188d4bdc20348e442278918c72e341f8566a2d5dc2ebfc29d82e79547cf6e5","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-f2e158f68fb81fbd15bbb81f3ccafd66d74a1375de6fd955ad40830589604079","qualified_name":"dite_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":1373,"column":97,"line":41},"start":{"byte":1139,"column":0,"line":39}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : Pow.{u_1, u_2} α β] (p : Prop) [inst_1 : Decidable p] (a : p → α)\n  (b : Not p → α) (c : β),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, u_2, u_1} α β α (@instHPow.{u_1, u_2} α β inst)\n      (@dite.{u_1 + 1} α p inst_1 (fun h => a h) fun h => b h) c)\n    (@dite.{u_1 + 1} α p inst_1 (fun h => @HPow.hPow.{u_1, u_2, u_1} α β α (@instHPow.{u_1, u_2} α β inst) (a h) c)\n      fun h => @HPow.hPow.{u_1, u_2, u_1} α β α (@instHPow.{u_1, u_2} α β inst) (b h) c)","statement_sha256":"73c44a5695b7c8dcb58113e1f116eee2a4bcb25a2420bf0c2d6942077bb47173","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-94c23f8192c5bfe4ab856b5414093f16a55af3507e0bbced1c1b020200d02657","qualified_name":"dite_smul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":954,"column":51,"line":35},"start":{"byte":905,"column":2,"line":35}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul.{u_2, u_1} β α] (p : Prop) [inst_1 : Decidable p] (a : α) (b : p → β)\n  (c : Not p → β),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{u_2, 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a)","statement_sha256":"cb466cb0db312d5116fe9e84ae23df5adfbfecf133dbc7902aa276ecc51f18b6","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-15932201c4647ebe417448b8b81bfd0ec0fcd76195b46feccce3b4767b8492d3","qualified_name":"dite_vadd","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":954,"column":51,"line":35},"start":{"byte":905,"column":2,"line":35}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : VAdd.{u_2, u_1} β α] (p : Prop) [inst_1 : Decidable p] (a : α) (b : p → β)\n  (c : Not p → β),\n  @Eq.{u_1 + 1} α\n    (@HVAdd.hVAdd.{u_2, u_1, u_1} β α α (@instHVAdd.{u_2, u_1} β α inst)\n      (@dite.{u_2 + 1} β p inst_1 (fun h => b h) fun h => c h) a)\n    (@dite.{u_1 + 1} α p inst_1 (fun h => @HVAdd.hVAdd.{u_2, u_1, u_1} β α α (@instHVAdd.{u_2, u_1} β α inst) (b h) a)\n      fun h => @HVAdd.hVAdd.{u_2, u_1, u_1} β α α (@instHVAdd.{u_2, u_1} β α inst) (c h) a)","statement_sha256":"db58bb8117c0398b94e09ead36d152db1ffe7913cd85e69ff85d673fe3bc8b74","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-d5e2697badb733e972fcbcf88ab2e24347bf4bbd97c1cbba097702fc907392de","qualified_name":"div_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17403,"column":52,"line":552},"start":{"byte":17336,"column":0,"line":551}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b)\n      c)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      a\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        b c))","statement_sha256":"07db351cf867ad9ac9bed1ea1b2a9f4a817c2078cf340c1e429253c660a7f6b0","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-2cd2f402686f39a9b3872f29d71ab3109f336c655556223bfc3262608afa567f","qualified_name":"div_div_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31819,"column":19,"line":944},"start":{"byte":31716,"column":0,"line":942}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      a\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        a b))\n    b","statement_sha256":"27f2cc549e7fcbfaae76dffbfb310a766adeba3250e7fdd8e3c0207f6424e07b","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-20ded4afbcb73c7af8cedce5ad2bd9993d86221d5841b507a3fc80e1349d99da","qualified_name":"div_div_cancel_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31920,"column":69,"line":947},"start":{"byte":31821,"column":0,"line":946}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        a b)\n      a)\n    (@Inv.inv.{u_3} G\n      (@InvOneClass.toInv.{u_3} G\n        (@DivInvOneMonoid.toInvOneClass.{u_3} G\n          (@DivisionMonoid.toDivInvOneMonoid.{u_3} G\n            (@DivisionCommMonoid.toDivisionMonoid.{u_3} G (@CommGroup.toDivisionCommMonoid.{u_3} G inst)))))\n      b)","statement_sha256":"e81db753f7ff5b6e4183b2be9fa7d6e3ea4f12d09bede0e2a5141fbb5f3dfa19","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-2ed0ade69d76d7636d87cd583fd0afa61339d7c26e09c0f47726692be4643dc9","qualified_name":"div_div_div_cancel_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":33737,"column":68,"line":993},"start":{"byte":33563,"column":0,"line":991}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        c a)\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        c b))\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      b a)","statement_sha256":"104ab4157e1abc1c5203de8df75bec89457063b976588275f5c5b9b9724b7860","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-9beb6262ce5beafc2d5e17703af91ae9824629019d4d0641bca48a7dcf17c5d1","qualified_name":"div_div_div_cancel_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24190,"column":58,"line":752},"start":{"byte":24025,"column":0,"line":750}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst)))\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a c)\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) b c))\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)","statement_sha256":"20d3a60652449ed339558bc44e4e9c4cbb23acde5ed8440b6fc58edcccb0b690","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-add0cd754b5c638b16bd73c747286651bee237cd071b41721e1e858ad5b0a948","qualified_name":"div_div_div_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18277,"column":71,"line":585},"start":{"byte":18191,"column":0,"line":584}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c d : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b)\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        c d))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a c)\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        b d))","statement_sha256":"d3b842fe80bb926d0691e4d04a631fe92fbadfb94df640027c0f15b407d3ff30","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-68df330e0c124c21b38390fcc3b4913ffa092a1103405d00ea50e4b46f520533","qualified_name":"div_div_div_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18189,"column":69,"line":582},"start":{"byte":18105,"column":0,"line":581}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c d : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b)\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        c d))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        a d)\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        b c))","statement_sha256":"d3c31b7bb89874e35107bad287430116bec5cbf6c312a7480523b5b30ccb130d","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-12a38a8e608a3c55df784a90a41a2fc89e07de24adcd7eef8c5ad0a545082bc9","qualified_name":"div_div_eq_mul_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16170,"column":63,"line":505},"start":{"byte":16092,"column":0,"line":504}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α 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b)","statement_sha256":"faabb97646cc40a22a6b4a99d6d9a60643053a9e11b7e441b3eafaa7e9783763","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-08e95bedff5844407107d42196b2375cd1d9b50b004b6535448d3526bb5e424c","qualified_name":"div_div_self'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31604,"column":60,"line":937},"start":{"byte":31516,"column":0,"line":936}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        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b)","statement_sha256":"f644c2910db1364c0ffaf49baa6885bd4cf47bbc0aa1e6ffeb361d817f418b0f","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-eea527d123fd961af8a1de5e71e191be43714a584f0fd68709bf7cfcdf76cf8c","qualified_name":"div_eq_iff_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24867,"column":100,"line":773},"start":{"byte":24752,"column":0,"line":772}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G 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inst)))))))","statement_sha256":"f788b205b35d1d2fcef5e7ab2b08b12e01a4acf5cd175f90ba2f06aac8db12fd","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-b72702d6b3ccf2fca30c5c5ab48c93ab5ad0b08564766821d6318e04208f3c19","qualified_name":"div_eq_mul_one_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":10846,"column":93,"line":344},"start":{"byte":10738,"column":0,"line":343}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : DivInvMonoid.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G (@HDiv.hDiv.{u_3, u_3, u_3} G G G (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} 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b))","statement_sha256":"f7b0a1aef751dfad7b78ab66b953a699e5e8c0cc720d312d80aba9afb799a985","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-b81120f57b48801a05dd79b59382c717363fc5d8496ce8a56a1b803a3547c908","qualified_name":"div_eq_of_eq_mul'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31067,"column":55,"line":920},"start":{"byte":30925,"column":0,"line":918}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        b c) →\n    @Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        a b)\n      c","statement_sha256":"50a02ab5ef801891e9bee61d74fb19079c4e659691c538ad441e5180dd43c158","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-fa1a23133d1ab146542ca994433c67001b32f78bda2ab505f3f754973edd263b","qualified_name":"div_eq_of_eq_mul''","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23111,"column":69,"line":721},"start":{"byte":23010,"column":0,"line":720}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        c b) →\n    @Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)\n      c","statement_sha256":"2bb2575949f8f01b2eda5abfae6a46471784f4c55cbaa246a5dd98446175ea9e","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-77f9d45b22ad069fe9e2f79500cc39834224a0ddc5085fa634f9c1793e4673f8","qualified_name":"div_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24307,"column":56,"line":756},"start":{"byte":24192,"column":0,"line":754}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G a b)","statement_sha256":"fe9bf41cc94ab435d2ae8dad1afd2898c374ff35989a3d7cb0f780b0a83befc7","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-c7cc29c9293c487131abefe0c0598779b76a757d6f12932fcf7f1465ccf9acba","qualified_name":"div_eq_one_of_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24354,"column":45,"line":758},"start":{"byte":24309,"column":0,"line":758}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  @Eq.{u_3 + 1} G a b →\n    @Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))))","statement_sha256":"80080184994c303e0a2a626403ee963aaa72eb7f90418d1406e914a41e35729f","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-4971756ef993a93842dbe9812b052fbaee89444825deefe8b6ac143f4993f294","qualified_name":"div_eq_self","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24614,"column":92,"line":767},"start":{"byte":24492,"column":0,"line":766}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)\n      a)\n    (@Eq.{u_3 + 1} G b\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))","statement_sha256":"81fd0588b7fa9b578b8495262638bf66401fef35ab6ad5451c2aff7642bfcdef","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-6edf36631c61df195b37955f89e83d5bd290cc481cd204a95f6e8fb9d2d9267d","qualified_name":"div_inv_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16256,"column":54,"line":508},"start":{"byte":16172,"column":0,"line":507}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))) a\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n        b))\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))\n      a b)","statement_sha256":"d904cfd803f382689956e1fd05dec48cd0c7fcad6397afdc459ef6e708e0d537","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-c05dcd3cffe4bff54fc6bf28277e4010e52abc60d2431daa336d8a3db3988aea","qualified_name":"div_left_inj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23539,"column":22,"line":736},"start":{"byte":23396,"column":0,"line":733}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) b a)\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) c a))\n    (@Eq.{u_3 + 1} G b c)","statement_sha256":"3ae01883868dca933ba633c69f2a646fbe143a1fcd486a96926488b53c0a5e47","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-198da85a3c698d5f5a64c61d14e6fe4dc89dd70975479132e77255c689762a8f","qualified_name":"div_left_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":22393,"column":50,"line":702},"start":{"byte":22143,"column":0,"line":698}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {b : G},\n  @Function.Injective.{u_3 + 1, u_3 + 1} G G fun a =>\n    @HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b","statement_sha256":"bc8e82bea7dc6ccbcb46d51ac770b6a9682b476c2feaae68f5ee8b2b67918db2","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-732ad0b37ba01775c7ac42e31841f78c568e30ff30529219a5c890740ae7da97","qualified_name":"div_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17472,"column":52,"line":555},"start":{"byte":17405,"column":0,"line":554}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b)\n      c)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      a\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        b c))","statement_sha256":"19c84a7f2c8fa36e5820ee8f750d52e876aea0db75c6984dde5c3a0cc04192ce","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-254586f7369f90b7dd9c27fe130e009317a3127b45db334ef211b85fd9d240d4","qualified_name":"div_mul_cancel_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":32674,"column":104,"line":966},"start":{"byte":32540,"column":0,"line":965}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        a b))\n    (@Inv.inv.{u_3} G\n      (@InvOneClass.toInv.{u_3} G\n        (@DivInvOneMonoid.toInvOneClass.{u_3} G\n          (@DivisionMonoid.toDivInvOneMonoid.{u_3} G\n            (@DivisionCommMonoid.toDivisionMonoid.{u_3} G (@CommGroup.toDivisionCommMonoid.{u_3} G inst)))))\n      b)","statement_sha256":"88510cdbecba447b15c3d3f102080c51e3659468e536e8c395fb83df29e2e323","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-79f0509368bd45fc119848d9814c99f501bf6af0c076c8db84e0181986d8fe97","qualified_name":"div_mul_cancel_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":22729,"column":104,"line":711},"start":{"byte":22595,"column":0,"line":710}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b a))\n    (@Inv.inv.{u_3} G\n      (@InvOneClass.toInv.{u_3} G\n        (@DivInvOneMonoid.toInvOneClass.{u_3} G\n          (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n      b)","statement_sha256":"1283b21c3c869727db78c7a6f53e958c1a5c212d1211ebb42137df074ad05f42","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-2b6dfd3221f9eda3d10f6efb1907c576d463fc9131b8bcd2f703d729cd73bc16","qualified_name":"div_mul_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18009,"column":55,"line":576},"start":{"byte":17939,"column":0,"line":575}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        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G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) b c))\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a c)","statement_sha256":"e0cdd9aaa579eb79b3155982a6ab0fa00f6a943a73aee7dceda016688d3a23e3","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-46114f72c8df2b4915fc1e3fa1628035f920f846e9956463ef1edb05964b784d","qualified_name":"div_mul_div_cancel'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":33419,"column":41,"line":985},"start":{"byte":33276,"column":0,"line":983}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        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b)","statement_sha256":"4d9eb5d85f102a3419a1c372e857a907bf64b995b42bafbc9df240d0c4bffb1a","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-3f904f4fb645891e76e96a326f2ec957bcce2a45d22e86c2e22b1acec0068d18","qualified_name":"div_mul_div_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18365,"column":71,"line":588},"start":{"byte":18279,"column":0,"line":587}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c d : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        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(@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        a c)\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        b d))","statement_sha256":"bebf150bc645501833f84578edc30a5629f844a533d73b904e363bc8b2279978","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-fc68009b8bdfe2de6139144c67dff1441e95e99e426ee01aea821b539d670994","qualified_name":"div_mul_eq_div_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17711,"column":63,"line":564},"start":{"byte":17633,"column":0,"line":563}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        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c)","statement_sha256":"ebde53ba0bf72bc59c9ece21c224da62b697b5822410a064bfca7e351bf8da2d","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-2126c06199cfeba1fe1f4c528f4051ca26afe9c293a4aef18811ff48a4c779ac","qualified_name":"div_mul_eq_div_div_swap","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16389,"column":52,"line":512},"start":{"byte":16258,"column":0,"line":510}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))) a\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))\n        b c))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))) a c)\n      b)","statement_sha256":"f899c519f46d8b7b3cc29442706a3704d95ebbedc43d52b224954426ba82bc74","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-f2c96babba27da61e47c50b8e94a8956f852c36ef1d050ff55e397ed0f0f5533","qualified_name":"div_mul_eq_div_mul_one_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18103,"column":77,"line":579},"start":{"byte":18011,"column":0,"line":578}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      a\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        b c))\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b)\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@InvOneClass.toOne.{u_1} α\n              (@DivInvOneMonoid.toInvOneClass.{u_1} α\n                (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))))\n        c))","statement_sha256":"dc8ab3efd1a2a0d3f4a392883d552747b7724c7c8dd05483eecff039b26bdb50","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-e17bfcb8b896a7c11d6b6f6991d803708c297a2cad158386c101281f5cd032c8","qualified_name":"div_mul_eq_mul_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17789,"column":61,"line":567},"start":{"byte":17713,"column":0,"line":566}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        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(@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        a c)\n      b)","statement_sha256":"4185107f44fd45156e3336de8a4333808311b85a710a901a404f35d435f545be","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-43c72eb4062247de76a6a6cbfc64041cce1dfe7966cd65d53a14a4f16e1429e1","qualified_name":"div_mul_mul_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":33274,"column":46,"line":981},"start":{"byte":33127,"column":0,"line":979}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G\n              (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        a c)\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        b c))\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G\n              (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n      a b)","statement_sha256":"f2be1a5436e37513e904876737b0952a86b98759950406a92aac7e09e4982448","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-0a2a54ceaf24ce57a129121f115d325dd8bf1470bc3d2e5ed2023bef4d8ae158","qualified_name":"div_ne_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24490,"column":22,"line":764},"start":{"byte":24405,"column":0,"line":762}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Ne.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))\n    (@Ne.{u_3 + 1} G a b)","statement_sha256":"65f6bbfe32f9a31917232a719d40f2be59460d6be72845933cef916bcaa7cc0c","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-81bf9dc4e37eaa8c480cbd6f5d73744383aa4993c5c45648c95c9fb175eab835","qualified_name":"div_ne_one_of_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":12082,"column":21,"line":391},"start":{"byte":11992,"column":0,"line":389}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a b : α},\n  @Ne.{u_1 + 1} α a b →\n    @Ne.{u_1 + 1} α\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))) a b)\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@MulOne.toOne.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))))","statement_sha256":"d3ee7fff20ea6d275958ffc8960fc8984645a0c586e491845f0b4f6214ea0d9c","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-a36f7028120a334deec3333b8fffe9cac3cfe17e47266805dbbad41e9ea2ad4d","qualified_name":"div_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11014,"column":63,"line":353},"start":{"byte":10921,"column":0,"line":352}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : DivInvOneMonoid.{u_3} G] (a : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@DivInvOneMonoid.toDivInvMonoid.{u_3} G inst))) a\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G (@InvOneClass.toOne.{u_3} G (@DivInvOneMonoid.toInvOneClass.{u_3} G inst)))))\n    a","statement_sha256":"709e0867ee2a1a4c4a8147f87a9b58de75d2a407230c9fd83c196d86571d0d07","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-b25ab6b3ef5f376c21e2883b262a38f83a2e1014dced2ba3fdf4087c30b55c12","qualified_name":"div_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18808,"column":46,"line":599},"start":{"byte":18666,"column":0,"line":597}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α) (n : Nat),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Nat α\n      (@instHPow.{u_1, 0} α Nat\n        (@NPow.toPow.{u_1} α\n          (@Monoid.toNPow.{u_1} α\n            (@DivInvMonoid.toMonoid.{u_1} α\n              (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b)\n      n)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HPow.hPow.{u_1, 0, u_1} α Nat α\n        (@instHPow.{u_1, 0} α Nat\n          (@NPow.toPow.{u_1} α\n            (@Monoid.toNPow.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))))\n        a n)\n      (@HPow.hPow.{u_1, 0, u_1} α Nat α\n        (@instHPow.{u_1, 0} α Nat\n          (@NPow.toPow.{u_1} α\n            (@Monoid.toNPow.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))))\n        b n))","statement_sha256":"2082d2dd00e99afdfb0ab80ee946f72cdf59557d3102eef78a83e68e1099d309","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-fbd8147347641ebb57ef3c8bc10c6d7e76f8edd1486bcffddf6d31a3619f7998","qualified_name":"div_right_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17334,"column":57,"line":549},"start":{"byte":17262,"column":0,"line":548}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b)\n      c)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a c)\n      b)","statement_sha256":"3b5830d01a749841f50678322d2a0060374982e5f355e10b77e07bbe40f15487","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-eabd9ca1a383de959cb0352e8a56b03bedea4fce55135872f1ac3d32b7b24eba","qualified_name":"div_right_inj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23394,"column":28,"line":731},"start":{"byte":23285,"column":0,"line":729}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a c))\n    (@Eq.{u_3 + 1} G b c)","statement_sha256":"189690a55040611fca98134a3caf192787f8b06e76e7374fccf034a4ceae2e5c","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-580230edbf3c17ddf0805513639f563e24fa5ad528d4552c1d3bdf0408f43400","qualified_name":"div_right_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":22593,"column":62,"line":708},"start":{"byte":22395,"column":0,"line":704}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {b : G},\n  @Function.Injective.{u_3 + 1, u_3 + 1} G G fun a =>\n    @HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) b a","statement_sha256":"6e21b018802b4dabdf922a20fc70db4a23160a1c2b02369486cb254259ab0fa0","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-191a76b2134941120f5c4f0375ee32512b74237ca7061c4ce69d5027d3bf9376","qualified_name":"div_zpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18955,"column":48,"line":603},"start":{"byte":18810,"column":0,"line":601}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α) (n : Int),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α\n          (@DivInvMonoid.toZPow.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b)\n      n)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HPow.hPow.{u_1, 0, u_1} α Int α\n        (@instHPow.{u_1, 0} α Int\n          (@ZPow.toPow.{u_1} α\n            (@DivInvMonoid.toZPow.{u_1} α\n              (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))\n        a n)\n      (@HPow.hPow.{u_1, 0, u_1} α Int α\n        (@instHPow.{u_1, 0} α Int\n          (@ZPow.toPow.{u_1} α\n            (@DivInvMonoid.toZPow.{u_1} α\n              (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))\n        b n))","statement_sha256":"ff4f512ee63b2b22ba02e883e9c33410abbff3ed0bdb655e8610064013d2cafc","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-6e6e5aecc7e4cdf695f29e035bfe94d3e827c9f55743735031342c37e9cadbe0","qualified_name":"eq_add_neg_iff_add_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21109,"column":13,"line":672},"start":{"byte":21098,"column":2,"line":672}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        b\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          c)))\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a c)\n      b)","statement_sha256":"f2df8c1c5521b06182fbbdf0218afe227eb3fcdf22d5d8f2f58f6024e0d854d7","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-6c10a662df8f8ba2e7b8e37c22066ef9457f9c351f59badcd9c1b8beffa53674","qualified_name":"eq_add_neg_of_add_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19455,"column":13,"line":622},"start":{"byte":19444,"column":2,"line":622}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a c)\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        b\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          c))","statement_sha256":"c6f9b189cdab6470fa38f51f424b65883e9f1a5b6ebf82ac31f7d0d127272c56","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-ee1d1f961806516046cdd620dc78f1b2a2a8234e3732445ade109f0ffcf43726","qualified_name":"eq_add_of_add_neg_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19837,"column":13,"line":634},"start":{"byte":19826,"column":2,"line":634}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          c))\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        b c)","statement_sha256":"d60f43ba2e016e27d073dc387bded1e808183a4d138599325ebc46e24c0f7c3f","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-86653cd9cd02a7a4704fdd2ec36afbd4e27499382410229a1f82089ccee48a63","qualified_name":"eq_add_of_neg_add_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19935,"column":13,"line":637},"start":{"byte":19924,"column":2,"line":637}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          b)\n        a)\n      c →\n    @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        b c)","statement_sha256":"c679966955c5ef994168c1da628fcf02c068822b204f779aa5aa9874ceab44d8","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-9de99e458912b25a80962ed71f021fde2cd3fee02e58b6cb54fd541bef2f5878","qualified_name":"eq_add_of_sub_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23126,"column":13,"line":723},"start":{"byte":23115,"column":2,"line":723}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) a c)\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        b c)","statement_sha256":"7fa57d6663aac9d621b04d71eceff1b0e98cf1e02d70ab34c9ad4f1e33a67dec","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-77f0f00c175775d5eb8ec68b93d16795b352b5a5c7f38b1adc349318d440a9d3","qualified_name":"eq_add_of_sub_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31305,"column":13,"line":929},"start":{"byte":31294,"column":2,"line":929}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        a b)\n      c →\n    @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        b c)","statement_sha256":"e9cda526a768c92a04cbec351f585520cbe234d77a309049cf4ce18aaed2e5f0","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-7aad7391bccfdeeb3863d402d5a0f36bdbde1e44c1cb041310e94b0020745088","qualified_name":"eq_div_iff_mul_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24750,"column":101,"line":770},"start":{"byte":24616,"column":0,"line":769}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) b c))\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a c)\n      b)","statement_sha256":"c001ab926ff690ac39281246cc2f77fbb608771a6862a149cc60d2e89a140337","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-030d3ddf64bcd2ffe2b2aba0741be177e72cd135a9779cafda20c6a1fc1e8a96","qualified_name":"eq_div_iff_mul_eq''","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":32049,"column":93,"line":950},"start":{"byte":31922,"column":0,"line":949}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        b c))\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        c a)\n      b)","statement_sha256":"45c0dd19fe21701a1cd20913c711278b26be849cce44451c20ec6e63d0dc81ad","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-1638f5038d50295c18ff26c9625467001a5ae4e2749a6f9b758e052b08a4a2d5","qualified_name":"eq_div_of_mul_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23008,"column":72,"line":718},"start":{"byte":22904,"column":0,"line":717}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a c)\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) b c)","statement_sha256":"432fde06438dfbd0a151361a97bcc48ef2cc64e1f5ba63d109234bace074a1d3","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-4684d4882eb9861927b8ed542f4abc05dfa24ef9bb3d57d1566ad93bd6c3be92","qualified_name":"eq_div_of_mul_eq''","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31290,"column":74,"line":927},"start":{"byte":31183,"column":0,"line":926}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        c a)\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        b c)","statement_sha256":"b46c7c714e0b4720a333b158eccd0db6703ed265b78580a5fb5a49c0e9e57c10","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-e181a0f61fbbf8f307ebe0c719837f588967f6cfb99bb46c1561b94a7ec87dbe","qualified_name":"eq_iff_eq_of_add_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9082,"column":13,"line":279},"start":{"byte":9071,"column":2,"line":279}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : AddCancelCommMonoid.{u_1} α] {a b c d : α},\n  @Eq.{u_1 + 1} α\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@AddCommMonoid.toAddMonoid.{u_1} α (@AddCancelCommMonoid.toAddCommMonoid.{u_1} α inst))))))\n        a b)\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@AddCommMonoid.toAddMonoid.{u_1} α (@AddCancelCommMonoid.toAddCommMonoid.{u_1} α inst))))))\n        c d) →\n    Iff (@Eq.{u_1 + 1} α a c) (@Eq.{u_1 + 1} α b d)","statement_sha256":"adc95656221c7456cf7f30c493afd187c6d4eadb670c0fffeb2a97b71a693f88","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-6708c1c2d1f97bf11fa4432462e06dbfb599f9d7b811c71d017e7ad2c2c618c3","qualified_name":"eq_iff_eq_of_div_eq_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24996,"column":36,"line":777},"start":{"byte":24869,"column":0,"line":775}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c d : G},\n  @Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) c d) →\n    Iff (@Eq.{u_3 + 1} G a b) (@Eq.{u_3 + 1} G c d)","statement_sha256":"388ec44b1dfd3bf3da64bb0cd4ed0fcf5af7b590eace5a7751feb4c836911698","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-820b95342205c5406c2f23a7b805362c5c05ab5151c3f134ec062181dfb2eb5d","qualified_name":"eq_iff_eq_of_mul_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9163,"column":94,"line":279},"start":{"byte":9069,"column":0,"line":279}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : CancelCommMonoid.{u_1} α] {a b c d : α},\n  @Eq.{u_1 + 1} α\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@CommMonoid.toMonoid.{u_1} α (@CancelCommMonoid.toCommMonoid.{u_1} α inst))))))\n        a b)\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@CommMonoid.toMonoid.{u_1} α (@CancelCommMonoid.toCommMonoid.{u_1} α inst))))))\n        c d) →\n    Iff (@Eq.{u_1 + 1} α a c) (@Eq.{u_1 + 1} α b d)","statement_sha256":"035dfaf52ebc1aaff8bb3fbe77983d15da9d534882ac70dea4ec9b76ac8b3c2c","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-512103554fadff963864add63686c2ff7fbaa573eefab2f3be5743dfbd392a56","qualified_name":"eq_iff_eq_of_sub_eq_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24882,"column":13,"line":775},"start":{"byte":24871,"column":2,"line":775}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c d : G},\n  @Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) a b)\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) c d) →\n    Iff (@Eq.{u_3 + 1} G a b) (@Eq.{u_3 + 1} G c d)","statement_sha256":"c836a3794fcc6b689ae3ba89a9e99856a78ef7eb50e3e718eb572704385606de","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-c31f799d0c6341ebdea2961d35d2433f7b16005fa73609ae17ddd0f162b72e02","qualified_name":"eq_inv_iff_mul_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20989,"column":28,"line":666},"start":{"byte":20886,"column":0,"line":664}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@Inv.inv.{u_3} G\n        (@InvOneClass.toInv.{u_3} G\n          (@DivInvOneMonoid.toInvOneClass.{u_3} G\n            (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n        b))\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))","statement_sha256":"4b99f61c640e3cef89cee7b58111a7fe00bebe50dd9170e8d13b5f4b2d830df5","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-59daeb64a29b8f7d89b9fd3da25ab63ec847d3fd127bb1f2b6abb83947910bb0","qualified_name":"eq_inv_mul_iff_mul_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21446,"column":94,"line":678},"start":{"byte":21273,"column":0,"line":676}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          b)\n        c))\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b a)\n      c)","statement_sha256":"88867b5c75846364f7e68351bad32906c66dbfa957a12740043bff431d83461f","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-f311d721f5bdabc0ec957e1020ea7107a6ffa8141d28d37477928f136d83807b","qualified_name":"eq_inv_mul_of_mul_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19636,"column":81,"line":626},"start":{"byte":19540,"column":0,"line":625}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b a)\n      c →\n    @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          b)\n        c)","statement_sha256":"f14aa87a97db9f52b560f20d3e94dcd81af7ba09120cd393c657e94a13bb790f","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-b4d755fb8ff738f227ec10500b2b867d7b45cb13b7c07b10556c17ef57ebd787","qualified_name":"eq_inv_of_mul_eq_one_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11343,"column":37,"line":369},"start":{"byte":11224,"column":0,"line":367}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))\n        a b)\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@MulOne.toOne.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α b (@Inv.inv.{u_1} α (@DivInvMonoid.toInv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)) a)","statement_sha256":"224bd169d6cecb19239dffb1343ff391b7de309919e326ed05f2c2e8664c635e","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-94fd3c26fd115df1bda386c2da942b9f3162cbdcbb0ca7b71d54732ce421b170","qualified_name":"eq_mul_inv_iff_mul_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21271,"column":96,"line":674},"start":{"byte":21096,"column":0,"line":672}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          c)))\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a c)\n      b)","statement_sha256":"a454582fbcb8be12194103d256a2d7244406e559a56ccc02c961af96cf3e96bc","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-5e525f17513303de27ef1bcd3c814b3fbfef11671d40ebfdd707729550e3a0c7","qualified_name":"eq_mul_inv_of_mul_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19538,"column":81,"line":623},"start":{"byte":19442,"column":0,"line":622}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a c)\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          c))","statement_sha256":"2b2cb60a3d3273190384b6a2c62d0bc195f9c1b82c81a59af439c466797f85b4","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-e57cf39c12a2c5301e9f2993d8a021cf292f6cc8f263388038bad7dbca04837a","qualified_name":"eq_mul_of_div_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23199,"column":71,"line":724},"start":{"byte":23113,"column":0,"line":723}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a c)\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b c)","statement_sha256":"d53374ee7bfad2cb9a8db7b2144e47c73d35ac1f58db8a3778a1a24d50b41c82","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-0cf218cbd14939cff268e50a251942c24397f06168d52f86b5c8abc64a832fa9","qualified_name":"eq_mul_of_div_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31380,"column":73,"line":930},"start":{"byte":31292,"column":0,"line":929}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        a b)\n      c →\n    @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        b c)","statement_sha256":"1a88e9ea10ff440be4259ff53e2c70b611ddb5eb1b8690d2a0b3357f736de937","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-f60483e3a3cd632bb291ca3519402c2eff6bede84b4293e6bcf31a9e3739df9a","qualified_name":"eq_mul_of_inv_mul_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20039,"column":102,"line":638},"start":{"byte":19922,"column":0,"line":637}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          b)\n        a)\n      c →\n    @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b c)","statement_sha256":"d0c09dfc3fcb5cf97600fd2ae04f233e76433896fa97146d60f902dafc648f9a","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-88a1520053566d143f0891ec2f2a361c7a20431bc537cf3d4216ff69c9c7621a","qualified_name":"eq_mul_of_mul_inv_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19920,"column":81,"line":635},"start":{"byte":19824,"column":0,"line":634}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          c))\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b c)","statement_sha256":"3fbae707a758fe371ea16d03f5b6a3a3072e7b4aecda30362deca87cf820cb29","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-deecc98a404b7e66112c0f85bade807eacfa666a7d417dabb114aa59556711d8","qualified_name":"eq_neg_add_iff_add_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21286,"column":13,"line":676},"start":{"byte":21275,"column":2,"line":676}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          b)\n        c))\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        b a)\n      c)","statement_sha256":"9e2cfc8ec35de601eab00754db47d703c4d0368e7117ca5f9187de7d94b097b3","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-a0da0dba6b5642809a7c41d62b9cfb3e05637c4c7371819328829c1203e92ba4","qualified_name":"eq_neg_add_of_add_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19553,"column":13,"line":625},"start":{"byte":19542,"column":2,"line":625}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        b a)\n      c →\n    @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          b)\n        c)","statement_sha256":"8f4582e27358eaf7ba3365f60e6e97f7a9fdbcba6da13f35ca503f863bb9e2e0","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-cf01b1d383752d2cf5e19f2b4f0289810f51b3637f678f44c0706f2e13b3f0e8","qualified_name":"eq_neg_iff_add_eq_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20899,"column":13,"line":664},"start":{"byte":20888,"column":2,"line":664}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@Neg.neg.{u_3} G\n        (@NegZeroClass.toNeg.{u_3} G\n          (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n        b))\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 0)\n        (@Zero.toOfNat0.{u_3} G\n          (@NegZeroClass.toZero.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst)))))))","statement_sha256":"7e41e82fa165e7f617353b27b62991008a4d81bc79a8413550bd140cb8507691","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-ac38c638fb29fecd60367aa94973189f318338872f7fdeae4859c268eec4a079","qualified_name":"eq_neg_of_add_eq_zero_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11237,"column":13,"line":367},"start":{"byte":11226,"column":2,"line":367}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))\n        a b)\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@AddZero.toZero.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α b\n      (@Neg.neg.{u_1} α (@SubNegMonoid.toNeg.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)) a)","statement_sha256":"0e6332e816f35ce2d4438ef8f5dae911e3a19fd8bd10f8838e92db935b785c20","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-c04b8ccf688cffcbc440d7028479ecc4232a828fcdb36986da0f0dca2f6195fa","qualified_name":"eq_of_add_neg_eq_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11887,"column":13,"line":386},"start":{"byte":11876,"column":2,"line":386}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))\n        a (@Neg.neg.{u_1} α (@SubNegMonoid.toNeg.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)) b))\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@AddZero.toZero.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α a b","statement_sha256":"a2c8a008d20a9717ca73c954d4d77246f73520334afd21e1648c1f360ea0c7b3","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-2e9bfc3da3fa1744424cac675d8c4f8ac6845186cb721bf8742870f58482cfd0","qualified_name":"eq_of_div_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11754,"column":76,"line":381},"start":{"byte":11611,"column":0,"line":379}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))) a b)\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@MulOne.toOne.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α a b","statement_sha256":"e3c48d666b51e6b703d4ac05f7231dd487b653897f215317e4cc4aea0cb165aa","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-cc1cc222d0811fd47b8ee92e2baff2c809f5687d60382f3c478895db6093c25d","qualified_name":"eq_of_inv_mul_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11872,"column":101,"line":384},"start":{"byte":11756,"column":0,"line":383}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))\n        (@Inv.inv.{u_1} α (@DivInvMonoid.toInv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)) a) b)\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@MulOne.toOne.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α a b","statement_sha256":"f2ed1e478b1e80f610d0874f7e01c82cc165d4f72aee00e48ea6e7d00c4f6d75","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-5863c9c33c56816908bbf9d5237462c1c88554c40ebda6d2525a8d7c420dcaa8","qualified_name":"eq_of_mul_inv_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11990,"column":101,"line":387},"start":{"byte":11874,"column":0,"line":386}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))\n        a (@Inv.inv.{u_1} α (@DivInvMonoid.toInv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)) b))\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@MulOne.toOne.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α a b","statement_sha256":"e107a54dc9fb64ac28c3e332637ed9aada56a9d7c849c9a870d094cb06034b2f","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-ebd3caea6b039da8718483d36dda94f85bb36d799c3b45d9db2cb220ee36ad0a","qualified_name":"eq_of_neg_add_eq_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11769,"column":13,"line":383},"start":{"byte":11758,"column":2,"line":383}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))\n        (@Neg.neg.{u_1} α (@SubNegMonoid.toNeg.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)) a) b)\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@AddZero.toZero.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α a b","statement_sha256":"924674580acec5c38f7fcd7cad0eb119f26043e510e864aa2d66da195b4fe364","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-b7cc3b797c93da00f591cf9197ab511c9a14a2561d7ab904c8a753928a3d73b0","qualified_name":"eq_of_one_div_eq_one_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14978,"column":48,"line":476},"start":{"byte":14848,"column":0,"line":474}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@InvOneClass.toOne.{u_1} α\n              (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))))\n        a)\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@InvOneClass.toOne.{u_1} α\n              (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))))\n        b) →\n    @Eq.{u_1 + 1} α a b","statement_sha256":"d24b095382662d60f2585ba21c3a75f72f5ab526d4c6b85f4e8992ede89c4986","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-2df51337b7f936fe5148a5a7c4a4e631e823540bff832ba37663d8f1bb805415","qualified_name":"eq_of_sub_eq_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11624,"column":13,"line":379},"start":{"byte":11613,"column":2,"line":379}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))) a b)\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@AddZero.toZero.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α a b","statement_sha256":"3f7badf994a7e8fdaec3d1d3d0ec689d161bb1ef396f8d6bb958842f646836f6","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-d516c6447e9d5de0fa1876e9530ecf6447530f2fcd260f1d56dac3fe9aae8a65","qualified_name":"eq_of_zero_sub_eq_zero_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14861,"column":13,"line":474},"start":{"byte":14850,"column":2,"line":474}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 0)\n          (@Zero.toOfNat0.{u_1} α\n            (@NegZeroClass.toZero.{u_1} α\n              (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst)))))\n        a)\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 0)\n          (@Zero.toOfNat0.{u_1} α\n            (@NegZeroClass.toZero.{u_1} α\n              (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst)))))\n        b) →\n    @Eq.{u_1 + 1} α a b","statement_sha256":"72d69de970e628f9e2a53d5e922c1fa8267cd1e91e8144aba0939bd7b6e7fd54","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-eeaad292c7a38bdf7151254f48811a4ad52006aa3532f881f1a6cce4ad8fd0d2","qualified_name":"eq_one_div_of_mul_eq_one_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11475,"column":43,"line":373},"start":{"byte":11345,"column":0,"line":371}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))\n        b a)\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@MulOne.toOne.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α b\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@MulOne.toOne.{u_1} α\n              (@MulOneClass.toMulOne.{u_1} α\n                (@Monoid.toMulOneClass.{u_1} α\n                  (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))))\n        a)","statement_sha256":"3d046e3693130104f043e5d32708b5fa5c63a419874eee7743af24c62c10f239","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-c8c98fc1d64e4ee7e2957494318bb0f2699546796dab2da7925ca7deeb7656cc","qualified_name":"eq_one_div_of_mul_eq_one_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11609,"column":44,"line":377},"start":{"byte":11477,"column":0,"line":375}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))\n        a b)\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@MulOne.toOne.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α b\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@MulOne.toOne.{u_1} α\n              (@MulOneClass.toMulOne.{u_1} α\n                (@Monoid.toMulOneClass.{u_1} α\n                  (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))))\n        a)","statement_sha256":"66845e45c3b2167b0bb1fed1db5464966e9af42e3606a9a5090642b5234ccf85","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-bf5e74ee172b9c4f91b05012bbdcff23fa8263fc708b38adf51414c7d1a9dd8f","qualified_name":"eq_one_iff_eq_one_of_mul_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3313,"column":45,"line":100},"start":{"byte":3163,"column":0,"line":98}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : MulOneClass.{u_4} M] {a b : M},\n  @Eq.{u_4 + 1} M\n      (@HMul.hMul.{u_4, u_4, u_4} M M M (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))\n        a b)\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))) →\n    Iff\n      (@Eq.{u_4 + 1} M a\n        (@OfNat.ofNat.{u_4} M (nat_lit 1)\n          (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))))\n      (@Eq.{u_4 + 1} M b\n        (@OfNat.ofNat.{u_4} M (nat_lit 1)\n          (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))))","statement_sha256":"b5256c3d638ba29f69cb55af2548fefccc751f1c3cf774af36028727fc7f3cff","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-aa6cc5f9f2d48d0e3150f249513b3ae60566ce1b0a1ce3739eed73cb9c8dc789","qualified_name":"eq_sub_iff_add_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24647,"column":31,"line":769},"start":{"byte":24618,"column":2,"line":769}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) b c))\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a c)\n      b)","statement_sha256":"b4ac732fd8509442fa0db91440435d40faa51fac05d23fe9456630cb12eebca3","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-267b9518c0e207bd007582cc232afc86ebab090b2b5fc88940eeb2c1d75539e6","qualified_name":"eq_sub_iff_add_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31954,"column":32,"line":949},"start":{"byte":31924,"column":2,"line":949}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G a\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        b c))\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        c a)\n      b)","statement_sha256":"3f39e216793f40be9a4987561779d80d58c5c442a1751dea717660ded1c2a322","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-d11a136ea6994f37a59c7f01348b454ea8ea5f835efaf3f3b44a6ddb3b41855f","qualified_name":"eq_sub_of_add_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":22934,"column":30,"line":717},"start":{"byte":22906,"column":2,"line":717}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a c)\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) b c)","statement_sha256":"4b2adecba070dda738bc08ade7b7d7e75d252bfaede34244b6dddb9818db3959","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-ac770b73eae04d0710dc23e655870e4f1913d8df4c713942339b1f876e24a26f","qualified_name":"eq_sub_of_add_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31214,"column":31,"line":926},"start":{"byte":31185,"column":2,"line":926}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        c a)\n      b →\n    @Eq.{u_3 + 1} G a\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        b c)","statement_sha256":"ea36388bf099880a8978e2ec77f228d7289011c52562cf5819981be1e8d476b2","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-1a0e8896d5be9ef5f0f2feb381914e7908dd549fa5cf5f0476e8f0983898fc74","qualified_name":"eq_zero_iff_eq_zero_of_add_eq_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3176,"column":13,"line":98},"start":{"byte":3165,"column":2,"line":98}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddZeroClass.{u_4} M] {a b : M},\n  @Eq.{u_4 + 1} M\n      (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst))) a b)\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst)))) →\n    Iff\n      (@Eq.{u_4 + 1} M a\n        (@OfNat.ofNat.{u_4} M (nat_lit 0)\n          (@Zero.toOfNat0.{u_4} M (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst)))))\n      (@Eq.{u_4 + 1} M b\n        (@OfNat.ofNat.{u_4} M (nat_lit 0)\n          (@Zero.toOfNat0.{u_4} M (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst)))))","statement_sha256":"f2c2f57d39131fc88f673ec435a048c1114e7e747a5ab2a5214aa68c142bcc20","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-6d9ca70b4c1d03b1b08b192f74481326f6f61ead059679bee41e8d7dc98dd62b","qualified_name":"eq_zero_sub_of_add_eq_zero_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11358,"column":13,"line":371},"start":{"byte":11347,"column":2,"line":371}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))\n        b a)\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@AddZero.toZero.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α b\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 0)\n          (@Zero.toOfNat0.{u_1} α\n            (@AddZero.toZero.{u_1} α\n              (@AddZeroClass.toAddZero.{u_1} α\n                (@AddMonoid.toAddZeroClass.{u_1} α\n                  (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))))))\n        a)","statement_sha256":"e3d0ff710651f872e73812285a964d3e028cb2a22303489ec0b9bb4157fca750","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-183a323b1a2d5472de9135d6b983d271edad1a53770dc387204f8d03a4b86e70","qualified_name":"eq_zero_sub_of_add_eq_zero_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11490,"column":13,"line":375},"start":{"byte":11479,"column":2,"line":375}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] {a b : α},\n  @Eq.{u_1 + 1} α\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))\n        a b)\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@AddZero.toZero.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))) →\n    @Eq.{u_1 + 1} α b\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 0)\n          (@Zero.toOfNat0.{u_1} α\n            (@AddZero.toZero.{u_1} α\n              (@AddZeroClass.toAddZero.{u_1} α\n                (@AddMonoid.toAddZeroClass.{u_1} α\n                  (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))))))\n        a)","statement_sha256":"f54c71ff9c2e6bfdee0d914839aca97b04deb26e6669f099fbab9b9311f7bde4","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-aa9db67fe0174a42f611367605aadd18cce40d89da5d1c9fa6a80cbbd5e9bebb","qualified_name":"hom_coe_nsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":37300,"column":13,"line":1064},"start":{"byte":37289,"column":2,"line":1064}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} {F : Type u_5} [inst : AddMonoid.{u_5} F] (c : F → M → M),\n  @Eq.{u_4 + 1} (M → M)\n      (c\n        (@OfNat.ofNat.{u_5} F (nat_lit 0)\n          (@Zero.toOfNat0.{u_5} F\n            (@AddZero.toZero.{u_5} F (@AddZeroClass.toAddZero.{u_5} F (@AddMonoid.toAddZeroClass.{u_5} F inst))))))\n      (@id.{u_4 + 1} M) →\n    (∀ (f g : F),\n        @Eq.{u_4 + 1} (M → M)\n          (c\n            (@HAdd.hAdd.{u_5, u_5, u_5} F F F\n              (@instHAdd.{u_5} F\n                (@AddZero.toAdd.{u_5} F (@AddZeroClass.toAddZero.{u_5} F (@AddMonoid.toAddZeroClass.{u_5} F inst))))\n              f g))\n          (@Function.comp.{u_4 + 1, u_4 + 1, u_4 + 1} M M M (c f) (c g))) →\n      ∀ (f : F) (n : Nat),\n        @Eq.{u_4 + 1} (M → M)\n          (c\n            (@HSMul.hSMul.{0, u_5, u_5} Nat F F\n              (@instHSMul.{0, u_5} Nat F (@NSMul.toSMul.{u_5} F (@AddMonoid.toNSMul.{u_5} F inst))) n f))\n          (@Nat.iterate.{u_4 + 1} M (c f) n)","statement_sha256":"e40188de4386b1278647aeb8324ecdce0f9f1187bc4e2c475ebe63f57890421d","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-bb0aa1ea6eb39705de1cdcd98964401c80033f14434e453a94bf3d24063da553","qualified_name":"hom_coe_nsmul._f","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":37300,"column":13,"line":1064},"start":{"byte":37289,"column":2,"line":1064}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} {F : Type u_5} [inst : AddMonoid.{u_5} F] (c : F → M → M),\n  @Eq.{u_4 + 1} (M → M)\n      (c\n        (@OfNat.ofNat.{u_5} F (nat_lit 0)\n          (@Zero.toOfNat0.{u_5} F\n            (@AddZero.toZero.{u_5} F (@AddZeroClass.toAddZero.{u_5} F (@AddMonoid.toAddZeroClass.{u_5} F inst))))))\n      (@id.{u_4 + 1} M) →\n    (∀ (f g : F),\n        @Eq.{u_4 + 1} (M → M)\n          (c\n            (@HAdd.hAdd.{u_5, u_5, u_5} F F F\n              (@instHAdd.{u_5} F\n                (@AddZero.toAdd.{u_5} F (@AddZeroClass.toAddZero.{u_5} F (@AddMonoid.toAddZeroClass.{u_5} F inst))))\n              f g))\n          (@Function.comp.{u_4 + 1, u_4 + 1, u_4 + 1} M M M (c f) (c g))) →\n      ∀ (f : F) (x : Nat)\n        (f_1 :\n          @Nat.below.{0}\n            (fun x =>\n              @Eq.{u_4 + 1} (M → M)\n                (c\n                  (@HSMul.hSMul.{0, u_5, u_5} Nat F F\n                    (@instHSMul.{0, u_5} Nat F (@NSMul.toSMul.{u_5} F (@AddMonoid.toNSMul.{u_5} F inst))) x f))\n                (@Nat.iterate.{u_4 + 1} M (c f) x))\n            x),\n        @Eq.{u_4 + 1} (M → M)\n          (c\n            (@HSMul.hSMul.{0, u_5, u_5} Nat F F\n              (@instHSMul.{0, u_5} Nat F (@NSMul.toSMul.{u_5} F (@AddMonoid.toNSMul.{u_5} F inst))) x f))\n          (@Nat.iterate.{u_4 + 1} M (c f) x)","statement_sha256":"2945e97d28874d2e2053bba052db95e031087efedab1466e06c529543c53ac1a","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-3900a6c2b406506123e0f37164eb78aa2fc4766042860126ceee69e737abbbef","qualified_name":"hom_coe_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":37582,"column":76,"line":1070},"start":{"byte":37210,"column":0,"line":1063}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} {F : Type u_5} [inst : Monoid.{u_5} F] (c : F → M → M),\n  @Eq.{u_4 + 1} (M → M)\n      (c\n        (@OfNat.ofNat.{u_5} F (nat_lit 1)\n          (@One.toOfNat1.{u_5} F\n            (@MulOne.toOne.{u_5} F (@MulOneClass.toMulOne.{u_5} F (@Monoid.toMulOneClass.{u_5} F inst))))))\n      (@id.{u_4 + 1} M) →\n    (∀ (f g : F),\n        @Eq.{u_4 + 1} (M → M)\n          (c\n            (@HMul.hMul.{u_5, u_5, u_5} F F F\n              (@instHMul.{u_5} F\n                (@MulOne.toMul.{u_5} F (@MulOneClass.toMulOne.{u_5} F (@Monoid.toMulOneClass.{u_5} F inst))))\n              f g))\n          (@Function.comp.{u_4 + 1, u_4 + 1, u_4 + 1} M M M (c f) (c g))) →\n      ∀ (f : F) (n : Nat),\n        @Eq.{u_4 + 1} (M → M)\n          (c\n            (@HPow.hPow.{u_5, 0, u_5} F Nat F\n              (@instHPow.{u_5, 0} F Nat (@NPow.toPow.{u_5} F (@Monoid.toNPow.{u_5} F inst))) f n))\n          (@Nat.iterate.{u_4 + 1} M (c f) n)","statement_sha256":"78eef61be944ac340f0ce4eeadb4e7b30f4014301ab98c9a463e0b9e7e428bc4","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-31826f688c66490ec601dd339dc92f2d284b4d303042e259a0203bcc90b39c2e","qualified_name":"inv_bijective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9563,"column":26,"line":294},"start":{"byte":9456,"column":0,"line":292}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : InvolutiveInv.{u_3} G],\n  @Function.Bijective.{u_3 + 1, u_3 + 1} G G (@Inv.inv.{u_3} G (@InvolutiveInv.toInv.{u_3} G inst))","statement_sha256":"ff2dd32bcb6af30ce33fb6077548705d0a39eb7cee0c4d4fcf8afd5c7e68fd77","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-ff8bb80898485e25a0fca4361ce9ee02b20ff8835c2b720ba9a329b1af7f33ad","qualified_name":"inv_comp_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":10108,"column":26,"line":316},"start":{"byte":10013,"column":0,"line":314}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (G : Type u_3) [inst : InvolutiveInv.{u_3} G],\n  @Eq.{u_3 + 1} (G → G)\n    (@Function.comp.{u_3 + 1, u_3 + 1, u_3 + 1} G G G (@Inv.inv.{u_3} G (@InvolutiveInv.toInv.{u_3} G inst))\n      (@Inv.inv.{u_3} G (@InvolutiveInv.toInv.{u_3} G inst)))\n    (@id.{u_3 + 1} G)","statement_sha256":"cea76c42df815147de07d50027d29e864edaee02233c539f493b3ef0d51b2558","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-6f78220cea4302cb5df3dd11f84ae17d98003d68a81856c6eb2fcd6c2fd4eeeb","qualified_name":"inv_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":12348,"column":49,"line":402},"start":{"byte":12269,"column":0,"line":401}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Inv.inv.{u_1} α (@DivInvMonoid.toInv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))) a b))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))) b a)","statement_sha256":"537a0139de4c906ae188fd08c5e9724b5db5f0a73a59b1ebe68fce6ed227ffa5","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-dcb1f260bd3b86fde6b5810997bc69d81155ab423a7865749366cc4bc8a41936","qualified_name":"inv_div'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16719,"column":60,"line":526},"start":{"byte":16644,"column":0,"line":525}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Inv.inv.{u_1} α\n      (@InvOneClass.toInv.{u_1} α\n        (@DivInvOneMonoid.toInvOneClass.{u_1} α\n          (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a)\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        b))","statement_sha256":"2fee66a5e46d609406ec97d160278972b0e35a366d8c7686834e36b0e0377b8a","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-1f3cf6c44a6453206497ff96343a1ca40d042795fc97ddec76e0aecd518d7681","qualified_name":"inv_div_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16944,"column":81,"line":534},"start":{"byte":16863,"column":0,"line":534}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a)\n      b)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        b)\n      a)","statement_sha256":"8205ed5d6b24fbfcc626a736f97a0e91495af2ffb1777e23b0dff1b2f26388a8","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-2fba98f47ce391f829db67defce74a92fc1c8cb48e27e094e49dff3dcdbf5df2","qualified_name":"inv_div_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17089,"column":56,"line":540},"start":{"byte":17018,"column":0,"line":539}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a)\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        b))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      b a)","statement_sha256":"4b88a70a13deca2d685cd0ab2d007ac5d4a8bab8d7c6ef48da19df151f95eb75","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-507e2b700d2cf8843e1c1f01631473ad31af255289f164c4dd1ce0b08d8d25f2","qualified_name":"inv_div_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":12267,"column":59,"line":399},"start":{"byte":12193,"column":0,"line":398}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} 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a))","statement_sha256":"1ce59670d8d81b96061ecb8fcbd4ff05b82d088e8237bbdfb9eaf913bd60ab2c","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-ac6d42b79b97c338e84e3b188a468d7b115b9c90cd1e78bbc88fc3dded523e40","qualified_name":"inv_eq_iff_eq_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9997,"column":23,"line":310},"start":{"byte":9902,"column":0,"line":308}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : InvolutiveInv.{u_3} G] {a b : G},\n  Iff (@Eq.{u_3 + 1} G (@Inv.inv.{u_3} G (@InvolutiveInv.toInv.{u_3} G inst) a) b)\n    (@Eq.{u_3 + 1} G a (@Inv.inv.{u_3} G (@InvolutiveInv.toInv.{u_3} G inst) b))","statement_sha256":"8247a72f81eb0e3528dd6d6301cc44edd0c2d95fbe534539d3734205b446a1cb","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-afbbaa5ef303400e6e2a34e864e4d137dfffa0e78cf1fef0f7419fd561fc7863","qualified_name":"inv_eq_iff_mul_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21094,"column":28,"line":670},"start":{"byte":20991,"column":0,"line":668}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@Inv.inv.{u_3} G\n        (@InvOneClass.toInv.{u_3} G\n          (@DivInvOneMonoid.toInvOneClass.{u_3} G\n            (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n        a)\n      b)\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))","statement_sha256":"35026715e02dc1a368f2ddc74a4a023aad086fef915c0cd9851b55114760a57e","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-67c2fbe28a7c79d22aac9afefe99b57f695b68a9b206fba80ff043da056285e3","qualified_name":"inv_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14661,"column":31,"line":464},"start":{"byte":14555,"column":0,"line":462}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a : α},\n  Iff\n    (@Eq.{u_1 + 1} α\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n        a)\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst))))))\n    (@Eq.{u_1 + 1} α a\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst))))))","statement_sha256":"cee6c234f6db10bfb7642406ee1b4e885d8ac43022d835ddea51614a4edde131","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-30205d0a2822d71d9ce436f77f361b273e95fdee977330e8f0c97b70a81c7cdf","qualified_name":"inv_inj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9900,"column":22,"line":306},"start":{"byte":9801,"column":0,"line":304}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : InvolutiveInv.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G (@Inv.inv.{u_3} G (@InvolutiveInv.toInv.{u_3} G inst) a)\n      (@Inv.inv.{u_3} G 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(@Inv.inv.{u_3} G (@InvolutiveInv.toInv.{u_3} G inst))","statement_sha256":"a9cd6d593273d765717aa9dadc803bfc40472992b035f57578f65c99bb5c8c16","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-dd7c16bb1ed83f880e0e7b4b8f50f766c48c358ac920d718a2db4ad1d89b17a9","qualified_name":"inv_inv_div_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17173,"column":67,"line":543},"start":{"byte":17091,"column":0,"line":542}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Inv.inv.{u_1} α\n      (@InvOneClass.toInv.{u_1} α\n        (@DivInvOneMonoid.toInvOneClass.{u_1} α\n          (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        (@Inv.inv.{u_1} α\n          (@InvOneClass.toInv.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α\n              (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n          a)\n        (@Inv.inv.{u_1} α\n          (@InvOneClass.toInv.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α\n              (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n          b)))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      a b)","statement_sha256":"81f2f3add8b91c9e9c4ea056b4c32ad0428fc44745bcaab7c67984ad09617915","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-d99efced4672279cfa6f1994a8111f189bbbbb38cff11f9bd2506b272f9725dc","qualified_name":"inv_involutive","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9454,"column":9,"line":290},"start":{"byte":9347,"column":0,"line":288}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : InvolutiveInv.{u_3} G],\n  @Function.Involutive.{u_3 + 1} G (@Inv.inv.{u_3} G (@InvolutiveInv.toInv.{u_3} G inst))","statement_sha256":"17766efee0e71f5631407d38b7b911ec9ac33afa11f81e19a768217229a0e9ec","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-7e2c126b58ea445527b11ece86be65ea0211367afe28eed5269a1f5d23bfcb76","qualified_name":"inv_mul'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17016,"column":55,"line":537},"start":{"byte":16946,"column":0,"line":536}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Inv.inv.{u_1} α\n      (@InvOneClass.toInv.{u_1} α\n        (@DivInvOneMonoid.toInvOneClass.{u_1} α\n          (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        a b))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a)\n      b)","statement_sha256":"53ac9763a99425268d3bae374564393763899d58ddb1879137d3be0bbc8d7d5b","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-826de7e075468ba5bfcd4babb7f1263e1a299324b274d22ebc2e0425c9163a8c","qualified_name":"inv_mul_eq_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16861,"column":54,"line":532},"start":{"byte":16792,"column":0,"line":531}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a)\n      b)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      b a)","statement_sha256":"2137251c207c54bf3af94ff6664971f5ef27d713e9287277804b2a9c88cfd1cc","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-458158eb67d8e548cd916cf72dadfbfe47df311749abeb4286a8bb4b7b32fb23","qualified_name":"inv_mul_eq_iff_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21621,"column":94,"line":682},"start":{"byte":21448,"column":0,"line":680}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          a)\n        b)\n      c)\n    (@Eq.{u_3 + 1} G b\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a c))","statement_sha256":"5310a48b057ed1f6e5f85fa15882f4c4c0d38d103092f0b93b27d1259372a761","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-005fe012952f648ba9507ec591e85c9b176e9e231755ee46feb3064e7a63b640","qualified_name":"inv_mul_eq_inv_mul_iff_mul_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":34295,"column":72,"line":1006},"start":{"byte":34116,"column":0,"line":1004}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] {a b c d : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G\n                (@DivisionCommMonoid.toDivisionMonoid.{u_3} G (@CommGroup.toDivisionCommMonoid.{u_3} G inst)))))\n          b)\n        a)\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G\n                (@DivisionCommMonoid.toDivisionMonoid.{u_3} G (@CommGroup.toDivisionCommMonoid.{u_3} G inst)))))\n          d)\n        c))\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        a d)\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        c b))","statement_sha256":"c3f60e7c08b165150bac492efb70483a5cde5708e813ca89199a442255cef5a3","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-0ef76ec9da49689190140067d45c571667c1dc60bd14e1bf998491b324371f15","qualified_name":"inv_mul_eq_of_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19729,"column":76,"line":629},"start":{"byte":19638,"column":0,"line":628}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G b\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a c) →\n    @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          a)\n        b)\n      c","statement_sha256":"74e4bda2027d3788cadb18c5ea48d886cf3c3301ba5c88be428163ee87af2f9e","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-c65ad574833d32942556d2f2dd693c4eb74295078daf625e94ce415ff481252c","qualified_name":"inv_mul_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":22014,"column":91,"line":692},"start":{"byte":21908,"column":0,"line":691}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          a)\n        b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G a b)","statement_sha256":"c4ec236310ac4b64c7c08caec030e3a50fb90c429c501b05e269708a100a38d7","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-64cdf4fc16656d80bed480ffd046b288c0208cde0605b506f2135b44fe9f58bf","qualified_name":"inv_ne_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14846,"column":16,"line":472},"start":{"byte":14766,"column":0,"line":470}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a : α},\n  Iff\n    (@Ne.{u_1 + 1} α\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n        a)\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst))))))\n    (@Ne.{u_1 + 1} α a\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst))))))","statement_sha256":"5be54ec92aae1d13bfc4159bfdc3e5a4956965f186de9d4e2588ec5dd8e754f9","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-530ae9957aae9810333a954519d651e229f869c2419fd35e8f1e71f3d567be82","qualified_name":"inv_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":13042,"column":66,"line":422},"start":{"byte":12837,"column":0,"line":419}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α) (n : Nat),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Nat α\n      (@instHPow.{u_1, 0} α Nat\n        (@NPow.toPow.{u_1} α\n          (@Monoid.toNPow.{u_1} α (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n        a)\n      n)\n    (@Inv.inv.{u_1} α\n      (@InvOneClass.toInv.{u_1} α\n        (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n      (@HPow.hPow.{u_1, 0, u_1} α Nat α\n        (@instHPow.{u_1, 0} α Nat\n          (@NPow.toPow.{u_1} α\n            (@Monoid.toNPow.{u_1} α (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))\n        a n))","statement_sha256":"467582c05dc4fbdffeeb96f4ee998ef85da44c687764a7a85de16c1819b6fa62","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-49cb814a58af2ec0461f00fc49053c845738d55e3abd86feb84a031704ab0d42","qualified_name":"inv_pow_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":26129,"column":50,"line":806},"start":{"byte":25948,"column":0,"line":804}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (a : G) {m n : Nat},\n  @LE.le.{0} Nat instLENat n m →\n    @Eq.{u_3 + 1} G\n      (@HPow.hPow.{u_3, 0, u_3} G Nat G\n        (@instHPow.{u_3, 0} G Nat\n          (@NPow.toPow.{u_3} G\n            (@Monoid.toNPow.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst)))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          a)\n        (@HSub.hSub.{0, 0, 0} Nat Nat Nat (@instHSub.{0} Nat instSubNat) m n))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          (@HPow.hPow.{u_3, 0, u_3} G Nat G\n            (@instHPow.{u_3, 0} G Nat\n              (@NPow.toPow.{u_3} G\n                (@Monoid.toNPow.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst)))))\n            a m))\n        (@HPow.hPow.{u_3, 0, u_3} G Nat G\n          (@instHPow.{u_3, 0} G Nat\n            (@NPow.toPow.{u_3} G\n              (@Monoid.toNPow.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst)))))\n          a n))","statement_sha256":"81b268d91d7895bc2b24992b134695ae8be0321969d7a03b10cc965a2f8674be","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-9d98e1d6f47d2afa316ee43fa4a5adca4f50c352f4ed4aec18cccb30c5c00949","qualified_name":"inv_surjective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9690,"column":27,"line":298},"start":{"byte":9565,"column":0,"line":296}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : InvolutiveInv.{u_3} G],\n  @Function.Surjective.{u_3 + 1, u_3 + 1} G G (@Inv.inv.{u_3} G (@InvolutiveInv.toInv.{u_3} G inst))","statement_sha256":"272b8caf7c71d348174b09274896c10350e29189716a0dbd5c3a838355a31d42","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-faf6690c85c349f3827cbfe9bb370a10225ddbfd625b1433ce093f14f270bd8a","qualified_name":"inv_unique","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":7682,"column":58,"line":226},"start":{"byte":7545,"column":0,"line":224}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : CommMonoid.{u_4} M] {x y z : M},\n  @Eq.{u_4 + 1} M\n      (@HMul.hMul.{u_4, u_4, u_4} M M M\n        (@instHMul.{u_4} M\n          (@MulOne.toMul.{u_4} M\n            (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M (@CommMonoid.toMonoid.{u_4} M inst)))))\n        x y)\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M\n          (@MulOne.toOne.{u_4} M\n            (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M (@CommMonoid.toMonoid.{u_4} M inst)))))) →\n    @Eq.{u_4 + 1} M\n        (@HMul.hMul.{u_4, u_4, u_4} M M M\n          (@instHMul.{u_4} M\n            (@MulOne.toMul.{u_4} M\n              (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M (@CommMonoid.toMonoid.{u_4} M inst)))))\n          x z)\n        (@OfNat.ofNat.{u_4} M (nat_lit 1)\n          (@One.toOfNat1.{u_4} M\n            (@MulOne.toOne.{u_4} M\n              (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M (@CommMonoid.toMonoid.{u_4} M inst)))))) →\n      @Eq.{u_4 + 1} M y z","statement_sha256":"ba8a4fd1969d13de8b93b1b954a6d97563dc4185b2d6df1a80e92598e84a29fa","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-3bb265e4a22b6762ea310c394c2ee0b60daf8b387f544fa223984cb10a022b3b","qualified_name":"inv_zpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":13950,"column":61,"line":445},"start":{"byte":13738,"column":0,"line":442}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α) (n : Int),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n        a)\n      n)\n    (@Inv.inv.{u_1} α\n      (@InvOneClass.toInv.{u_1} α\n        (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n      (@HPow.hPow.{u_1, 0, u_1} α Int α\n        (@instHPow.{u_1, 0} α Int\n          (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n        a n))","statement_sha256":"b41e9f0e07de941682c9f6582f0c6e2710cf71efa9126d02f334f50db0453e53","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-71d02e0d974858fea0165a6740f68aa1662dac0af39be542b0d9ff244fa080ce","qualified_name":"inv_zpow'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14081,"column":88,"line":448},"start":{"byte":13952,"column":0,"line":447}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α) (n : Int),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n        a)\n      n)\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      a (@Neg.neg.{0} Int Int.instNegInt n))","statement_sha256":"47d24444cb8d6a75038d42f2cf5a6d0e9031696f9425d7e45fa2ff9c7513d06b","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-e99127e1aedb6e96c1689827d5a87f341b831a628887ee4a4f05e19dc2ee431d","qualified_name":"ite_add_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":2870,"column":13,"line":88},"start":{"byte":2859,"column":2,"line":88}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddZeroClass.{u_4} M] {P : Prop} [inst_1 : Decidable P] {a b : M},\n  @Eq.{u_4 + 1} M\n    (@ite.{u_4 + 1} M P inst_1\n      (@HAdd.hAdd.{u_4, u_4, 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inst))))))","statement_sha256":"bfeb8eb140072ac8f288146e94c0558fff4fd9af561789316a1ca6ac98d1b730","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-d9040f62da5b54443fa829e52245d42fd6a55f31785131db3cd6e3ac2bde722d","qualified_name":"ite_mul_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3008,"column":29,"line":91},"start":{"byte":2857,"column":0,"line":88}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : MulOneClass.{u_4} M] {P : Prop} [inst_1 : Decidable P] {a b : M},\n  @Eq.{u_4 + 1} M\n    (@ite.{u_4 + 1} M P inst_1\n      (@HMul.hMul.{u_4, u_4, u_4} M M M (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))\n        a b)\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))))\n    (@HMul.hMul.{u_4, u_4, u_4} M M M (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))\n      (@ite.{u_4 + 1} M P inst_1 a\n        (@OfNat.ofNat.{u_4} M (nat_lit 1)\n          (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))))\n      (@ite.{u_4 + 1} M P inst_1 b\n        (@OfNat.ofNat.{u_4} M (nat_lit 1)\n          (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst))))))","statement_sha256":"e61fa96030179ce040ae578caa88f57beb001bf6b9abd759c7ff00914319194d","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-9950d9b6307dd68c47c338114ff21e8ba15e89be3f8ac366c63a4b4c4922d48a","qualified_name":"ite_one_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3161,"column":29,"line":96},"start":{"byte":3010,"column":0,"line":93}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : MulOneClass.{u_4} M] {P : Prop} [inst_1 : Decidable P] {a b : M},\n  @Eq.{u_4 + 1} M\n    (@ite.{u_4 + 1} M P inst_1\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst))))\n      (@HMul.hMul.{u_4, u_4, u_4} M M M (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))\n        a b))\n    (@HMul.hMul.{u_4, u_4, u_4} M M M (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))\n      (@ite.{u_4 + 1} M P inst_1\n        (@OfNat.ofNat.{u_4} M (nat_lit 1)\n          (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst))))\n        a)\n      (@ite.{u_4 + 1} M P inst_1\n        (@OfNat.ofNat.{u_4} M (nat_lit 1)\n          (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst))))\n        b))","statement_sha256":"6ac23c615e8e036e8dd4e491f7674c8184665fdaacf0f9c993ca642637674e67","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-ad80355f7121967d094a693274faee52c6e3dd9a8873c61a3137602e936d9979","qualified_name":"ite_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":1757,"column":77,"line":49},"start":{"byte":1567,"column":0,"line":47}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : Pow.{u_1, u_2} α β] (p : Prop) [inst_1 : Decidable p] (a b : α) (c : β),\n  @Eq.{u_1 + 1} α (@HPow.hPow.{u_1, u_2, u_1} α β α (@instHPow.{u_1, u_2} α β inst) (@ite.{u_1 + 1} α p inst_1 a b) c)\n    (@ite.{u_1 + 1} α p inst_1 (@HPow.hPow.{u_1, u_2, u_1} α β α (@instHPow.{u_1, u_2} α β inst) a c)\n      (@HPow.hPow.{u_1, u_2, u_1} α β α (@instHPow.{u_1, u_2} α β inst) b c))","statement_sha256":"7e03135ef493567a3105d524dedd85381c9009e7d39273bf6e31fb8b12515f45","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-abbd40e86137e8c2e2a63a82338b16ec42746c3f2973517e05779703892c82b4","qualified_name":"ite_smul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":1425,"column":50,"line":43},"start":{"byte":1377,"column":2,"line":43}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul.{u_2, u_1} β α] (p : Prop) [inst_1 : Decidable p] (a : α) (b c : β),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{u_2, u_1, u_1} β α α (@instHSMul.{u_2, u_1} β α inst) (@ite.{u_2 + 1} β p inst_1 b c) a)\n    (@ite.{u_1 + 1} α p inst_1 (@HSMul.hSMul.{u_2, u_1, u_1} β α α (@instHSMul.{u_2, u_1} β α inst) b a)\n      (@HSMul.hSMul.{u_2, u_1, u_1} β α α (@instHSMul.{u_2, u_1} β α inst) c a))","statement_sha256":"5b66697a49554af6f2d438def49c1ba480b370ccb8d1f2fad805a34dce3cd94e","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-913c18b26b1e18ed902e4d17a3a20d3844aabd93f40434d65d5b887d35b2723f","qualified_name":"ite_vadd","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":1425,"column":50,"line":43},"start":{"byte":1377,"column":2,"line":43}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : VAdd.{u_2, u_1} β α] (p : Prop) [inst_1 : Decidable p] (a : α) (b c : β),\n  @Eq.{u_1 + 1} α\n    (@HVAdd.hVAdd.{u_2, u_1, u_1} β α α (@instHVAdd.{u_2, u_1} β α inst) (@ite.{u_2 + 1} β p inst_1 b c) a)\n    (@ite.{u_1 + 1} α p inst_1 (@HVAdd.hVAdd.{u_2, u_1, u_1} β α α (@instHVAdd.{u_2, u_1} β α inst) b a)\n      (@HVAdd.hVAdd.{u_2, u_1, u_1} β α α (@instHVAdd.{u_2, u_1} β α inst) c a))","statement_sha256":"7325f64860d7f8d01631e5f1275c61811c1e2b567e2abbc39daf023a68f449d2","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-e93eda313916f774068b433ba820bce4df67de07b91b3cfe5c74dafe2d05a5d2","qualified_name":"ite_zero_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3023,"column":13,"line":93},"start":{"byte":3012,"column":2,"line":93}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddZeroClass.{u_4} M] {P : Prop} [inst_1 : Decidable P] {a b : M},\n  @Eq.{u_4 + 1} M\n    (@ite.{u_4 + 1} M P inst_1\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst))))\n      (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst))) a b))\n    (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n      (@instHAdd.{u_4} M (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst)))\n      (@ite.{u_4 + 1} M P inst_1\n        (@OfNat.ofNat.{u_4} M (nat_lit 0)\n          (@Zero.toOfNat0.{u_4} M (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst))))\n        a)\n      (@ite.{u_4 + 1} M P inst_1\n        (@OfNat.ofNat.{u_4} M (nat_lit 0)\n          (@Zero.toOfNat0.{u_4} M (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M inst))))\n        b))","statement_sha256":"d2b9ab40478e75add1e237f8f77bd90614ff850998107c8e01195cf652bea0e5","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-649f66d4e1b1bbb50ff4f8dec98980fb2e8ac14025751d625085d5617f54f2da","qualified_name":"leftInverse_add_left_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":25165,"column":13,"line":783},"start":{"byte":25154,"column":2,"line":783}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (c : G),\n  @Function.LeftInverse.{u_3 + 1, u_3 + 1} G G\n    (fun x =>\n      @HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        x c)\n    fun x =>\n    @HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) x 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fun a => @Neg.neg.{u_3} G (@InvolutiveNeg.toNeg.{u_3} G inst) a","statement_sha256":"05e41621065e1f844c9dca2f8c9571ce94be1d3bd998400901cd4982951299d2","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-2f81fb8a8d45830ccba12a80666624502f2e340b7ac9bde818c0162b878c369e","qualified_name":"leftInverse_neg_add_add_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":25480,"column":13,"line":792},"start":{"byte":25469,"column":2,"line":792}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (c : G),\n  @Function.LeftInverse.{u_3 + 1, u_3 + 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x","statement_sha256":"18146e552070a262e81f10478637681e7e89bc884a57f20b0fc53cab1001cb1c","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-b921c60df401072b3e2269982b236cfb35a7e37a232cb94dc68cb861a114dcdc","qualified_name":"leftInverse_sub_add_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":25011,"column":13,"line":779},"start":{"byte":25000,"column":2,"line":779}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (c : G),\n  @Function.LeftInverse.{u_3 + 1, u_3 + 1} G G\n    (fun x =>\n      @HSub.hSub.{u_3, u_3, u_3} G G G\n     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c","statement_sha256":"4ea5b5769017d862951031c380ec32d10d254c5d2ca51b3d30c02bf215a66031","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-2ac14cda2e818623f21e0e7837072bb6ee6adcc7f8e35ec661215c4a783d2c25","qualified_name":"left_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":8194,"column":28,"line":243},"start":{"byte":8168,"column":2,"line":243}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M]\n  [@IsLeftCancelAdd.{u_4} M\n      (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M 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inst)))]\n  {a b : M},\n  Iff\n    (@Eq.{u_4 + 1} M a\n      (@HMul.hMul.{u_4, u_4, u_4} M M M\n        (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))\n        a b))\n    (@Eq.{u_4 + 1} M b\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M\n          (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M 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(@AddMonoid.toAddZeroClass.{u_4} M inst)))]\n  {a b : M},\n  Iff\n    (@Ne.{u_4 + 1} M a\n      (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M\n          (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n        a b))\n    (@Ne.{u_4 + 1} M b\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M\n          (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M 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inst)))]\n  {a b : M},\n  Iff\n    (@Ne.{u_4 + 1} M a\n      (@HMul.hMul.{u_4, u_4, u_4} M M M\n        (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))\n        a b))\n    (@Ne.{u_4 + 1} M b\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M\n          (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))))","statement_sha256":"e9e3a8cb52fa516190aec58b136f50d5985b5546a5a4ab1b50bdb53916f4b905","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-1c57230c419670d8066401264f0005d8471ff8b33759b2056f20afae26c460eb","qualified_name":"mul_comm_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17937,"column":57,"line":573},"start":{"byte":17865,"column":0,"line":572}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a b)\n      c)\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      a\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        c b))","statement_sha256":"c1ca73a360bfa245da924c408772209f77b31bb3cf9a746eeedf6817fac60a4a","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-2ca904157091dcddfc2a4cb08ee0d2498ce456b1c01a8a2a5d1c306766c080de","qualified_name":"mul_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":10736,"column":97,"line":341},"start":{"byte":10624,"column":0,"line":340}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : DivInvMonoid.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G inst)))))\n      a (@HDiv.hDiv.{u_3, u_3, u_3} G G G (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G inst)) b c))\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G inst))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G inst)))))\n        a b)\n      c)","statement_sha256":"fd2c536c9ded19d705ddbefff907d0b5b9bf00ac951047efa12a7184149fbc60","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-1f4520d0235f9f9e5cab06563539883db490b09fb6687105f7eea6d7f945ded5","qualified_name":"mul_div_assoc'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":10622,"column":28,"line":338},"start":{"byte":10515,"column":0,"line":336}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : DivInvMonoid.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G inst)))))\n      a (@HDiv.hDiv.{u_3, u_3, u_3} G G G (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G inst)) b c))\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G inst))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G inst)))))\n        a b)\n      c)","statement_sha256":"fd2c536c9ded19d705ddbefff907d0b5b9bf00ac951047efa12a7184149fbc60","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-5d3e7b47ce463e4be602246aa93b903ea9bf37d046dad454eeb5457d1e401f7e","qualified_name":"mul_div_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":32538,"column":45,"line":963},"start":{"byte":32406,"column":0,"line":961}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G\n              (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n      a\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        b a))\n    b","statement_sha256":"174838ed8bb79f75da9aa0c158d63dcc92ed4071f36390056f80d4ab43033b5b","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-8a44aec1e3585279f4f8c1628e4600c7a12624726ad586fd5ff82cdb04842994","qualified_name":"mul_div_cancel_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":32404,"column":100,"line":959},"start":{"byte":32274,"column":0,"line":958}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        a b)\n      a)\n    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(@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        a b)\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        a c))\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G\n              (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n      b c)","statement_sha256":"6368749c4aad7ff9bf950dd923cab639100c3a770790781647f89149ccf5246f","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-9e4201a3de3f099ca4f4b03ec96de5010451666173fab8d51a0aae5ce10e1e87","qualified_name":"mul_div_left_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17553,"column":64,"line":558},"start":{"byte":17474,"column":0,"line":557}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      a\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        b c))\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      b\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a c))","statement_sha256":"8cc75e38f0d15a79940be22112c9439c9308cca66ad71848707102607933dd1d","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-728838fde6ee2294f123733981afc75bf1ab7728ac98e87d8092db87f9714058","qualified_name":"mul_div_mul_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18453,"column":71,"line":591},"start":{"byte":18367,"column":0,"line":590}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c d : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        a b)\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        c d))\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a c)\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        b d))","statement_sha256":"e263029764fe42aba97a34bb98c013422e9fded42134543140e24db23ec08160","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-577c765a39a90ca3e9823a2bc29d550e1fdfe253e4ab3284a732782f4373c2cf","qualified_name":"mul_div_mul_left_eq_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31181,"column":6,"line":924},"start":{"byte":31069,"column":0,"line":922}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        c a)\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        c b))\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G\n        (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n      a b)","statement_sha256":"8d11c30c52c0001f8480ade30d5245f02e218121dca7b6e7410481ddf6c544b3","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-259c65e56728b39f8791fe6c16d6601783c759f1c0b8cc87e343c405d150b111","qualified_name":"mul_div_mul_right_eq_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":22902,"column":64,"line":715},"start":{"byte":22731,"column":0,"line":713}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst)))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a c)\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b c))\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) a b)","statement_sha256":"2221fe0ff6ec85aac78272b64159a3413fd9fa30850754e74a60ba0a34b09bb3","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-ebe84f0854a6eda2457b07ba17c5b729f3235ff68fc9806dee90f6abe5aa5136","qualified_name":"mul_div_right_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17631,"column":61,"line":561},"start":{"byte":17555,"column":0,"line":560}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        a b)\n      c)\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a c)\n      b)","statement_sha256":"3d8b132eaf164c8001bfaf4c56f97329fea6df954cfb8ed543900f5fceb50cdc","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-08a9c5b7d9109900c0036c1b521484f0451bc08bde82bbf366a3694b0478e6b2","qualified_name":"mul_eq_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":8164,"column":36,"line":241},"start":{"byte":7999,"column":0,"line":238}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M]\n  [@IsLeftCancelMul.{u_4} M\n      (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst)))]\n  {a b : M},\n  Iff\n    (@Eq.{u_4 + 1} M\n      (@HMul.hMul.{u_4, u_4, u_4} M M M\n        (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))\n        a b)\n      a)\n    (@Eq.{u_4 + 1} M b\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M\n          (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))))","statement_sha256":"3cc18a7ecabfe7ed5015e23a429bef26a2334e7920894e326372a52358da916b","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-f50d79225078070e3d26fabffd0099724fcdaedbf0d2fc14b7f0f904efb35498","qualified_name":"mul_eq_of_eq_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23283,"column":67,"line":727},"start":{"byte":23201,"column":0,"line":726}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))) c b) →\n    @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a b)\n      c","statement_sha256":"4df698c23f42e367d369556bbc5789aaff1a40dc27c819764288247f25e0dc04","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-22032a68659381d8e020d226acb7adfdb62d5982057dc3885f54cfbb4ef86a44","qualified_name":"mul_eq_of_eq_div'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31514,"column":57,"line":934},"start":{"byte":31382,"column":0,"line":932}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G b\n      (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n        (@instHDiv.{u_3} G\n          (@DivInvMonoid.toDiv.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst))))\n        c a) →\n    @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        a b)\n      c","statement_sha256":"3c29a865838818da47fe159b4304adb69b60a4a77cf99be4a01b955311363ffc","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-f241263aa95469e4f81f4b849119b44c52be1b40e9d9dbcb205d4b241c903ece","qualified_name":"mul_eq_of_eq_inv_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20151,"column":95,"line":641},"start":{"byte":20041,"column":0,"line":640}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G b\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          a)\n        c) →\n    @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a b)\n      c","statement_sha256":"d780ddab43de9f22b5d0344f6298779681d6e0c09035a387311443953e98e95b","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-1c7ddc7535003923f28ae57a1c9a0db83a1b585a3bdeba492c5cc69bdd6ab80f","qualified_name":"mul_eq_of_eq_mul_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20244,"column":76,"line":644},"start":{"byte":20153,"column":0,"line":643}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        c\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          b)) →\n    @Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a b)\n      c","statement_sha256":"1086a782e1aa77303b23fbbcb95cc8514828d39d9178452c789b577350d1dbed","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-a49bd5ea212828702f8b9ae4e8fd55d9f7ca2012dd319df0b168417999cccdfa","qualified_name":"mul_eq_one_iff_eq_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20391,"column":70,"line":648},"start":{"byte":20246,"column":0,"line":646}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G a\n      (@Inv.inv.{u_3} G\n        (@InvOneClass.toInv.{u_3} G\n          (@DivInvOneMonoid.toInvOneClass.{u_3} G\n            (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n        b))","statement_sha256":"a82c3723a1ee370dad9ecc5b8332364be48e9b1e3fbe094c7aa846b08d769507","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-078816a12e1f322463a08cf8550aa4d70f9effaa0d42288f30afc1766001453b","qualified_name":"mul_eq_one_iff_eq_inv'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20701,"column":37,"line":657},"start":{"byte":20520,"column":0,"line":654}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G b\n      (@Inv.inv.{u_3} G\n        (@InvOneClass.toInv.{u_3} G\n          (@DivInvOneMonoid.toInvOneClass.{u_3} G\n            (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n        a))","statement_sha256":"768fd03d9d584c0b6cd83d3702328452f7429b2730f00a3255e9d694266ff100","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-7cfd1ede58f385ffdaf4a3a74ed2836746a4114165d61054306263926b7e696d","qualified_name":"mul_eq_one_iff_inv_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20518,"column":47,"line":652},"start":{"byte":20393,"column":0,"line":650}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G\n      (@Inv.inv.{u_3} G\n        (@InvOneClass.toInv.{u_3} G\n          (@DivInvOneMonoid.toInvOneClass.{u_3} G\n            (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n        a)\n      b)","statement_sha256":"f22bbd420ba20fd4c08847d01281d872340ff65168d8566d9ce2e440ce576b4b","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-79e8c7e1aed1bb86d2a7bb158e2eb9acbad6c6c964c8d223c75cc2339be7f9e4","qualified_name":"mul_eq_one_iff_inv_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":20884,"column":37,"line":662},"start":{"byte":20703,"column":0,"line":659}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G\n      (@Inv.inv.{u_3} G\n        (@InvOneClass.toInv.{u_3} G\n          (@DivInvOneMonoid.toInvOneClass.{u_3} G\n            (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n        b)\n      a)","statement_sha256":"d85d9bfe1fdb1c9557c5df60b783e7b12dd150118de316991437eb0ff3edfc02","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-a390d0338c0da263c311c544b038b44a50009ea299a0d486aa50816f380acc18","qualified_name":"mul_eq_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":8700,"column":37,"line":262},"start":{"byte":8533,"column":0,"line":259}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M]\n  [@IsRightCancelMul.{u_4} M\n      (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst)))]\n  {a b : M},\n  Iff\n    (@Eq.{u_4 + 1} M\n      (@HMul.hMul.{u_4, u_4, u_4} M M M\n        (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))\n        a b)\n      b)\n    (@Eq.{u_4 + 1} M a\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M\n          (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))))","statement_sha256":"447931448fc8d79c4e9e0dd2c3278cd2a5af4aca52387c91949e6ee033c83b25","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-3f7423ba24308ef4d58b9f24e6f023d65ae382f88e4715dcde5406cea6be4a2f","qualified_name":"mul_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16642,"column":59,"line":523},"start":{"byte":16560,"column":0,"line":522}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Inv.inv.{u_1} α\n      (@InvOneClass.toInv.{u_1} α\n        (@DivInvOneMonoid.toInvOneClass.{u_1} α\n          (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        a b))\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        a)\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α\n            (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        b))","statement_sha256":"f7030cd362156531eaad3ac18e21db8f4d2b8acdc1bcc779591da7f35d1f86f6","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-796c2f88773755f5982de2123eb96d9d9cb0fb2d7594d914503bfb909a2a5b80","qualified_name":"mul_inv_eq_iff_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21798,"column":96,"line":686},"start":{"byte":21623,"column":0,"line":684}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          b))\n      c)\n    (@Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        c b))","statement_sha256":"2133e89784f967c92926bc17416f6ecae18bcac9797d2bacf550e3360f36cf00","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-d3cb0dc220a608696f3fd505d663b9f5030e1e53d2f7cc2aa627394f16b1f9c8","qualified_name":"mul_inv_eq_mul_inv_iff_mul_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":34114,"column":72,"line":1002},"start":{"byte":33935,"column":0,"line":1000}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] {a b c d : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        a\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G\n                (@DivisionCommMonoid.toDivisionMonoid.{u_3} G (@CommGroup.toDivisionCommMonoid.{u_3} G inst)))))\n          b))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        c\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G\n                (@DivisionCommMonoid.toDivisionMonoid.{u_3} G (@CommGroup.toDivisionCommMonoid.{u_3} G inst)))))\n          d)))\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        a d)\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        c b))","statement_sha256":"028aec50b013ae2856b4464509839e7c229f526fdbc92b1a7bc2a83a59d9e28d","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-7a2baf9f66936a520f72b4dcf9ba990ae3508346e34f8c0adc9b563664cb678a","qualified_name":"mul_inv_eq_of_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19822,"column":76,"line":632},"start":{"byte":19731,"column":0,"line":631}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          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c","statement_sha256":"857c729b9b028bfe9e4bf9b7d0a36627c4b5a3c857b416cf8d156dd504ff389c","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-2f4bf84e77d4913749e6c6d84a8ccb405c10499b4884aedaa5e624dd38b2bc7c","qualified_name":"mul_inv_eq_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21906,"column":91,"line":689},"start":{"byte":21800,"column":0,"line":688}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          b))\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G a b)","statement_sha256":"8dbd06e7aa3f655f7499c5a1129291edcee4bfc6bb18b6f5c94b92dced77ba21","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-ece505ea6d15ba1042342e1b3697432698d813bba4de269ccbe56be8eedec6c4","qualified_name":"mul_inv_mul_mul_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24023,"column":63,"line":748},"start":{"byte":23852,"column":0,"line":746}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        a\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n          b))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n        b c))\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n      a c)","statement_sha256":"d3872ade212555c914c44c3694b49fd572d561d1edd40edd123d3482977f8a38","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-a1edb37b681d56e0858bc1767671a04e358a545cb130035dfbdaa9e0f6714fcf","qualified_name":"mul_left_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3669,"column":43,"line":118},"start":{"byte":3543,"column":0,"line":116}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommSemigroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G (@CommMagma.toMul.{u_3} G 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c))","statement_sha256":"d8d2f625d8cdd345bebdc34caf9c86fb6cd953d4c5bad6ce5f29f51a881ca6c0","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-e4985fdbd7278ee6d1262d8d0837172adf2bd3434ad2ba15ab284db2c01a9019","qualified_name":"mul_left_iterate","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6129,"column":49,"line":185},"start":{"byte":5954,"column":0,"line":182}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M] (a : M) (n : Nat),\n  @Eq.{u_4 + 1} (M → M)\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HMul.hMul.{u_4, u_4, u_4} 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x","statement_sha256":"202012ae50182c7f32f31c553c0de2dfed2f76a3e585f798e8edeafdbaf55a7e","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-04b51e617266f5aa253d54473d3d5528d68f05ec6b0125cee4da8e2db77583a1","qualified_name":"mul_left_iterate_apply","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6544,"column":78,"line":194},"start":{"byte":6311,"column":0,"line":192}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M] {n : Nat} (a b : M),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HMul.hMul.{u_4, u_4, u_4} 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b)","statement_sha256":"f790df26b75d757674ecd5b8c2f1a9a85bbf1ac41cdd4243b742162b5f548dd6","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-d6f1a62cd605e10fae397ae51b10f24d7bb9c4cdec5fabb562702a326ea15c60","qualified_name":"mul_left_iterate_apply_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":7169,"column":76,"line":209},"start":{"byte":7078,"column":0,"line":208}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M] {n : Nat} (a : M),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HMul.hMul.{u_4, u_4, 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n)","statement_sha256":"10bbfb608e9637647ffceb1c675ad99506d34c16aa9bccc1a380af67e779881f","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-487f735da153ffa2b234d953cef45b20019ab9be6b6559d0b967534265a9844e","qualified_name":"mul_left_iterate_apply_self","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6928,"column":39,"line":202},"start":{"byte":6785,"column":0,"line":200}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M] (a : M) (n : Nat),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HMul.hMul.{u_4, u_4, u_4} M M M\n          (@instHMul.{u_4} M\n            (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))\n          a x)\n      n a)\n    (@HPow.hPow.{u_4, 0, u_4} M Nat M (@instHPow.{u_4, 0} M Nat (@NPow.toPow.{u_4} M (@Monoid.toNPow.{u_4} M inst))) a\n      (@HAdd.hAdd.{0, 0, 0} Nat Nat Nat (@instHAdd.{0} Nat instAddNat) n\n        (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))","statement_sha256":"d21f282ef7d99fcc485c68e60c2ef9856d5cfffa16ca49fea7a6a618fa662a2b","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-645dabeafd6d1ec5fb900bcb4a1383525b712096822ece3f19ec856e764715ad","qualified_name":"mul_left_surjective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19291,"column":53,"line":616},"start":{"byte":19162,"column":0,"line":614}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (a : G),\n  @Function.Surjective.{u_3 + 1, u_3 + 1} G G fun x =>\n    @HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n      a x","statement_sha256":"c3f1f4dee49e51b0b399953fe23cd34d11f0acf10f4d8f58634ba16f74173df4","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-1afb412ca6066bc9fdfcff3242b02a442a7c6fa4b9284fc869e9496b7d669cf6","qualified_name":"mul_mul_div_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":33125,"column":32,"line":977},"start":{"byte":32992,"column":0,"line":975}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        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(@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n      a b)","statement_sha256":"f8e4d9362b34d828f5cee40bdf7761b7dbddbdd00f3e2213d4bc7370675970c2","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-474e9ec9238a401e85e94a205b2dfa26bc39dd422cb6910760804fc156f75344","qualified_name":"mul_mul_inv_cancel'_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":32990,"column":45,"line":973},"start":{"byte":32857,"column":0,"line":971}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G\n              (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n      a\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G\n          (@MulOne.toMul.{u_3} G\n            (@MulOneClass.toMulOne.{u_3} G\n              (@Monoid.toMulOneClass.{u_3} G\n                (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G (@CommGroup.toGroup.{u_3} G inst)))))))\n        b\n        (@Inv.inv.{u_3} G\n          (@InvOneClass.toInv.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G\n                (@DivisionCommMonoid.toDivisionMonoid.{u_3} G (@CommGroup.toDivisionCommMonoid.{u_3} G inst)))))\n          a)))\n    b","statement_sha256":"3a69daa0b209cb512164cb991d69b060fb6dc3ebacc32931c5259df50f39b15d","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-0e3db718c20279ea2c52627d5b8240910b55ab571f1260eb35b7ec89f98bc9cf","qualified_name":"mul_mul_inv_mul_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23850,"column":58,"line":744},"start":{"byte":23684,"column":0,"line":742}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        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u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        (@MulOne.toMul.{u_3} G\n          (@MulOneClass.toMulOne.{u_3} G\n            (@Monoid.toMulOneClass.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n      a c)","statement_sha256":"37a5572792523353a39fb0981f3ea41c63850e6916871dd9cc6cede5ea1ff8bd","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-ad249d73ea742e99c0d2b7ec1482efca045258c585c9a8458243867fb45b8768","qualified_name":"mul_mul_mul_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3926,"column":38,"line":126},"start":{"byte":3792,"column":0,"line":124}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommSemigroup.{u_3} G] (a b c d : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G (@CommMagma.toMul.{u_3} G (@CommSemigroup.toCommMagma.{u_3} G inst)))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G (@CommMagma.toMul.{u_3} G (@CommSemigroup.toCommMagma.{u_3} G inst))) a b)\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G (@CommMagma.toMul.{u_3} G (@CommSemigroup.toCommMagma.{u_3} G inst))) c d))\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G (@CommMagma.toMul.{u_3} G (@CommSemigroup.toCommMagma.{u_3} G inst)))\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G (@CommMagma.toMul.{u_3} G (@CommSemigroup.toCommMagma.{u_3} G inst))) a c)\n      (@HMul.hMul.{u_3, u_3, u_3} G G G\n        (@instHMul.{u_3} G (@CommMagma.toMul.{u_3} G (@CommSemigroup.toCommMagma.{u_3} G inst))) b d))","statement_sha256":"c642e24f28428f06f54d5c8d180b634e37a369f326d31ef309f13b79fe0cea48","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-90da324263d61c1b8aadaa06c0826e38d5dcec2492b5f8364cdcb1fac85c3970","qualified_name":"mul_mul_mul_comm'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":4028,"column":7,"line":130},"start":{"byte":3928,"column":0,"line":128}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommSemigroup.{u_3} G] (a b c d : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G 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d)","statement_sha256":"3c196818ac19fef41eac905124f65e6c9982408f706b20b338090c413ff5bcbb","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-521056982127b12c3d542b613e742da1e2b389b9b77665ac9071a61460790f19","qualified_name":"mul_ne_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":8349,"column":64,"line":248},"start":{"byte":8270,"column":0,"line":247}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M]\n  [@IsLeftCancelMul.{u_4} M\n      (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst)))]\n  {a b : M},\n  Iff\n    (@Ne.{u_4 + 1} M\n      (@HMul.hMul.{u_4, u_4, u_4} M M M\n        (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))\n        a b)\n      a)\n    (@Ne.{u_4 + 1} M b\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M\n          (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M 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y)","statement_sha256":"b605fc45b061a76841edda888634a76c593f1491382b2aa610e1f2cd58a64a35","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-d623e72e4bb222519674120138ffd981e728b5812193dc359afa607ecf11810b","qualified_name":"mul_one_eq_id","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3473,"column":16,"line":108},"start":{"byte":3395,"column":0,"line":106}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : MulOneClass.{u_4} M],\n  @Eq.{u_4 + 1} (M → M)\n    (fun x =>\n      @HMul.hMul.{u_4, u_4, u_4} M M M (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M 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M)","statement_sha256":"19d01f7d942dab2ddb6f22da50b7a83b3379b6fa1d73585281e6d108e47fecca","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-58c35a50817a23f8158fae29c4abb15c7e3a8175f3ea922913eaa743a8ed3881","qualified_name":"mul_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":7904,"column":79,"line":230},"start":{"byte":7684,"column":0,"line":228}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : CommMonoid.{u_4} M] (a b : M) (n : Nat),\n  @Eq.{u_4 + 1} M\n    (@HPow.hPow.{u_4, 0, u_4} M Nat M\n      (@instHPow.{u_4, 0} M Nat (@NPow.toPow.{u_4} M 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n))","statement_sha256":"bfd964f39c7250a2f772816927d6ae33ff8424305c5ce0f5b73f285823139fc0","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-71018af9a4d75e8e2b8e29af26e476dd816dfb7e11cb605946deca37cea98170","qualified_name":"mul_pow_sub_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":5074,"column":71,"line":161},"start":{"byte":4893,"column":0,"line":159}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M] {n : Nat},\n  @Ne.{1} Nat n (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n    ∀ (a : M),\n      @Eq.{u_4 + 1} 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n)","statement_sha256":"a219ec729d388dcab95f59d4927869d37a73447b9a4a89b0ef9171c281901c62","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-0ff5fbd5e8801b978d063538e54559e555c9027b7693208e3099b36008d2069a","qualified_name":"mul_right_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3790,"column":39,"line":122},"start":{"byte":3671,"column":0,"line":120}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : CommSemigroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G (@CommMagma.toMul.{u_3} G 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b)","statement_sha256":"9aabd17accbfb6f6554d9e25462ec9fa724b7a2c834d992c0bb3e7601e783949","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-12c794ff8272a21554628b03196e40edd08f3fde6aa226aa8f9d5242976041fc","qualified_name":"mul_right_iterate","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6309,"column":51,"line":190},"start":{"byte":6131,"column":0,"line":187}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M] (a : M) (n : Nat),\n  @Eq.{u_4 + 1} (M → M)\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HMul.hMul.{u_4, u_4, u_4} M M M\n          (@instHMul.{u_4} M\n            (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))\n          x a)\n      n)\n    fun x =>\n    @HMul.hMul.{u_4, u_4, u_4} M M M\n      (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst)))) x\n      (@HPow.hPow.{u_4, 0, u_4} M Nat M (@instHPow.{u_4, 0} M Nat (@NPow.toPow.{u_4} M (@Monoid.toNPow.{u_4} M inst))) a\n        n)","statement_sha256":"d1c0bf0005ef15d94d25de3172543d0febee3e7fadd6939d7de39a9c9cbe3766","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-c7ad20c68ec0777847d998de105cb6be1bedf019f66e71c630300dd20da0a787","qualified_name":"mul_right_iterate_apply","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":6783,"column":79,"line":198},"start":{"byte":6546,"column":0,"line":196}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M] {n : Nat} (a b : M),\n  @Eq.{u_4 + 1} M\n    (@Nat.iterate.{u_4 + 1} M\n      (fun x =>\n        @HMul.hMul.{u_4, u_4, 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a))","statement_sha256":"d27d303413921365c0e2c1ee6d31391cc79f897c051b8465cec5bc9225f376b9","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-529b565b2d961fd7778bd485e1749ff82c7c04ce7df55b782d1cb30429f17199","qualified_name":"mul_self_zpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":27353,"column":39,"line":835},"start":{"byte":27214,"column":0,"line":833}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (a : G) (n : Int),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        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n))","statement_sha256":"b16587c0ab3986d8f50fa8ea176d0cb10b7ad92689e38f6380f869e489b8176e","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-5787949fabb5df9f3362d4a2bbea5306ad13625c0ad98168a4f79d70cb7ba993","qualified_name":"mul_zpow_neg_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":13736,"column":45,"line":440},"start":{"byte":13561,"column":0,"line":438}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        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1)))))","statement_sha256":"b0f87efa3c29c9062dd86cf43124ac387760b374b88dd59d293bf38dc3bf472b","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-dec813579db0be723e03ace45f9e7146c02c84a20074eff779b7f259ea2942fd","qualified_name":"mul_zsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":15854,"column":23,"line":496},"start":{"byte":15833,"column":2,"line":496}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a : α) (m n : Int),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{0, u_1, u_1} Int α α\n      (@instHSMul.{0, u_1} Int α\n   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a))","statement_sha256":"b71b791a58b29e87b66a07e7761b79e076efcda5729d1cb6088ae2070f3ff67d","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-07c2200ab1dd40a95c2b6bcd26531afb54dd65cd2c75aa9d8badbd1988c608ce","qualified_name":"multiplicative_of_symm_of_total","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":35636,"column":78,"line":1037},"start":{"byte":34722,"column":0,"line":1022}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : Monoid.{u_2} β] (p r : α → α → Prop) [@Std.Total.{u_1 + 1} α r] (f : α → α → β)\n  [@Std.Symm.{u_1 + 1} α p],\n  (∀ {a b : α},\n      p a b →\n        @Eq.{u_2 + 1} β\n          (@HMul.hMul.{u_2, u_2, u_2} β β β\n            (@instHMul.{u_2} β\n              (@MulOne.toMul.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst))))\n            (f a b) (f b a))\n          (@OfNat.ofNat.{u_2} β (nat_lit 1)\n            (@One.toOfNat1.{u_2} β\n              (@MulOne.toOne.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst)))))) →\n    (∀ {a b c : α},\n        r a b →\n          r b c →\n            p a b →\n              p b c →\n                p a c →\n                  @Eq.{u_2 + 1} β (f a c)\n                    (@HMul.hMul.{u_2, u_2, u_2} β β β\n                      (@instHMul.{u_2} β\n                        (@MulOne.toMul.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst))))\n                      (f a b) (f b c))) →\n      ∀ {a b c : α},\n        p a b →\n          p b c →\n            p a c →\n              @Eq.{u_2 + 1} β (f a c)\n                (@HMul.hMul.{u_2, u_2, u_2} β β β\n                  (@instHMul.{u_2} β\n                    (@MulOne.toMul.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst))))\n                  (f a b) (f b c))","statement_sha256":"d2a7784c8ebe02b1ee801334f4b332af120e11282d689f649ea88212b92e4264","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-14423032a51f96f089cf464e800fc3fc468098fe0377771b05126c1cd0c24e3f","qualified_name":"multiplicative_of_symmetric_of_total","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":35910,"column":77,"line":1042},"start":{"byte":35742,"column":0,"line":1041}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : Monoid.{u_2} β] (p r : α → α → Prop) [@Std.Total.{u_1 + 1} α r] (f : α → α → β)\n  [@Std.Symm.{u_1 + 1} α p],\n  (∀ {a b : α},\n      p a b →\n        @Eq.{u_2 + 1} β\n          (@HMul.hMul.{u_2, u_2, u_2} β β β\n            (@instHMul.{u_2} β\n              (@MulOne.toMul.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst))))\n            (f a b) (f b a))\n          (@OfNat.ofNat.{u_2} β (nat_lit 1)\n            (@One.toOfNat1.{u_2} β\n              (@MulOne.toOne.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst)))))) →\n    (∀ {a b c : α},\n        r a b →\n          r b c →\n            p a b →\n              p b c →\n                p a c →\n                  @Eq.{u_2 + 1} β (f a c)\n                    (@HMul.hMul.{u_2, u_2, u_2} β β β\n                      (@instHMul.{u_2} β\n                        (@MulOne.toMul.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst))))\n                      (f a b) (f b c))) →\n      ∀ {a b c : α},\n        p a b →\n          p b c →\n            p a c →\n              @Eq.{u_2 + 1} β (f a c)\n                (@HMul.hMul.{u_2, u_2, u_2} β β β\n                  (@instHMul.{u_2} β\n                    (@MulOne.toMul.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst))))\n                  (f a b) (f b c))","statement_sha256":"d2a7784c8ebe02b1ee801334f4b332af120e11282d689f649ea88212b92e4264","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-c8761d2aa555bf8fcd02c72f9838899061209548df4f9cd39cd071eebf310892","qualified_name":"multiplicative_of_total","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":37188,"column":51,"line":1059},"start":{"byte":35912,"column":0,"line":1044}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : Monoid.{u_2} β] (r : α → α → Prop) [@Std.Total.{u_1 + 1} α r] (f : α → α → β)\n  (p : α → Prop),\n  (∀ {a b : α},\n      p a →\n        p b →\n          @Eq.{u_2 + 1} β\n            (@HMul.hMul.{u_2, u_2, u_2} β β β\n              (@instHMul.{u_2} β\n                (@MulOne.toMul.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst))))\n              (f a b) (f b a))\n            (@OfNat.ofNat.{u_2} β (nat_lit 1)\n              (@One.toOfNat1.{u_2} β\n                (@MulOne.toOne.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst)))))) →\n    (∀ {a b c : α},\n        r a b →\n          r b c →\n            p a →\n              p b →\n                p c →\n                  @Eq.{u_2 + 1} β (f a c)\n                    (@HMul.hMul.{u_2, u_2, u_2} β β β\n                      (@instHMul.{u_2} β\n                        (@MulOne.toMul.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst))))\n                      (f a b) (f b c))) →\n      ∀ {a b c : α},\n        p a →\n          p b →\n            p c →\n              @Eq.{u_2 + 1} β (f a c)\n                (@HMul.hMul.{u_2, u_2, u_2} β β β\n                  (@instHMul.{u_2} β\n                    (@MulOne.toMul.{u_2} β (@MulOneClass.toMulOne.{u_2} β (@Monoid.toMulOneClass.{u_2} β inst))))\n                  (f a b) (f b c))","statement_sha256":"69e914f5e3a895b240882a52e3da53d4974f68791b72e836ba032d179419e47a","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-802f1655f4a195e304387ef9aa75cd56d39e0138946607865ccd98ff9bda9680","qualified_name":"natAbs_nsmul_eq_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":25683,"column":49,"line":797},"start":{"byte":25636,"column":2,"line":797}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a : G} {n : Int},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HSMul.hSMul.{0, u_3, u_3} Nat G G\n        (@instHSMul.{0, u_3} Nat G\n          (@NSMul.toSMul.{u_3} G\n            (@AddMonoid.toNSMul.{u_3} G (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst)))))\n        n.natAbs a)\n      (@OfNat.ofNat.{u_3} G (nat_lit 0)\n        (@Zero.toOfNat0.{u_3} G\n          (@NegZeroClass.toZero.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        n a)\n      (@OfNat.ofNat.{u_3} G (nat_lit 0)\n        (@Zero.toOfNat0.{u_3} G\n          (@NegZeroClass.toZero.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst)))))))","statement_sha256":"b10f586e4a569fc4344b236493e5ca2d8c37a67c11720ad52e258a074730e4a3","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-5ae080254ab69bafb9bcf9568df7a2b470e72952138fd3f16a4e4dd455fe95fe","qualified_name":"natCast_sub_natCast_zsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":27665,"column":39,"line":843},"start":{"byte":27628,"column":2,"line":843}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G) (m n : Nat),\n  @Eq.{u_3 + 1} G\n    (@HSMul.hSMul.{0, u_3, u_3} Int G G\n      (@instHSMul.{0, u_3} Int G\n        (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n      (@HSub.hSub.{0, 0, 0} Int Int Int (@instHSub.{0} Int Int.instSub) (@Nat.cast.{0} Int instNatCastInt m)\n        (@Nat.cast.{0} Int instNatCastInt n))\n      a)\n    (@HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst)))\n      (@HSMul.hSMul.{0, u_3, u_3} Nat G G\n        (@instHSMul.{0, u_3} Nat G\n          (@NSMul.toSMul.{u_3} G\n            (@AddMonoid.toNSMul.{u_3} G (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst)))))\n        m a)\n      (@HSMul.hSMul.{0, u_3, u_3} Nat G G\n        (@instHSMul.{0, u_3} Nat G\n          (@NSMul.toSMul.{u_3} G\n            (@AddMonoid.toNSMul.{u_3} G (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst)))))\n        n a))","statement_sha256":"e5526ef4271f81a0b3750a489fcca99ddd754f380ef9baa0549f52760bc3fbdf","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-fa37f8156cca02e9fd54573f005d5b0399ce6bc60e512215c42b1755e32b32e1","qualified_name":"natCast_sub_one_zsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":27842,"column":35,"line":847},"start":{"byte":27809,"column":2,"line":847}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G) (n : Nat),\n  @Eq.{u_3 + 1} G\n    (@HSMul.hSMul.{0, u_3, u_3} Int G G\n      (@instHSMul.{0, u_3} Int G\n        (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n      (@HSub.hSub.{0, 0, 0} Int Int Int (@instHSub.{0} Int Int.instSub) (@Nat.cast.{0} Int instNatCastInt n)\n        (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))\n      a)\n    (@HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst)))\n      (@HSMul.hSMul.{0, u_3, u_3} Nat G G\n        (@instHSMul.{0, u_3} Nat G\n          (@NSMul.toSMul.{u_3} G\n            (@AddMonoid.toNSMul.{u_3} G (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst)))))\n        n a)\n      a)","statement_sha256":"dfdc1082a185647c591f067fd3427852cc538e95426a558053cba3f2916e9025","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-ca6dddf1b38b07e52133b226fcd93d1108b03cd433d8bb28cc2d491b3fd2b775","qualified_name":"ne_iff_ne_of_add_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9177,"column":13,"line":280},"start":{"byte":9166,"column":2,"line":280}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : AddCancelCommMonoid.{u_1} α] {a b c d : α},\n  @Eq.{u_1 + 1} α\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@AddCommMonoid.toAddMonoid.{u_1} α (@AddCancelCommMonoid.toAddCommMonoid.{u_1} α inst))))))\n        a b)\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@AddCommMonoid.toAddMonoid.{u_1} α (@AddCancelCommMonoid.toAddCommMonoid.{u_1} α inst))))))\n        c d) →\n    Iff (@Ne.{u_1 + 1} α a c) (@Ne.{u_1 + 1} α b d)","statement_sha256":"2bdc9e434a0e7d00218db22b174bbe0f28ad485f1378c4256e94d2003e00e05b","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-a40cbc6f354a9ac12467c9ada6e824c35d728e082fa67c22da70f2a337568dc3","qualified_name":"ne_iff_ne_of_mul_eq_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9262,"column":98,"line":280},"start":{"byte":9164,"column":0,"line":280}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : CancelCommMonoid.{u_1} α] {a b c d : α},\n  @Eq.{u_1 + 1} α\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@CommMonoid.toMonoid.{u_1} α (@CancelCommMonoid.toCommMonoid.{u_1} α inst))))))\n        a b)\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@CommMonoid.toMonoid.{u_1} α (@CancelCommMonoid.toCommMonoid.{u_1} α inst))))))\n        c d) →\n    Iff (@Ne.{u_1 + 1} α a c) (@Ne.{u_1 + 1} α b d)","statement_sha256":"75d879f09bd9458fc28f80c071a412e3f95393df8629e00a45fe104123dc67c0","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-36b6e5d484789669996a5c2b8f6ea91484c2c63e2344500351a6f2e7224f6f21","qualified_name":"neg_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16581,"column":21,"line":522},"start":{"byte":16562,"column":2,"line":522}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Neg.neg.{u_1} α\n      (@NegZeroClass.toNeg.{u_1} α\n        (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n          (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α\n                  (@SubtractionMonoid.toSubNegMonoid.{u_1} α\n                    (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))))\n        a b))\n    (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n      (@instHAdd.{u_1} α\n        (@AddZero.toAdd.{u_1} α\n          (@AddZeroClass.toAddZero.{u_1} α\n            (@AddMonoid.toAddZeroClass.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α\n                (@SubtractionMonoid.toSubNegMonoid.{u_1} α\n                  (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))))\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a)\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        b))","statement_sha256":"37455a57e4c65941f0a7cd723dba87a6d9beb31ed69e1790ee92fd896a107f72","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-58c1ca0381810d4c9075dac0fcbbdd6cf5382ea8e72e4ecf2a94875b8cd4f44f","qualified_name":"neg_add'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16959,"column":13,"line":536},"start":{"byte":16948,"column":2,"line":536}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Neg.neg.{u_1} α\n      (@NegZeroClass.toNeg.{u_1} α\n        (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n          (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α\n                  (@SubtractionMonoid.toSubNegMonoid.{u_1} α\n                    (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))))\n        a b))\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        (@SubNegMonoid.toSub.{u_1} α\n          (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a)\n      b)","statement_sha256":"d473f6b9bb27bfe8b24b946a06a1612735a00b4bde236248be45bee781cc611c","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-6ceac455d2d93a038f84d9e1d7e01838a6b6b87149e996c7801a56bcb5f885eb","qualified_name":"neg_add_eq_iff_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21461,"column":13,"line":680},"start":{"byte":21450,"column":2,"line":680}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          a)\n        b)\n      c)\n    (@Eq.{u_3 + 1} G b\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a c))","statement_sha256":"522a248ac1901617ddf396ec989aeb1ee6f1562a7ba4a3d29113983e423150ac","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-59dd30f717e9f62f92cebfa8241f74cdd9758a0887db8c5949982ffa691766ce","qualified_name":"neg_add_eq_neg_add_iff_add_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":34129,"column":13,"line":1004},"start":{"byte":34118,"column":2,"line":1004}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c d : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G\n                (@SubtractionCommMonoid.toSubtractionMonoid.{u_3} G\n                  (@AddCommGroup.toDivisionAddCommMonoid.{u_3} G inst)))))\n          b)\n        a)\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G\n                (@SubtractionCommMonoid.toSubtractionMonoid.{u_3} G\n                  (@AddCommGroup.toDivisionAddCommMonoid.{u_3} G inst)))))\n          d)\n        c))\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        a d)\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        c b))","statement_sha256":"d072d5fd1be0a15e2d8140a0a4cf61faa7cfa7f602398b86dc253ffbdd9ca2c5","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-b80ef32d84ddba126d77010c904abac7237115489dd9362ae0d435bc45344351","qualified_name":"neg_add_eq_of_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19651,"column":13,"line":628},"start":{"byte":19640,"column":2,"line":628}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G b\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a c) →\n    @Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          a)\n        b)\n      c","statement_sha256":"147b5a52ac3155b8661cfb7ce04c2f2e98b5399a9e7a4003aaa3f18660b70850","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-8b2e345b8ca4b532f8f87915304b24e7ab8c5703040ba813f88006b378ff9944","qualified_name":"neg_add_eq_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16805,"column":13,"line":531},"start":{"byte":16794,"column":2,"line":531}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n      (@instHAdd.{u_1} α\n        (@AddZero.toAdd.{u_1} α\n          (@AddZeroClass.toAddZero.{u_1} α\n            (@AddMonoid.toAddZeroClass.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α\n                (@SubtractionMonoid.toSubNegMonoid.{u_1} α\n                  (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))))\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a)\n      b)\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        (@SubNegMonoid.toSub.{u_1} α\n          (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      b a)","statement_sha256":"d03b7efa9e09c9147f5a31296ddbd63bd64d942fdb13f0be6daaa4c7573222c7","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-b1434410ba400dde774f3ddda979b635e62a9c1c820eb9d223c5dd847428d303","qualified_name":"neg_add_eq_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":21921,"column":13,"line":691},"start":{"byte":21910,"column":2,"line":691}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        (@Neg.neg.{u_3} G\n          (@NegZeroClass.toNeg.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n          a)\n        b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 0)\n        (@Zero.toOfNat0.{u_3} G\n          (@NegZeroClass.toZero.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst)))))))\n    (@Eq.{u_3 + 1} G a b)","statement_sha256":"55de63cf879180976c82e90b2b0974fe69d9dba758b1951bc2f3b77623a23582","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-0782efd2e6e54c7098b55d8e739f0d0a6cf1fdc51eedab471c624cde0d4cb3fc","qualified_name":"neg_bijective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9469,"column":13,"line":292},"start":{"byte":9458,"column":2,"line":292}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : InvolutiveNeg.{u_3} G],\n  @Function.Bijective.{u_3 + 1, u_3 + 1} G G (@Neg.neg.{u_3} G (@InvolutiveNeg.toNeg.{u_3} G 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G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@Neg.neg.{u_3} G\n        (@NegZeroClass.toNeg.{u_3} G\n          (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n        a)\n      b)\n    (@Eq.{u_3 + 1} G\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        a b)\n      (@OfNat.ofNat.{u_3} G (nat_lit 0)\n        (@Zero.toOfNat0.{u_3} G\n          (@NegZeroClass.toZero.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G 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(@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst)))\n        a)\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@NegZeroClass.toZero.{u_1} α\n            (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst))))))\n    (@Eq.{u_1 + 1} α a\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@NegZeroClass.toZero.{u_1} α\n            (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst))))))","statement_sha256":"dc34c1aa9077ad9881ee47d19bf1f5b88a416558cd7dee4012261c8103019b1c","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-505f2254c28890525428ec6f3ad1dc79c551cbd4a963230edd2a990f1870dab9","qualified_name":"neg_inj","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9829,"column":28,"line":304},"start":{"byte":9803,"column":2,"line":304}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : InvolutiveNeg.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G (@Neg.neg.{u_3} G (@InvolutiveNeg.toNeg.{u_3} G inst) a)\n      (@Neg.neg.{u_3} G 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inst))))))","statement_sha256":"8e86dc2fe53834d56eb16b0b835a64be20865ef2d60afbd4b28d1b735b471bee","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-a77ecda32f9d72c9da87346426a62c1de2a5dfca068e69fe2d1b48ce3184677e","qualified_name":"neg_neg_sub_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17104,"column":13,"line":542},"start":{"byte":17093,"column":2,"line":542}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Neg.neg.{u_1} α\n      (@NegZeroClass.toNeg.{u_1} α\n        (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n          (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α\n          (@SubNegMonoid.toSub.{u_1} α\n            (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        (@Neg.neg.{u_1} α\n          (@NegZeroClass.toNeg.{u_1} α\n            (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α\n                (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n          a)\n        (@Neg.neg.{u_1} α\n          (@NegZeroClass.toNeg.{u_1} α\n            (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α\n                (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n          b)))\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      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@Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{0, u_1, u_1} Nat α α\n      (@instHSMul.{0, u_1} Nat α\n        (@NSMul.toSMul.{u_1} α\n          (@AddMonoid.toNSMul.{u_1} α\n            (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))))\n      x\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst)))\n        a))\n    (@Neg.neg.{u_1} α\n      (@NegZeroClass.toNeg.{u_1} α\n        (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst)))\n      (@HSMul.hSMul.{0, u_1, u_1} Nat α α\n        (@instHSMul.{0, u_1} Nat α\n          (@NSMul.toSMul.{u_1} α\n            (@AddMonoid.toNSMul.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))))\n        x a))","statement_sha256":"c8f8d6faa31f8fc9847c9cae8bf18cf8a11724a37380054eb69b83d19f6ff0f9","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-161de2aa978bf3b6ed7d4db1b8538804c31f83f43f01d721b06711f373188e7d","qualified_name":"neg_one_zsmul_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":13592,"column":31,"line":438},"start":{"byte":13563,"column":2,"line":438}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{0, u_1, u_1} Int α α\n      (@instHSMul.{0, u_1} Int α\n        (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n      (@Neg.neg.{0} Int Int.instNegInt (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))))\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))\n        a b))\n    (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n      (@instHAdd.{u_1} α\n        (@AddZero.toAdd.{u_1} α\n          (@AddZeroClass.toAddZero.{u_1} α\n            (@AddMonoid.toAddZeroClass.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))\n      (@HSMul.hSMul.{0, u_1, u_1} Int α α\n        (@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n        (@Neg.neg.{0} Int Int.instNegInt (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1)))) b)\n      (@HSMul.hSMul.{0, u_1, u_1} Int α α\n        (@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n        (@Neg.neg.{0} Int Int.instNegInt (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1)))) a))","statement_sha256":"e58d60a72c443920b5517fcf2c73790d72f6b33fa5bc1b2e13ed45bb431a0425","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-1fbcab9deffc9ee2946e2263c43fb2f9f6cdfa3eb1214824475e35dc8612ffa1","qualified_name":"neg_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":12297,"column":28,"line":401},"start":{"byte":12271,"column":2,"line":401}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Neg.neg.{u_1} α (@SubNegMonoid.toNeg.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))) a b))\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))) b a)","statement_sha256":"12080ab41c337e5f1f41ee1284580d4b3498d81dbfa6e57da74740e1d6f8af31","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-e20749221d700142aaffff5c1123f725161c231e6b88be494f09faad662c1b44","qualified_name":"neg_sub'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16657,"column":13,"line":525},"start":{"byte":16646,"column":2,"line":525}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@Neg.neg.{u_1} α\n      (@NegZeroClass.toNeg.{u_1} α\n        (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n          (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α\n          (@SubNegMonoid.toSub.{u_1} α\n            (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a b))\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        (@SubNegMonoid.toSub.{u_1} α\n          (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a)\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        b))","statement_sha256":"8ec3f4fd7da01a81ee39c56b9730ae2356034773f0b3116a057a2a1797551583","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-887ddd294f51b7916d3c0bebb62f4a314b6466157176055ccd444aae2a56eee2","qualified_name":"neg_sub_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16876,"column":13,"line":534},"start":{"byte":16865,"column":2,"line":534}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        (@SubNegMonoid.toSub.{u_1} α\n          (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a)\n      b)\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        (@SubNegMonoid.toSub.{u_1} α\n          (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        b)\n      a)","statement_sha256":"e86eba6d53e6c6eccfda6f3ef0fd6ff37861dab3ef92595f4cc187c0ad505edd","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-ca273749e1a14aff1d6f2fc6a692af56271eeaad26250017088026ea47ab4f27","qualified_name":"neg_sub_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":12206,"column":13,"line":398},"start":{"byte":12195,"column":2,"line":398}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))\n      (@Neg.neg.{u_1} α (@SubNegMonoid.toNeg.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)) a) b)\n    (@Neg.neg.{u_1} α (@SubNegMonoid.toNeg.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))\n      (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n        (@instHAdd.{u_1} α\n          (@AddZero.toAdd.{u_1} α\n            (@AddZeroClass.toAddZero.{u_1} α\n              (@AddMonoid.toAddZeroClass.{u_1} α\n                (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))))\n        b a))","statement_sha256":"17000b13cc93ad707ee4b3c1ba97dc9fc78ab0b0f374675fbe9f1d239e96f532","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-68967fe5a4d5979c50892de2ab9898f2c85412a1e59469e374e76cfdd40c4c44","qualified_name":"neg_sub_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17031,"column":13,"line":539},"start":{"byte":17020,"column":2,"line":539}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        (@SubNegMonoid.toSub.{u_1} α\n          (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        a)\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        b))\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        (@SubNegMonoid.toSub.{u_1} α\n          (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      b a)","statement_sha256":"3b300830c0a9898fd891e3f65465ecf5b4024f500f17231a0b30eddd454c9f5c","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-e6669840a4bfb77f7b2c27d9d65b5461a154edaa23837cf57bac0b6e1dfa6d57","qualified_name":"neg_surjective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":9593,"column":28,"line":296},"start":{"byte":9567,"column":2,"line":296}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : InvolutiveNeg.{u_3} G],\n  @Function.Surjective.{u_3 + 1, u_3 + 1} G G (@Neg.neg.{u_3} G (@InvolutiveNeg.toNeg.{u_3} G inst))","statement_sha256":"658d38812ea1e959e4f95db9b790068f0db165d099f27a9fec31e0d5665a78f1","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-e3781d001742132bd3fe7d43a8d208c79b0ebea0e478b5672606ba8829c52cc7","qualified_name":"neg_unique","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":7558,"column":13,"line":224},"start":{"byte":7547,"column":2,"line":224}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddCommMonoid.{u_4} M] {x y z : M},\n  @Eq.{u_4 + 1} M\n      (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M\n          (@AddZero.toAdd.{u_4} M\n            (@AddZeroClass.toAddZero.{u_4} M\n              (@AddMonoid.toAddZeroClass.{u_4} M (@AddCommMonoid.toAddMonoid.{u_4} M inst)))))\n        x y)\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M\n          (@AddZero.toZero.{u_4} M\n            (@AddZeroClass.toAddZero.{u_4} M\n              (@AddMonoid.toAddZeroClass.{u_4} M (@AddCommMonoid.toAddMonoid.{u_4} M inst)))))) →\n    @Eq.{u_4 + 1} M\n        (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n          (@instHAdd.{u_4} M\n            (@AddZero.toAdd.{u_4} M\n              (@AddZeroClass.toAddZero.{u_4} M\n                (@AddMonoid.toAddZeroClass.{u_4} M (@AddCommMonoid.toAddMonoid.{u_4} M inst)))))\n          x z)\n        (@OfNat.ofNat.{u_4} M (nat_lit 0)\n          (@Zero.toOfNat0.{u_4} M\n            (@AddZero.toZero.{u_4} M\n              (@AddZeroClass.toAddZero.{u_4} M\n                (@AddMonoid.toAddZeroClass.{u_4} M (@AddCommMonoid.toAddMonoid.{u_4} M inst)))))) →\n      @Eq.{u_4 + 1} M y z","statement_sha256":"4c555b0eaf13b3b874d480354769954a1e7fc941f7f901619e66cbe0f68d38f3","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-5430f8b748d6ef8a7447af56183017ade3c5fb677da0dc609244cb8851e6b8a2","qualified_name":"neg_zsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":13349,"column":38,"line":430},"start":{"byte":13313,"column":2,"line":430}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a : α) (n : Int),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{0, u_1, u_1} Int α α\n      (@instHSMul.{0, u_1} Int α\n        (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n      (@Neg.neg.{0} Int Int.instNegInt n) a)\n    (@Neg.neg.{u_1} α\n      (@NegZeroClass.toNeg.{u_1} α\n        (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst)))\n      (@HSMul.hSMul.{0, u_1, u_1} Int α α\n        (@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n        n a))","statement_sha256":"04c84c4deef90a44bfbb559445be9d429ab2d85fc8cb47ef3dcab418b49745b5","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-5692589990f93815204a8ca8f4fbc629faa7d14768df00b7c1a7dea38ae09d52","qualified_name":"nsmul_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":7707,"column":23,"line":228},"start":{"byte":7686,"column":2,"line":228}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddCommMonoid.{u_4} M] (a b : M) (n : Nat),\n  @Eq.{u_4 + 1} M\n    (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n      (@instHSMul.{0, u_4} Nat M\n        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(@HSMul.hSMul.{0, u_4, u_4} Nat M M\n              (@instHSMul.{0, u_4} Nat M\n                (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M (@AddCommMonoid.toAddMonoid.{u_4} M inst))))\n              x b)))\n      x),\n  @Eq.{u_4 + 1} M\n    (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n      (@instHSMul.{0, u_4} Nat M\n        (@NSMul.toSMul.{u_4} M (@AddMonoid.toNSMul.{u_4} M (@AddCommMonoid.toAddMonoid.{u_4} M inst))))\n      x\n      (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M\n          (@AddZero.toAdd.{u_4} M\n            (@AddZeroClass.toAddZero.{u_4} M\n              (@AddMonoid.toAddZeroClass.{u_4} M (@AddCommMonoid.toAddMonoid.{u_4} M inst)))))\n        a b))\n    (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n      (@instHAdd.{u_4} M\n        (@AddZero.toAdd.{u_4} M\n          (@AddZeroClass.toAddZero.{u_4} M\n            (@AddMonoid.toAddZeroClass.{u_4} M (@AddCommMonoid.toAddMonoid.{u_4} M inst)))))\n      (@HSMul.hSMul.{0, u_4, u_4} Nat M M\n        (@instHSMul.{0, 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x","statement_sha256":"35b37aef11e475a0f6d078ceb1b972281cc7e7814edb28594097941068a7d82d","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-7ed3c813a2f995231ed9b74757163a96754a1ebad970e6640e6aed40b73a70b7","qualified_name":"nsmul_iterate._f","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":7298,"column":13,"line":214},"start":{"byte":7287,"column":2,"line":214}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M] (k x : Nat)\n  (f :\n    @Nat.below.{0}\n      (fun x =>\n        @Eq.{u_4 + 1} (M → M)\n          (@Nat.iterate.{u_4 + 1} 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x_1","statement_sha256":"185354c6f4e21b9d63e419b05e99d20f12044565f818c1fb26f9275422771bc8","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-b88d380af73dda9cb8cda3349ec49567e2b211c817cfd4688cc51ea9c9c3ba36","qualified_name":"nsmul_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18689,"column":23,"line":597},"start":{"byte":18668,"column":2,"line":597}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b : α) (n : Nat),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{0, u_1, u_1} Nat α α\n      (@instHSMul.{0, u_1} Nat α\n  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(@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))))\n        n a)\n      (@HSMul.hSMul.{0, u_1, u_1} Nat α α\n        (@instHSMul.{0, u_1} Nat α\n          (@NSMul.toSMul.{u_1} α\n            (@AddMonoid.toNSMul.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α\n                (@SubtractionMonoid.toSubNegMonoid.{u_1} α\n                  (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))))\n        n b))","statement_sha256":"206c6690857346786ab5e4c0f325742e93489219e304ed948c736421caf57fd0","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-d2a0bb6784c3688c76ea840cc69d51e89ffd026b355950006bb533d6c1538e4f","qualified_name":"nsmul_zero_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14111,"column":28,"line":450},"start":{"byte":14085,"column":2,"line":450}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a : α) (n : Nat),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{0, u_1, u_1} Nat α α\n      (@instHSMul.{0, u_1} Nat α\n   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(@instHSMul.{0, u_1} Nat α\n          (@NSMul.toSMul.{u_1} α\n            (@AddMonoid.toNSMul.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))))\n        n a))","statement_sha256":"64efd7d11112864dd73913dde9ab697012f893b17854bc0e506d35a591fc729d","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-cbf987ed30b52be24ebbaba44e909d3063a76d8db85f47dd77b52e5ecd1dc059","qualified_name":"one_add_zsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":27118,"column":27,"line":830},"start":{"byte":27093,"column":2,"line":830}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G) (n : Int),\n  @Eq.{u_3 + 1} G\n    (@HSMul.hSMul.{0, u_3, u_3} Int G G\n      (@instHSMul.{0, u_3} Int G\n        (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n      (@HAdd.hAdd.{0, 0, 0} Int Int Int (@instHAdd.{0} Int Int.instAdd)\n        (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))) n)\n      a)\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      a\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        n a))","statement_sha256":"f42d83282d9494a5a5e5e1eceb858d1686972b31292c2105f059f50aa008372c","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-7c1d7c620ae042b6e2570e2dbaab3b369b43b776d157b84efe8d0b7c40a61597","qualified_name":"one_div_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":12417,"column":52,"line":405},"start":{"byte":12350,"column":0,"line":404}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@MulOne.toOne.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))) a b))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))) b a)","statement_sha256":"ca12baba725b63ae972345d98f27bce20fb015c27ee1704f3517906c56bcc2e4","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-b718766ae8c806798a04acd5dd80543a6c7645805bcdc072df1c34235e3fdcc1","qualified_name":"one_div_mul_eq_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17863,"column":57,"line":570},"start":{"byte":17791,"column":0,"line":569}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@InvOneClass.toOne.{u_1} α\n              (@DivInvOneMonoid.toInvOneClass.{u_1} α\n                (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))))\n        a)\n      b)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      b a)","statement_sha256":"d961a651b51855a744c43ac42211ee0ee56d6796d7982a7aa94ffdae586ea87e","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-0698cb7424bf288b073238076af84b21d021dc9aa15ed49e334d1be1496f1e1c","qualified_name":"one_div_mul_one_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17260,"column":70,"line":546},"start":{"byte":17175,"column":0,"line":545}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α\n                (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@InvOneClass.toOne.{u_1} α\n              (@DivInvOneMonoid.toInvOneClass.{u_1} α\n                (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))))\n        a)\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α\n          (@DivInvMonoid.toDiv.{u_1} α\n            (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@InvOneClass.toOne.{u_1} α\n              (@DivInvOneMonoid.toInvOneClass.{u_1} α\n                (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))))\n        b))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α\n        (@DivInvMonoid.toDiv.{u_1} α\n          (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α\n              (@DivisionMonoid.toDivInvOneMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst))))))\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α\n                  (@DivisionMonoid.toDivInvMonoid.{u_1} α (@DivisionCommMonoid.toDivisionMonoid.{u_1} α inst)))))))\n        a b))","statement_sha256":"dff455a1cbae0a9d480b4388916afc2b2c45bf99eb2f33a03b9c8c63bc1bc1b3","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-d7e66eafa298368714ed4b09b135f1f93b1b6a653d57e0857f1a83e08bb89801","qualified_name":"one_div_mul_one_div_rev","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":12191,"column":74,"line":396},"start":{"byte":12102,"column":0,"line":395}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HMul.hMul.{u_1, u_1, u_1} α α α\n      (@instHMul.{u_1} α\n        (@MulOne.toMul.{u_1} α\n          (@MulOneClass.toMulOne.{u_1} α\n            (@Monoid.toMulOneClass.{u_1} α\n              (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@MulOne.toOne.{u_1} α\n              (@MulOneClass.toMulOne.{u_1} α\n                (@Monoid.toMulOneClass.{u_1} α\n                  (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))))\n        a)\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@MulOne.toOne.{u_1} α\n              (@MulOneClass.toMulOne.{u_1} α\n                (@Monoid.toMulOneClass.{u_1} α\n                  (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))))\n        b))\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@MulOne.toOne.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))))\n      (@HMul.hMul.{u_1, u_1, u_1} α α α\n        (@instHMul.{u_1} α\n          (@MulOne.toMul.{u_1} α\n            (@MulOneClass.toMulOne.{u_1} α\n              (@Monoid.toMulOneClass.{u_1} α\n                (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))))\n        b a))","statement_sha256":"2c8744b83c804f1d979b64d67d8ab3f31c6e782302fbcac73ee4826852873ae0","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-bde6978e2fae5e558af5d9e801dffe8d498551902dfde60714f27e9e2b53f73b","qualified_name":"one_div_one","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":11083,"column":11,"line":357},"start":{"byte":11016,"column":0,"line":355}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : DivInvOneMonoid.{u_3} G],\n  @Eq.{u_3 + 1} G\n    (@HDiv.hDiv.{u_3, u_3, u_3} G G G\n      (@instHDiv.{u_3} G (@DivInvMonoid.toDiv.{u_3} G (@DivInvOneMonoid.toDivInvMonoid.{u_3} G inst)))\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G (@InvOneClass.toOne.{u_3} G (@DivInvOneMonoid.toInvOneClass.{u_3} G inst))))\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G (@InvOneClass.toOne.{u_3} G (@DivInvOneMonoid.toInvOneClass.{u_3} G inst)))))\n    (@OfNat.ofNat.{u_3} G (nat_lit 1)\n      (@One.toOfNat1.{u_3} G (@InvOneClass.toOne.{u_3} G (@DivInvOneMonoid.toInvOneClass.{u_3} G inst))))","statement_sha256":"e5252bb8a9ff971204a8ca1e0a5739b267fe53f018d89d8601208761dd590dff","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-133aa74d9c4736cbb6e1a4257d38363bc4a14ebb5bad035c863177ee561e3d6e","qualified_name":"one_div_one_div","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":12486,"column":52,"line":408},"start":{"byte":12419,"column":0,"line":407}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α),\n  @Eq.{u_1 + 1} α\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α 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a","statement_sha256":"0e9e5ec2887d99844960f04024ceefef2012ab705d7dfadb5dba437744f16471","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-e13bfb40c75c679a7955ba24700febb4d949e37b1db9ad7b6cf9da4ad13efa36","qualified_name":"one_div_pow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14210,"column":97,"line":451},"start":{"byte":14083,"column":0,"line":450}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α) (n : Nat),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Nat α\n      (@instHPow.{u_1, 0} α Nat\n        (@NPow.toPow.{u_1} α\n          (@Monoid.toNPow.{u_1} α (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@InvOneClass.toOne.{u_1} α\n              (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))))\n        a)\n      n)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))))\n      (@HPow.hPow.{u_1, 0, u_1} α Nat α\n        (@instHPow.{u_1, 0} α Nat\n          (@NPow.toPow.{u_1} α\n            (@Monoid.toNPow.{u_1} α (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))\n        a n))","statement_sha256":"9dd0a9adb054119603768231b57cb4b0712f6dc02eb0e7a17abebdfa49f0a900","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-851c6fe507e397c99c156612acac7f82d77c5b6860075b709b04efd60336b15a","qualified_name":"one_div_zpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14341,"column":99,"line":454},"start":{"byte":14212,"column":0,"line":453}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α) (n : Int),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n        (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 1)\n          (@One.toOfNat1.{u_1} α\n            (@InvOneClass.toOne.{u_1} α\n              (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))))\n        a)\n      n)\n    (@HDiv.hDiv.{u_1, u_1, u_1} α α α\n      (@instHDiv.{u_1} α (@DivInvMonoid.toDiv.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))))\n      (@HPow.hPow.{u_1, 0, u_1} α Int α\n        (@instHPow.{u_1, 0} α Int\n          (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n        a n))","statement_sha256":"8034173c0d6ec1922c8e03c2921213fcb83af471e45b8b4e15e6301a392052a0","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-8eaaeae1ee480bd738eecc3c695c9858544a0f421eb4d640bb312304ff685717","qualified_name":"one_eq_inv","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14764,"column":26,"line":468},"start":{"byte":14663,"column":0,"line":466}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] {a : α},\n  Iff\n    (@Eq.{u_1 + 1} α\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))))\n      (@Inv.inv.{u_1} α\n        (@InvOneClass.toInv.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n        a))\n    (@Eq.{u_1 + 1} α a\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst))))))","statement_sha256":"19e11db2beb896c59e421a0f6244ce305009bdb23bda5742d0c7ca1464ae4369","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-5f4eebffbd84353031949d57d512dccbdb9cde1364b2f688c6f154a217817344","qualified_name":"one_mul_eq_id","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":3393,"column":16,"line":104},"start":{"byte":3315,"column":0,"line":102}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : MulOneClass.{u_4} M],\n  @Eq.{u_4 + 1} (M → M)\n    (fun x =>\n      @HMul.hMul.{u_4, u_4, u_4} M M M (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst)))\n        (@OfNat.ofNat.{u_4} M (nat_lit 1)\n          (@One.toOfNat1.{u_4} M (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M inst))))\n        x)\n    (@id.{u_4 + 1} M)","statement_sha256":"ed1c7b9fd9bc15060ae3cf027da57975b7d61bfaacad5201ec0d4b671b2ab82b","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-a62271e8d81f2a972c52443f84096e97690482c452d3540958dd839b54dd9b2d","qualified_name":"one_sub_natCast_zsmul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":28009,"column":35,"line":851},"start":{"byte":27976,"column":2,"line":851}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G) (n : Nat),\n  @Eq.{u_3 + 1} G\n    (@HSMul.hSMul.{0, u_3, u_3} Int G G\n      (@instHSMul.{0, u_3} Int G\n        (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n      (@HSub.hSub.{0, 0, 0} Int Int Int (@instHSub.{0} Int Int.instSub)\n        (@OfNat.ofNat.{0} Int (nat_lit 1) (@instOfNat (nat_lit 1))) (@Nat.cast.{0} Int instNatCastInt n))\n      a)\n    (@HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) a\n      (@HSMul.hSMul.{0, u_3, u_3} Nat G G\n        (@instHSMul.{0, u_3} Nat G\n          (@NSMul.toSMul.{u_3} G\n            (@AddMonoid.toNSMul.{u_3} G (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst)))))\n        n a))","statement_sha256":"fcf79b54d65dc857ea8f2dffe636d227ab45264880c4489670923de4f1030639","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-8c7c030d46f34b23b39214f8706a5a0cee6d2e8676b06b168852ea9a3a0cc462","qualified_name":"one_zpow","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":13309,"column":56,"line":428},"start":{"byte":13127,"column":0,"line":425}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (n : Int),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      (@OfNat.ofNat.{u_1} α (nat_lit 1)\n        (@One.toOfNat1.{u_1} α\n          (@InvOneClass.toOne.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))))\n      n)\n    (@OfNat.ofNat.{u_1} α (nat_lit 1)\n      (@One.toOfNat1.{u_1} α\n        (@InvOneClass.toOne.{u_1} α\n          (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))))","statement_sha256":"9f01aa671419a2ee730e3b37a08805228c15459936fb82985514bd4f65bf5f9b","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-792f97c5d238d33e3f0bb556278c50625bd31552e16cf35a92728cd8418c4e4c","qualified_name":"pow_boole","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":4563,"column":94,"line":149},"start":{"byte":4391,"column":0,"line":147}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M] (P : Prop) [inst_1 : Decidable P] (a : M),\n  @Eq.{u_4 + 1} M\n    (@HPow.hPow.{u_4, 0, u_4} M Nat M (@instHPow.{u_4, 0} M Nat (@NPow.toPow.{u_4} M (@Monoid.toNPow.{u_4} M inst))) a\n      (@ite.{1} Nat P inst_1 (@OfNat.ofNat.{0} Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))\n        (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))\n    (@ite.{u_4 + 1} M P inst_1 a\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M\n          (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))))","statement_sha256":"a56f1836874ef8dfdc5d3886ac1f17ba6b46f4dc14c9f3ab62e2850a8dc66d56","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-069ed98bd69d402f1479a105c782456e2b7eae7801524b67a33da6327c2b53a4","qualified_name":"pow_dite","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":1137,"column":97,"line":37},"start":{"byte":903,"column":0,"line":35}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : Pow.{u_1, u_2} α β] (p : Prop) [inst_1 : Decidable p] (a : α) (b : p → β)\n  (c : Not p → β),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, u_2, u_1} α β α (@instHPow.{u_1, u_2} α β inst) a\n      (@dite.{u_2 + 1} β p inst_1 (fun h => b h) fun h => c h))\n    (@dite.{u_1 + 1} α p inst_1 (fun h => @HPow.hPow.{u_1, u_2, u_1} α β α (@instHPow.{u_1, u_2} α β inst) a (b h))\n      fun h => @HPow.hPow.{u_1, u_2, u_1} α β α (@instHPow.{u_1, u_2} α β inst) a (c 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(@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))\n        (@HPow.hPow.{u_4, 0, u_4} M Nat M (@instHPow.{u_4, 0} M Nat (@NPow.toPow.{u_4} M (@Monoid.toNPow.{u_4} M inst)))\n          a (@HSub.hSub.{0, 0, 0} Nat Nat Nat (@instHSub.{0} Nat instSubNat) n m))\n        (@HPow.hPow.{u_4, 0, u_4} M Nat M (@instHPow.{u_4, 0} M Nat (@NPow.toPow.{u_4} M (@Monoid.toNPow.{u_4} M inst)))\n          a m))\n      (@HPow.hPow.{u_4, 0, u_4} M Nat M (@instHPow.{u_4, 0} M Nat (@NPow.toPow.{u_4} M (@Monoid.toNPow.{u_4} M inst))) a\n        n)","statement_sha256":"a767fd50e81c0f3f8daae7151e6da68a7082e07af02f0b5af59e21200696d4c2","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-bc0ee4e950f5069635b48a5d2de1a880da2c233169de9988e9442a35f0c3be52","qualified_name":"pow_sub_one_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":5256,"column":70,"line":165},"start":{"byte":5076,"column":0,"line":163}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : Monoid.{u_4} M] {n : Nat},\n  @Ne.{1} Nat n (@OfNat.ofNat.{0} Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) →\n    ∀ (a : M),\n      @Eq.{u_4 + 1} 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[@IsRightCancelAdd.{u_4} M\n      (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst)))]\n  {a b : M},\n  Iff\n    (@Eq.{u_4 + 1} M b\n      (@HAdd.hAdd.{u_4, u_4, u_4} M M M\n        (@instHAdd.{u_4} M\n          (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M inst))))\n        a b))\n    (@Eq.{u_4 + 1} M a\n      (@OfNat.ofNat.{u_4} M (nat_lit 0)\n        (@Zero.toOfNat0.{u_4} M\n          (@AddZero.toZero.{u_4} M (@AddZeroClass.toAddZero.{u_4} M (@AddMonoid.toAddZeroClass.{u_4} M 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inst))))))","statement_sha256":"93d517015debf1b6d10f7fc98d1b1e93f1b9a661932345dd487164334a63c955","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-b2cabef9b8b6cba90bda10a0d6f5c02d3698909a1d4818baaeca0a067bde8e6a","qualified_name":"right_ne_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":8904,"column":13,"line":271},"start":{"byte":8893,"column":2,"line":271}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {M : Type u_4} [inst : AddMonoid.{u_4} M]\n  [@IsRightCancelAdd.{u_4} M\n      (@AddZero.toAdd.{u_4} M (@AddZeroClass.toAddZero.{u_4} M 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inst)))]\n  {a b : M},\n  Iff\n    (@Ne.{u_4 + 1} M b\n      (@HMul.hMul.{u_4, u_4, u_4} M M M\n        (@instHMul.{u_4} M (@MulOne.toMul.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))\n        a b))\n    (@Ne.{u_4 + 1} M a\n      (@OfNat.ofNat.{u_4} M (nat_lit 1)\n        (@One.toOfNat1.{u_4} M\n          (@MulOne.toOne.{u_4} M (@MulOneClass.toMulOne.{u_4} M (@Monoid.toMulOneClass.{u_4} M inst))))))","statement_sha256":"18edf40f3740d6dc0e8457dde001c0ce269ff604ac636dc80b732888aacc6724","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-69702896e64dbc09398612b578830eff8bf2364a4a74bbccde9d848057ecbac5","qualified_name":"smul_dite","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":1190,"column":51,"line":39},"start":{"byte":1141,"column":2,"line":39}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul.{u_2, u_1} β α] (p : Prop) [inst_1 : Decidable p] (a : p → α)\n  (b : Not p → α) (c : β),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{u_2, u_1, u_1} β α α (@instHSMul.{u_2, u_1} β α inst) c\n      (@dite.{u_1 + 1} α p inst_1 (fun h => a h) fun h => b h))\n    (@dite.{u_1 + 1} α p inst_1 (fun h => @HSMul.hSMul.{u_2, u_1, u_1} β α α (@instHSMul.{u_2, u_1} β α inst) c (a h))\n      fun h => @HSMul.hSMul.{u_2, u_1, u_1} β α α (@instHSMul.{u_2, u_1} β α inst) c (b h))","statement_sha256":"d992a9f7115b12c802b7eadd1af8d47101f8e1372dd4c91a9f386312e41685d2","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-265445e760e48666f199317e7dba0770e92d68e5d48b51c8b2e7f86a88a7bc30","qualified_name":"smul_ite","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":1617,"column":50,"line":47},"start":{"byte":1569,"column":2,"line":47}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul.{u_2, u_1} β α] (p : Prop) [inst_1 : Decidable p] (a b : α) (c : β),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{u_2, u_1, u_1} β α α (@instHSMul.{u_2, u_1} β α inst) c (@ite.{u_1 + 1} α p inst_1 a b))\n    (@ite.{u_1 + 1} α p inst_1 (@HSMul.hSMul.{u_2, u_1, u_1} β α α (@instHSMul.{u_2, u_1} β α inst) c a)\n      (@HSMul.hSMul.{u_2, u_1, u_1} β α α (@instHSMul.{u_2, u_1} β α inst) c b))","statement_sha256":"b6cae81befa1f34de6993aae17f9275aeb10f50b7b6df80c83ddfaff0c53528f","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-9c8190a1171d0231b3579e9955f472decd7ec060c2786ea98fbfb12d6e91651e","qualified_name":"sub_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":17418,"column":13,"line":554},"start":{"byte":17407,"column":2,"line":554}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b c : α),\n  @Eq.{u_1 + 1} α\n    (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n      (@instHAdd.{u_1} α\n        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α inst))))\n        b c))","statement_sha256":"7ac3a9a1f7f9d15c12614d6903eac5ec5b4778015c8496d9b53151541c503edb","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-a04ab43624370ef261b173196ec7e909cdb7a63fe7c406a67f7bbfdc9c4ac147","qualified_name":"sub_add_add_cancel","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":33155,"column":28,"line":979},"start":{"byte":33129,"column":2,"line":979}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} 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(@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G\n                (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n      a b)","statement_sha256":"5b36c8be4622dea5bbdf119ac8137aa3d02de81261b8d47f0ffe07e76e720110","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-e8566be550fcf8273739e115c926dab0f86f341f4fab45e277a9ad670cea4f29","qualified_name":"sub_add_cancel_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":32568,"column":28,"line":965},"start":{"byte":32542,"column":2,"line":965}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G\n        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b)","statement_sha256":"08747072c90e390d0d32a656d27de4b1df7709324e5867b9e73bb3c8f799f2dc","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-989542765f0f6d9e1c53827f247f9ccba1170fb16c3d71080d45d0d0ce15a091","qualified_name":"sub_add_cancel_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":22623,"column":28,"line":710},"start":{"byte":22597,"column":2,"line":710}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a b : G),\n  @Eq.{u_3 + 1} G\n    (@HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G 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b))","statement_sha256":"2ff02fc1cdd9dfe2395aea4850fa91cffffb10e1ac7e90b7888233f4a5e11da3","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-bc87901238278bb594dea32fe76aace27694f7e07c80170365773e67982f38fa","qualified_name":"sub_eq_iff_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":32172,"column":13,"line":955},"start":{"byte":32161,"column":2,"line":955}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        a b)\n      c)\n    (@Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        a c)\n      b)","statement_sha256":"6414dbd66ff5ec2a7415efafcf163eb3bca9006df76d271e84f65ca1b2f111ab","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-b17d05470c3cf42a16badae95a7f1125616e5abeecb6316be313532cfebee828","qualified_name":"sub_eq_iff_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24765,"column":13,"line":772},"start":{"byte":24754,"column":2,"line":772}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G 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b))","statement_sha256":"9e20d11c5b39c8ce71768489d3650f4dc11572db9572bdb2c416a663062d62c5","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-5fd313ceaf371e1b407459c2280503ae30939740253719ec527b4b5b8e01e48c","qualified_name":"sub_eq_iff_eq_add'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":32064,"column":13,"line":952},"start":{"byte":32053,"column":2,"line":952}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        a b)\n      c)\n    (@Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        b c))","statement_sha256":"abe9d2fb500faa1e75aed84c1523dd69de996893a4739fe14ebfa5a1c3fde5a7","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-80578d422d6b46e3591d3943077e33820699e98d6f702c1be1bd5697f33c427c","qualified_name":"sub_eq_neg_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16734,"column":13,"line":528},"start":{"byte":16723,"column":2,"line":528}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b : α),\n  @Eq.{u_1 + 1} α\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        (@SubNegMonoid.toSub.{u_1} α\n          (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n      a b)\n    (@HAdd.hAdd.{u_1, u_1, u_1} α α α\n      (@instHAdd.{u_1} α\n        (@AddZero.toAdd.{u_1} α\n          (@AddZeroClass.toAddZero.{u_1} α\n            (@AddMonoid.toAddZeroClass.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α\n                (@SubtractionMonoid.toSubNegMonoid.{u_1} α\n                  (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))))\n      (@Neg.neg.{u_1} α\n        (@NegZeroClass.toNeg.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α\n            (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst))))\n        b)\n      a)","statement_sha256":"c4693b1bdea750bf3bb274553b2cd43d3de34d857906becabc5e56b7ad75c386","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-51d6333972686f101b3f2ba0503e249ffd4ec96fbdcb164f04db66283d9efdc0","qualified_name":"sub_eq_neg_self","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":19068,"column":28,"line":611},"start":{"byte":19042,"column":2,"line":611}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G 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inst)))))))","statement_sha256":"667ef55a602355542f5e749e6a8e5fac27d953b9dc5d5e5918f9280322c55573","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-d6a8f94ae1e064d3adf273890ef74f3a22dce2728f71ccaceefae773eef52d07","qualified_name":"sub_eq_of_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":23040,"column":30,"line":720},"start":{"byte":23012,"column":2,"line":720}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n        c b) →\n    @Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) a b)\n      c","statement_sha256":"9bf5ef94e604e4353b59676e08b8221183ee0ee441b293f98019ee94d1a3b668","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-62f9cbff93139a67e63cadc500cda3ff9eaad0c315415f489f96f90db2dced97","qualified_name":"sub_eq_of_eq_add'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":30938,"column":13,"line":918},"start":{"byte":30927,"column":2,"line":918}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c : G},\n  @Eq.{u_3 + 1} G a\n      (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n        (@instHAdd.{u_3} G\n          (@AddZero.toAdd.{u_3} G\n            (@AddZeroClass.toAddZero.{u_3} G\n              (@AddMonoid.toAddZeroClass.{u_3} G\n                (@SubNegMonoid.toAddMonoid.{u_3} G\n                  (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        b c) →\n    @Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        a b)\n      c","statement_sha256":"ec74a3a77425bd4b3307267ec33d1ea57bf0ec16b11260c7a5bc9f01c2ff11b2","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-bd30fb440ab96613ff0407cd9c6939366489228d72b552667aee2c3586428781","qualified_name":"sub_eq_self","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24520,"column":28,"line":766},"start":{"byte":24494,"column":2,"line":766}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))) a b)\n      a)\n    (@Eq.{u_3 + 1} G b\n      (@OfNat.ofNat.{u_3} G (nat_lit 0)\n        (@Zero.toOfNat0.{u_3} G\n          (@NegZeroClass.toZero.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst)))))))","statement_sha256":"1feb92e6bab81e09926c8cef3923f78a8a07c1ca85ae963235959f9453d3b60d","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-14372740bbee0e816952710080fa9fe912a25da081d32c6c0358c5aad35c6d64","qualified_name":"sub_eq_sub_add_sub","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":31619,"column":13,"line":939},"start":{"byte":31608,"column":2,"line":939}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G\n        (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n      a b)\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G\n                (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        c b)\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} G\n          (@SubNegMonoid.toSub.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst))))\n        a c))","statement_sha256":"71136e907243e07f47e7c812e0f248b1ab336ef4a2aed50729f548d06ca8621d","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-fb17ca0572874f1bbde60c7a2be2ee806e11542adb6c1d1f0c5d0ef46a11a847","qualified_name":"sub_eq_sub_iff_add_eq_add","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":33752,"column":13,"line":995},"start":{"byte":33741,"column":2,"line":995}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddCommGroup.{u_3} G] {a b c d : G},\n  Iff\n    (@Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} 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(@AddGroup.toSubNegMonoid.{u_3} G (@AddCommGroup.toAddGroup.{u_3} G inst)))))))\n        c b))","statement_sha256":"2c048b92f90b66227a2626317ac39bce6f7f93a70f2651e4605e68ae88a0c90d","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-c3a84ac3fbb13abe3f595801e21f8cca60b1e78cf3ad87d2fccba560b0078fc7","qualified_name":"sub_eq_sub_iff_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":12501,"column":13,"line":410},"start":{"byte":12490,"column":2,"line":410}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a b c : α) {d : α},\n  Iff\n    (@Eq.{u_1 + 1} α\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))) a b)\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))) c d))\n    (@Eq.{u_1 + 1} α\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))) b a)\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))) d 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b)","statement_sha256":"ad567b760afbf9fc860827686c3ea1d9bced07b58b277fbecd20e0101cffdc50","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-e3d6326ceb39085c0547e5a48d3df88a4c2a1a67cd2148382466150860101066","qualified_name":"sub_eq_zero_of_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24403,"column":47,"line":760},"start":{"byte":24356,"column":0,"line":760}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {a b : G},\n  @Eq.{u_3 + 1} G a b →\n    @Eq.{u_3 + 1} G\n      (@HSub.hSub.{u_3, u_3, u_3} G G G\n        (@instHSub.{u_3} 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c)","statement_sha256":"a06ef4cfa1edc3c487f56de23c9d84e482769a0a61f1b05bee462256ec375a11","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-7183dd0e265760c087ded99b6f84747cd7e140354f6b25b821c135791f1850e1","qualified_name":"sub_left_injective","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":22156,"column":13,"line":698},"start":{"byte":22145,"column":2,"line":698}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {b : G},\n  @Function.Injective.{u_3 + 1, u_3 + 1} G G fun a =>\n    @HSub.hSub.{u_3, u_3, u_3} G G G\n      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a)","statement_sha256":"e73639a1f53183c46dfd57957889fafc609a29251a35284de58ce1840467d69e","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-5b289bdc0472a13912f5e8006b61743aec5809cf7d8f4e95e17a6beb1f3d0506","qualified_name":"sub_sub_sub_cancel_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":24053,"column":28,"line":750},"start":{"byte":24027,"column":2,"line":750}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a b c : G),\n  @Eq.{u_3 + 1} G\n    (@HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G 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d))","statement_sha256":"f2a351888540f9727a8f956cc070d144dacb31476788941370c257b341f28955","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-d7a2387e3ea5a59376bfadcca3974023ff3e47affcbb6660a1651c3eb66c5170","qualified_name":"sub_sub_sub_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":18118,"column":13,"line":581},"start":{"byte":18107,"column":2,"line":581}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionCommMonoid.{u_1} α] (a b c d : α),\n  @Eq.{u_1 + 1} α\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α\n        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c))","statement_sha256":"aa59896b606fea475c596acf00845220203d645fc0a3e0ed69e7c78d8345225e","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-bd0e424309cb34026de9e5300b5716dfeca489fb425ca6c95f5ab16a32a427ac","qualified_name":"sub_zero","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":10949,"column":28,"line":352},"start":{"byte":10923,"column":2,"line":352}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : SubNegZeroMonoid.{u_3} G] (a : G),\n  @Eq.{u_3 + 1} G\n    (@HSub.hSub.{u_3, u_3, u_3} G G G\n      (@instHSub.{u_3} G (@SubNegMonoid.toSub.{u_3} G 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h))","statement_sha256":"4755f2afe467eb499c5c50ebe794ac24ddc0b357d8e24498eeb78756d331bcea","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-3872e9b393a80a860e2d62e43cb1840a4bba2ed28ad26776658103adde80e0f9","qualified_name":"vadd_ite","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":1617,"column":50,"line":47},"start":{"byte":1569,"column":2,"line":47}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {β : Type u_2} [inst : VAdd.{u_2, u_1} β α] (p : Prop) [inst_1 : Decidable p] (a b : α) (c : β),\n  @Eq.{u_1 + 1} α\n    (@HVAdd.hVAdd.{u_2, u_1, u_1} β α α 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M)","statement_sha256":"f07a03babf0c0ebd5c9d5d0111589a4bf0e7086eec6c502fd6d8f8474cf5605a","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-2fe2024b92cea7aa3e111f6deb79cb454ca2213771f015f3a05bf13ca6eb3964","qualified_name":"zero_eq_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14691,"column":28,"line":466},"start":{"byte":14665,"column":2,"line":466}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] {a : α},\n  Iff\n    (@Eq.{u_1 + 1} α\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n        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a)","statement_sha256":"b94a2f28c3614e112cf20660c60af365a91537b15d263b4b3a4fc0b457a7e692","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-9965f65781b4b853af0c882f68321b4a20dc0d4403a6fc8d80e55394b7b2c791","qualified_name":"zpow_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":16072,"column":101,"line":500},"start":{"byte":15956,"column":0,"line":499}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α) (m n : Int),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      (@HPow.hPow.{u_1, 0, u_1} α Int α\n        (@instHPow.{u_1, 0} α Int\n          (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n        a m)\n      n)\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      (@HPow.hPow.{u_1, 0, u_1} α Int α\n        (@instHPow.{u_1, 0} α Int\n          (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n        a n)\n      m)","statement_sha256":"467215e59e9f85d5093ce6e6edf7bd40bfce08594d621ad03909d8ed15041270","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-48188094ffd85d3362a5c05e365d3f30fadf1807568d90ea427b0b70ceb07d3b","qualified_name":"zpow_eq_inv_pow_natAbs","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14535,"column":78,"line":458},"start":{"byte":14343,"column":0,"line":456}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α) {n : Int},\n  @LE.le.{0} Int Int.instLEInt n (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))) →\n    @Eq.{u_1 + 1} α\n      (@HPow.hPow.{u_1, 0, u_1} α Int α\n        (@instHPow.{u_1, 0} α Int\n          (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n        a n)\n      (@HPow.hPow.{u_1, 0, u_1} α Nat α\n        (@instHPow.{u_1, 0} α Nat\n          (@NPow.toPow.{u_1} α\n            (@Monoid.toNPow.{u_1} α (@DivInvMonoid.toMonoid.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst)))))\n        (@Inv.inv.{u_1} α\n          (@InvOneClass.toInv.{u_1} α\n            (@DivInvOneMonoid.toInvOneClass.{u_1} α (@DivisionMonoid.toDivInvOneMonoid.{u_1} α inst)))\n          a)\n        n.natAbs)","statement_sha256":"8673ae22923e572b853dae01f8838adc793911692645c966e855f7c9464d5224","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-36f373db391fdb67ebfcb17fa69b77be72bdacf53aeaf9ffc141e52323042e4c","qualified_name":"zpow_eq_zpow_emod","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":28740,"column":54,"line":866},"start":{"byte":28509,"column":0,"line":862}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {x : G} (m : Int) {n : Int},\n  @Eq.{u_3 + 1} G\n      (@HPow.hPow.{u_3, 0, u_3} G Int G\n        (@instHPow.{u_3, 0} G Int\n          (@ZPow.toPow.{u_3} G (@DivInvMonoid.toZPow.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))\n        x n)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))) →\n    @Eq.{u_3 + 1} G\n      (@HPow.hPow.{u_3, 0, u_3} G Int G\n        (@instHPow.{u_3, 0} G Int\n          (@ZPow.toPow.{u_3} G (@DivInvMonoid.toZPow.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))\n        x m)\n      (@HPow.hPow.{u_3, 0, u_3} G Int G\n        (@instHPow.{u_3, 0} G Int\n          (@ZPow.toPow.{u_3} G (@DivInvMonoid.toZPow.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))\n        x (@HMod.hMod.{0, 0, 0} Int Int Int (@instHMod.{0} Int Int.instMod) m n))","statement_sha256":"6ac71e219b840c25db2bba7fca8d49bc06adc6d63f6187e8332ca7d863c81112","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-caa352677435fa77f4ed5967265ef3b83bb986a087b5eaba5f65a32d85856007","qualified_name":"zpow_eq_zpow_emod'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":28880,"column":65,"line":869},"start":{"byte":28742,"column":0,"line":868}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {x : G} (m : Int) {n : Nat},\n  @Eq.{u_3 + 1} G\n      (@HPow.hPow.{u_3, 0, u_3} G Nat G\n        (@instHPow.{u_3, 0} G Nat\n          (@NPow.toPow.{u_3} G\n            (@Monoid.toNPow.{u_3} G (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst)))))\n        x n)\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))) →\n    @Eq.{u_3 + 1} G\n      (@HPow.hPow.{u_3, 0, u_3} G Int G\n        (@instHPow.{u_3, 0} G Int\n          (@ZPow.toPow.{u_3} G (@DivInvMonoid.toZPow.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))\n        x m)\n      (@HPow.hPow.{u_3, 0, u_3} G Int G\n        (@instHPow.{u_3, 0} G Int\n          (@ZPow.toPow.{u_3} G (@DivInvMonoid.toZPow.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))\n        x (@HMod.hMod.{0, 0, 0} Int Int Int (@instHMod.{0} Int Int.instMod) m (@Nat.cast.{0} Int instNatCastInt n)))","statement_sha256":"39a81d0f757a761f7e990591dbe5f147f6fd0437dee27a1b6ef2fdbeab0c5692","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-5fc9f91c1c011532b8cee7accb2d30739eb44012ce1c502fc1c5b8a7d6f0a136","qualified_name":"zpow_induction_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":29966,"column":20,"line":891},"start":{"byte":29092,"column":0,"line":876}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {g : G} {P : G → Prop},\n  P\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))) →\n    (∀ (a : G),\n        P a →\n          P\n            (@HMul.hMul.{u_3, u_3, u_3} G G G\n              (@instHMul.{u_3} G\n                (@MulOne.toMul.{u_3} G\n                  (@MulOneClass.toMulOne.{u_3} G\n                    (@Monoid.toMulOneClass.{u_3} G\n                      (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n              g a)) →\n      (∀ (a : G),\n          P a →\n            P\n              (@HMul.hMul.{u_3, u_3, u_3} G G G\n                (@instHMul.{u_3} G\n                  (@MulOne.toMul.{u_3} G\n                    (@MulOneClass.toMulOne.{u_3} G\n                      (@Monoid.toMulOneClass.{u_3} G\n                        (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n                (@Inv.inv.{u_3} G\n                  (@InvOneClass.toInv.{u_3} G\n                    (@DivInvOneMonoid.toInvOneClass.{u_3} G\n                      (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n                  g)\n                a)) →\n        ∀ (n : Int),\n          P\n            (@HPow.hPow.{u_3, 0, u_3} G Int G\n              (@instHPow.{u_3, 0} G Int\n                (@ZPow.toPow.{u_3} G (@DivInvMonoid.toZPow.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))\n              g n)","statement_sha256":"f1c1953767f6f6318ed80da188fc907e24f62bc3eb3f846786bc60ffe3f4a21b","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-4f1f71fd5f90f9c934de6c519f0c9467d0eeabb806f220632155d8431b9a606e","qualified_name":"zpow_induction_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":30783,"column":20,"line":908},"start":{"byte":29968,"column":0,"line":893}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] {g : G} {P : G → Prop},\n  P\n      (@OfNat.ofNat.{u_3} G (nat_lit 1)\n        (@One.toOfNat1.{u_3} G\n          (@InvOneClass.toOne.{u_3} G\n            (@DivInvOneMonoid.toInvOneClass.{u_3} G\n              (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst)))))) →\n    (∀ (a : G),\n        P a →\n          P\n            (@HMul.hMul.{u_3, u_3, u_3} G G G\n              (@instHMul.{u_3} G\n                (@MulOne.toMul.{u_3} G\n                  (@MulOneClass.toMulOne.{u_3} G\n                    (@Monoid.toMulOneClass.{u_3} G\n                      (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n              a g)) →\n      (∀ (a : G),\n          P a →\n            P\n              (@HMul.hMul.{u_3, u_3, u_3} G G G\n                (@instHMul.{u_3} G\n                  (@MulOne.toMul.{u_3} G\n                    (@MulOneClass.toMulOne.{u_3} G\n                      (@Monoid.toMulOneClass.{u_3} G\n                        (@DivInvMonoid.toMonoid.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))))\n                a\n                (@Inv.inv.{u_3} G\n                  (@InvOneClass.toInv.{u_3} G\n                    (@DivInvOneMonoid.toInvOneClass.{u_3} G\n                      (@DivisionMonoid.toDivInvOneMonoid.{u_3} G (@Group.toDivisionMonoid.{u_3} G inst))))\n                  g))) →\n        ∀ (n : Int),\n          P\n            (@HPow.hPow.{u_3, 0, u_3} G Int G\n              (@instHPow.{u_3, 0} G Int\n                (@ZPow.toPow.{u_3} G (@DivInvMonoid.toZPow.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))\n              g n)","statement_sha256":"00bbdaad59f40fbc8f0a71cb6e7f8a6c0f76f7b4b862e468577e38a02ef2fda9","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-ac12f9626c64d6f8da5cc5323283e6dfb246a21b825eb105209681379cb48943","qualified_name":"zpow_iterate","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":29090,"column":65,"line":874},"start":{"byte":28882,"column":0,"line":871}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (k : Int) (n : Nat),\n  @Eq.{u_3 + 1} (G → G)\n    (@Nat.iterate.{u_3 + 1} G\n      (fun x =>\n        @HPow.hPow.{u_3, 0, u_3} G Int G\n          (@instHPow.{u_3, 0} G Int\n            (@ZPow.toPow.{u_3} G (@DivInvMonoid.toZPow.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))\n          x k)\n      n)\n    fun x =>\n    @HPow.hPow.{u_3, 0, u_3} G Int G\n      (@instHPow.{u_3, 0} G Int\n        (@ZPow.toPow.{u_3} G (@DivInvMonoid.toZPow.{u_3} G (@Group.toDivInvMonoid.{u_3} G inst))))\n      x (@HPow.hPow.{0, 0, 0} Int Nat Int (@instHPow.{0, 0} Int Nat (@instPowNat.{0} Int Int.instNatPow)) k n)","statement_sha256":"78b3fe5704e31e4208fa8c9a780055b43be609fd3b5ddda832d9481816892556","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-3f1bc926efcded3c0b18f56ba16333431510bb11b4d12f0ab873f9783f2293a0","qualified_name":"zpow_mul","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":15829,"column":7,"line":494},"start":{"byte":15158,"column":0,"line":481}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α) (m n : Int),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      a (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul) m n))\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      (@HPow.hPow.{u_1, 0, u_1} α Int α\n        (@instHPow.{u_1, 0} α Int\n          (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n        a m)\n      n)","statement_sha256":"8dbe5a63439b5549ce1696e84989297d25ad53be825b917cb39b3fb7c5457a74","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-2c45ec4f58f7c9246662333f0f5b69aaa50a5c2c1713a01ab3f4bda04878b53c","qualified_name":"zpow_mul'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":15954,"column":98,"line":497},"start":{"byte":15831,"column":0,"line":496}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : DivisionMonoid.{u_1} α] (a : α) (m n : Int),\n  @Eq.{u_1 + 1} α\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      a (@HMul.hMul.{0, 0, 0} Int Int Int (@instHMul.{0} Int Int.instMul) m n))\n    (@HPow.hPow.{u_1, 0, u_1} α Int α\n      (@instHPow.{u_1, 0} α Int\n        (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n      (@HPow.hPow.{u_1, 0, u_1} α Int α\n        (@instHPow.{u_1, 0} α Int\n          (@ZPow.toPow.{u_1} α (@DivInvMonoid.toZPow.{u_1} α (@DivisionMonoid.toDivInvMonoid.{u_1} α inst))))\n        a n)\n      m)","statement_sha256":"e779080ba8af299b76201a9cbe17967ce23cef0428746d5a34b7d934a2135268","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-d6ce63312b6ccd6aaab2bcbaefa96177b0814dbe942de79c21e98d586fbbe855","qualified_name":"zpow_mul_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":28277,"column":43,"line":856},"start":{"byte":28141,"column":0,"line":855}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : Group.{u_3} G] (a : G) (m n : Int),\n  @Eq.{u_3 + 1} G\n    (@HMul.hMul.{u_3, u_3, u_3} G G G\n      (@instHMul.{u_3} G\n        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 (@HSMul.hSMul.{0, u_1, u_1} Int α α\n        (@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α\n            (@SubNegMonoid.toZSMul.{u_1} α\n              (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))\n        n a)\n      (@HSMul.hSMul.{0, u_1, u_1} Int α α\n        (@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α\n            (@SubNegMonoid.toZSMul.{u_1} α\n              (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))\n        n b))","statement_sha256":"c250a922d0058aa1e61c5fda53cf2aef7112c2ed70b21f8017e1fe6b3029d7b1","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-9ec640b3645e7ddd37dc7b5307643985c619efd4c3f8b59ee7fd3b58a9fcd456","qualified_name":"zsmul_add_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":28154,"column":13,"line":855},"start":{"byte":28143,"column":2,"line":855}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (a : G) (m n : Int),\n  @Eq.{u_3 + 1} G\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        m a)\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        n a))\n    (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n      (@instHAdd.{u_3} G\n        (@AddZero.toAdd.{u_3} G\n          (@AddZeroClass.toAddZero.{u_3} G\n            (@AddMonoid.toAddZeroClass.{u_3} G\n              (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        n a)\n      (@HSMul.hSMul.{0, u_3, u_3} Int G G\n        (@instHSMul.{0, u_3} Int G\n          (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n        m a))","statement_sha256":"9a3eb7a022a8e94265d2db437fad4e9a3e124341558eae2580969a004dba3334","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-9b676ce28f80bfc68bae33d30fa1d453b8700cb08371200ed297bea2f671c8f0","qualified_name":"zsmul_comm","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":15969,"column":13,"line":499},"start":{"byte":15958,"column":2,"line":499}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a : α) (m n : Int),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{0, u_1, u_1} Int α α\n      (@instHSMul.{0, u_1} Int α\n        (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n      n\n      (@HSMul.hSMul.{0, u_1, u_1} Int α α\n        (@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n        m a))\n    (@HSMul.hSMul.{0, u_1, u_1} Int α α\n      (@instHSMul.{0, u_1} Int α\n        (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n      m\n      (@HSMul.hSMul.{0, u_1, u_1} Int α α\n        (@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n        n a))","statement_sha256":"faf3ac0ada1db5ae3151a93cf9e1f55aaa81e05b724c51d2b4b9156a900ba8c8","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-3e774f2434e079e9a70512904b9a8fa0ee07b63ab89a65448ed32c0c112331b4","qualified_name":"zsmul_eq_neg_nsmul_natAbs","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":14356,"column":13,"line":456},"start":{"byte":14345,"column":2,"line":456}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a : α) {n : Int},\n  @LE.le.{0} Int Int.instLEInt n (@OfNat.ofNat.{0} Int (nat_lit 0) (@instOfNat (nat_lit 0))) →\n    @Eq.{u_1 + 1} α\n      (@HSMul.hSMul.{0, u_1, u_1} Int α α\n        (@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n        n a)\n      (@HSMul.hSMul.{0, u_1, u_1} Nat α α\n        (@instHSMul.{0, u_1} Nat α\n          (@NSMul.toSMul.{u_1} α\n            (@AddMonoid.toNSMul.{u_1} α\n              (@SubNegMonoid.toAddMonoid.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))))\n        n.natAbs\n        (@Neg.neg.{u_1} α\n          (@NegZeroClass.toNeg.{u_1} α\n            (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst)))\n          a))","statement_sha256":"295f0482c796d72a4136a47c20077ec46c7eb894db188c0b7c4df9240b370949","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-56b14751fec15654362199d928311e68fa6269153074c97cf53b4f6f05e025de","qualified_name":"zsmul_induction_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":29556,"column":40,"line":881},"start":{"byte":29317,"column":2,"line":879}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {g : G} {P : G → Prop},\n  P\n      (@OfNat.ofNat.{u_3} G (nat_lit 0)\n        (@Zero.toOfNat0.{u_3} G\n          (@NegZeroClass.toZero.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst)))))) →\n    (∀ (a : G),\n        P a →\n          P\n            (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n              (@instHAdd.{u_3} G\n                (@AddZero.toAdd.{u_3} G\n                  (@AddZeroClass.toAddZero.{u_3} G\n                    (@AddMonoid.toAddZeroClass.{u_3} G\n                      (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n              g a)) →\n      (∀ (a : G),\n          P a →\n            P\n              (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n                (@instHAdd.{u_3} G\n                  (@AddZero.toAdd.{u_3} G\n                    (@AddZeroClass.toAddZero.{u_3} G\n                      (@AddMonoid.toAddZeroClass.{u_3} G\n                        (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n                (@Neg.neg.{u_3} G\n                  (@NegZeroClass.toNeg.{u_3} G\n                    (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n                      (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n                  g)\n                a)) →\n        ∀ (n : Int),\n          P\n            (@HSMul.hSMul.{0, u_3, u_3} Int G G\n              (@instHSMul.{0, u_3} Int G\n                (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n              n g)","statement_sha256":"73b60fd3e93f2d27aace74b9556cccb0efd3d52bc69aabf543ab08d52a7fb5d2","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-77f7ca15c0ea5fdd806070a30d6a3dc9ca33e0df608479871e16e5abace40ea0","qualified_name":"zsmul_induction_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":30436,"column":45,"line":898},"start":{"byte":30195,"column":2,"line":896}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] {g : G} {P : G → Prop},\n  P\n      (@OfNat.ofNat.{u_3} G (nat_lit 0)\n        (@Zero.toOfNat0.{u_3} G\n          (@NegZeroClass.toZero.{u_3} G\n            (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n              (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst)))))) →\n    (∀ (a : G),\n        P a →\n          P\n            (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n              (@instHAdd.{u_3} G\n                (@AddZero.toAdd.{u_3} G\n                  (@AddZeroClass.toAddZero.{u_3} G\n                    (@AddMonoid.toAddZeroClass.{u_3} G\n                      (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n              a g)) →\n      (∀ (a : G),\n          P a →\n            P\n              (@HAdd.hAdd.{u_3, u_3, u_3} G G G\n                (@instHAdd.{u_3} G\n                  (@AddZero.toAdd.{u_3} G\n                    (@AddZeroClass.toAddZero.{u_3} G\n                      (@AddMonoid.toAddZeroClass.{u_3} G\n                        (@SubNegMonoid.toAddMonoid.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))))\n                a\n                (@Neg.neg.{u_3} G\n                  (@NegZeroClass.toNeg.{u_3} G\n                    (@SubNegZeroMonoid.toNegZeroClass.{u_3} G\n                      (@SubtractionMonoid.toSubNegZeroMonoid.{u_3} G (@AddGroup.toSubtractionMonoid.{u_3} G inst))))\n                  g))) →\n        ∀ (n : Int),\n          P\n            (@HSMul.hSMul.{0, u_3, u_3} Int G G\n              (@instHSMul.{0, u_3} Int G\n                (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n              n g)","statement_sha256":"bd366167d3c91c9e495eb2911beda42ae7fc09ab3012c8258a40b3308db768de","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-4ec4edbccad2745ecad04fecbef1ad21f3bffd669e2b54f2d506f7271fa57a06","qualified_name":"zsmul_iterate","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":28895,"column":13,"line":871},"start":{"byte":28884,"column":2,"line":871}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (k : Int) (n : Nat),\n  @Eq.{u_3 + 1} (G → G)\n    (@Nat.iterate.{u_3 + 1} G\n      (fun x =>\n        @HSMul.hSMul.{0, u_3, u_3} Int G G\n          (@instHSMul.{0, u_3} Int G\n            (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n          k x)\n      n)\n    fun x =>\n    @HSMul.hSMul.{0, u_3, u_3} Int G G\n      (@instHSMul.{0, u_3} Int G\n        (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n      (@HPow.hPow.{0, 0, 0} Int Nat Int (@instHPow.{0, 0} Int Nat (@instPowNat.{0} Int Int.instNatPow)) k n) x","statement_sha256":"e6749acc2c002ed822f3dc8a156bc195a49cf3775ea06823167d11ffbbeeac57","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-b22b1b61388198bffae2cdce10f8cc45f0f264b58aa681d6b3c9c44cd4379628","qualified_name":"zsmul_iterate._f","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":28895,"column":13,"line":871},"start":{"byte":28884,"column":2,"line":871}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {G : Type u_3} [inst : AddGroup.{u_3} G] (k : Int) (x : Nat)\n  (f :\n    @Nat.below.{0}\n      (fun x =>\n        @Eq.{u_3 + 1} (G → G)\n          (@Nat.iterate.{u_3 + 1} G\n            (fun x =>\n              @HSMul.hSMul.{0, u_3, u_3} Int G G\n                (@instHSMul.{0, u_3} Int G\n                  (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n                k x)\n            x)\n          fun x_1 =>\n          @HSMul.hSMul.{0, u_3, u_3} Int G G\n            (@instHSMul.{0, u_3} Int G\n              (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n            (@HPow.hPow.{0, 0, 0} Int Nat Int (@instHPow.{0, 0} Int Nat (@instPowNat.{0} Int Int.instNatPow)) k x) x_1)\n      x),\n  @Eq.{u_3 + 1} (G → G)\n    (@Nat.iterate.{u_3 + 1} G\n      (fun x =>\n        @HSMul.hSMul.{0, u_3, u_3} Int G G\n          (@instHSMul.{0, u_3} Int G\n            (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n          k x)\n      x)\n    fun x_1 =>\n    @HSMul.hSMul.{0, u_3, u_3} Int G G\n      (@instHSMul.{0, u_3} Int G\n        (@ZSMul.toSMul.{u_3} G (@SubNegMonoid.toZSMul.{u_3} G (@AddGroup.toSubNegMonoid.{u_3} G inst))))\n      (@HPow.hPow.{0, 0, 0} Int Nat Int (@instHPow.{0, 0} Int Nat (@instPowNat.{0} Int Int.instNatPow)) k x) x_1","statement_sha256":"e31c21b0424ec1976e76b1d3d2366a76434c9c5f4d53df2cba6adb1d86035680","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-fe715fdff5cc1a62a3e86551c5bfb5c137db43de4110b15f99488e8e161f5be3","qualified_name":"zsmul_neg","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":13761,"column":23,"line":442},"start":{"byte":13740,"column":2,"line":442}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a : α) (n : Int),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{0, u_1, u_1} Int α α\n      (@instHSMul.{0, u_1} Int α\n        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a))","statement_sha256":"110f681e116e83aa2e3d4b77c0ea353fddb6e76ed8b6111686e02dce68b7b16a","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-7bbfa4c7bb386b7670c6c189c8d07103041cd584b7ed48944ac53f791f379bca","qualified_name":"zsmul_neg'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Algebra.Group.Basic","path":"Mathlib/Algebra/Group/Basic.lean","range":{"end":{"byte":13991,"column":39,"line":447},"start":{"byte":13954,"column":2,"line":447}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : SubtractionMonoid.{u_1} α] (a : α) (n : Int),\n  @Eq.{u_1 + 1} α\n    (@HSMul.hSMul.{0, u_1, u_1} Int α α\n      (@instHSMul.{0, u_1} Int α\n       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(@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α\n            (@SubNegMonoid.toZSMul.{u_1} α\n              (@SubtractionMonoid.toSubNegMonoid.{u_1} α (@SubtractionCommMonoid.toSubtractionMonoid.{u_1} α inst)))))\n        n 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(@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n      n\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@NegZeroClass.toZero.{u_1} α\n            (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst))))))\n    (@OfNat.ofNat.{u_1} α (nat_lit 0)\n      (@Zero.toOfNat0.{u_1} α\n        (@NegZeroClass.toZero.{u_1} α\n          (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α 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α\n        (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n      n\n      (@HSub.hSub.{u_1, u_1, u_1} α α α\n        (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))\n        (@OfNat.ofNat.{u_1} α (nat_lit 0)\n          (@Zero.toOfNat0.{u_1} α\n            (@NegZeroClass.toZero.{u_1} α\n              (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst)))))\n        a))\n    (@HSub.hSub.{u_1, u_1, u_1} α α α\n      (@instHSub.{u_1} α (@SubNegMonoid.toSub.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst)))\n      (@OfNat.ofNat.{u_1} α (nat_lit 0)\n        (@Zero.toOfNat0.{u_1} α\n          (@NegZeroClass.toZero.{u_1} α\n            (@SubNegZeroMonoid.toNegZeroClass.{u_1} α (@SubtractionMonoid.toSubNegZeroMonoid.{u_1} α inst)))))\n      (@HSMul.hSMul.{0, u_1, u_1} Int α α\n        (@instHSMul.{0, u_1} Int α\n          (@ZSMul.toSMul.{u_1} α (@SubNegMonoid.toZSMul.{u_1} α (@SubtractionMonoid.toSubNegMonoid.{u_1} α inst))))\n        n 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