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c","statement_sha256":"2ea16517ca1546013ed4156ea224ffc673ceba7d4ea33c3f8fb623d08d8ec20a","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-464448cdd150a0d4c61c2f75c4192689b571fa7e74f1a89c71e649242558495c","qualified_name":"Function.Injective.linearOrder","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":26290,"column":16,"line":711},"start":{"byte":25229,"column":0,"line":693}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u_2} →\n  {β : Type u_3} →\n    [inst : LinearOrder.{u_3} β] →\n      [inst_1 : LE.{u_2} α] →\n        [inst_2 : LT.{u_2} α] →\n          [inst_3 : Max.{u_2} 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(@PartialOrder.toPreorder.{u_3} β (@LinearOrder.toPartialOrder.{u_3} β inst)))\n                                    (f x) (f y))\n                                  (@LT.lt.{u_2} α inst_2 x y)) →\n                              (∀ (x y : α),\n                                  @Eq.{u_3 + 1} β (f (@Min.min.{u_2} α inst_4 x y))\n                                    (@Min.min.{u_3} β (@LinearOrder.toMin.{u_3} β inst) (f x) (f y))) →\n                                (∀ (x y : α),\n                                    @Eq.{u_3 + 1} β (f (@Max.max.{u_2} α inst_3 x y))\n                                      (@Max.max.{u_3} β (@LinearOrder.toMax.{u_3} β inst) (f x) (f y))) →\n                                  (∀ (x y : α),\n                                      @Eq.{1} Ordering\n                                        (@Ord.compare.{u_3} β (@LinearOrder.toOrd.{u_3} β inst) (f x) (f y))\n                                        (@Ord.compare.{u_2} α inst_5 x y)) →\n                                    LinearOrder.{u_2} α","statement_sha256":"d073876c3b8137a936b26aebe357686631ae862c649579870f405581c745fa39","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-b213ab70b562a185a8da11b490dbf3e4b4ceb0d27b4291998a43d363feb4b333","qualified_name":"Function.Injective.partialOrder","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":25227,"column":72,"line":691},"start":{"byte":24753,"column":0,"line":682}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u_2} →\n  {β : Type u_3} →\n    [inst : PartialOrder.{u_3} β] →\n      [inst_1 : LE.{u_2} α] →\n        [inst_2 : LT.{u_2} α] →\n          (f : α → β) →\n            @Function.Injective.{u_2 + 1, u_3 + 1} α β f →\n              (∀ {x y : α},\n                  Iff (@LE.le.{u_3} β (@Preorder.toLE.{u_3} β (@PartialOrder.toPreorder.{u_3} β inst)) (f x) (f y))\n                    (@LE.le.{u_2} α inst_1 x y)) →\n                (∀ {x y : α},\n                    Iff (@LT.lt.{u_3} β (@Preorder.toLT.{u_3} β (@PartialOrder.toPreorder.{u_3} β inst)) (f x) (f y))\n                      (@LT.lt.{u_2} α inst_2 x y)) →\n                  PartialOrder.{u_2} α","statement_sha256":"dc16e38e14adfa6620ea625357837a4a4eae68c61c86cc4adfb0a805cae18adc","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-afb71e0355306351ed041c232ac33d29a55139af0ec3cf1548026a9c620dbd78","qualified_name":"Function.Injective.preorder","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":24751,"column":49,"line":680},"start":{"byte":24259,"column":0,"line":670}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u_2} →\n  {β : Type u_3} →\n    [inst : Preorder.{u_3} β] →\n      [inst_1 : LE.{u_2} α] →\n        [inst_2 : LT.{u_2} α] →\n          (f : α → β) →\n            (∀ {x y : α}, Iff (@LE.le.{u_3} β (@Preorder.toLE.{u_3} β inst) (f x) (f y)) (@LE.le.{u_2} α inst_1 x y)) →\n              (∀ {x y : α},\n                  Iff (@LT.lt.{u_3} β (@Preorder.toLT.{u_3} β inst) (f x) (f y)) (@LT.lt.{u_2} α inst_2 x y)) →\n                Preorder.{u_2} 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(@Preorder.toLE.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) b a → @Eq.{u_1 + 1} α a b","statement_sha256":"bd11071736091023173f8beb10fbc9f2c9a05eecab6c64a6476cb3dddf79f12b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-bdd48269f114a9886f34cdade8e2a028e18d0d4062999577a664e2fbd59935bf","qualified_name":"LE.le.antisymm'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7174,"column":19,"line":212},"start":{"byte":7157,"column":2,"line":212}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : PartialOrder.{u_1} α] {a b : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} 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PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b →\n    Not (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b) → @Eq.{u_2 + 1} α a 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PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a →\n    Or (@Eq.{u_2 + 1} α a b) (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a)","statement_sha256":"cf31096192f6b389ee60b4abda8ec254b5997d495dd436bf05e8422428c4ba2a","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-7e6b038fd787f83b5d110baded40b0afa9ba4d96f9e3781c7626b70e8f5fd161","qualified_name":"LE.le.eq_or_lt_dec","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":9231,"column":77,"line":263},"start":{"byte":9154,"column":0,"line":263}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α}\n  [@DecidableLE.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst))],\n  @LE.le.{u_2} α 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(@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a →\n    Or (@Eq.{u_2 + 1} α a b) (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a)","statement_sha256":"dffe636ff6b0ca8fd75c06ef2c2596b2401b377ce4f036aec7d2062a156367e9","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-6930696efbd62e26fb8ceb7319a01c8a1adfe2efd718296bd4d813a93a711730","qualified_name":"LE.le.ge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":2275,"column":69,"line":74},"start":{"byte":2206,"column":0,"line":74}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LE.{u_2} α] {a b : α}, @LE.le.{u_2} α inst a b → @GE.ge.{u_2} α inst b a","statement_sha256":"02d6f4a00df9e12e86521d506c6d722f70511fdfbf90c0d1b25f856c0101c77d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-0146b9f6f6b885b699cc2948f835649fd27cf94c3f2ea65c614057dd747473ee","qualified_name":"LE.le.ge_iff_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7573,"column":78,"line":222},"start":{"byte":7473,"column":0,"line":221}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b →\n    Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a) (@Eq.{u_2 + 1} α a b)","statement_sha256":"652a65739bc5e7a7153692ce4f82cc325761583dc515d167f0ac9f4e18dfea35","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-19201c9ca26d0424b3dc3f8785161a65df42ff430282643f7237d5f5ef7701ca","qualified_name":"LE.le.ge_iff_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7493,"column":20,"line":221},"start":{"byte":7475,"column":2,"line":221}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a →\n    Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b) (@Eq.{u_2 + 1} α a b)","statement_sha256":"0bb491673bcc87f61d736ad4353e33af5bb0eaee850a40dc98154dbcb7860070","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-5f4962a0b203a8796572cf0f80465bd8482660c8cc3d7e5ab930c390e69b8dc7","qualified_name":"LE.le.ge_or_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":11575,"column":93,"line":319},"start":{"byte":11462,"column":0,"line":318}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))\n      a b →\n    ∀ (c : α),\n      Or\n        (@LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c)\n        (@LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c b)","statement_sha256":"35f059d20c91c0fef222c69a5f1e934808e191e2c72a9e0bd5891294b0df2207","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-332eb73c4cf2299750f79deac030f8136afb6a7054e8dad3cd939807c6a1080d","qualified_name":"LE.le.ge_or_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":11460,"column":94,"line":316},"start":{"byte":11346,"column":0,"line":315}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))\n      a b →\n    ∀ (c : α),\n      Or\n        (@LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c)\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c b)","statement_sha256":"7a14f6cec89321ad6cdc87467015adf1c2fe7e188e89d3a9d217245046f5595b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-fd854e4c460c7ba007528b69f4addfefca7ca28c575dd6b3d2b125e7e5dddd85","qualified_name":"LE.le.gt_or_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":11344,"column":91,"line":313},"start":{"byte":11233,"column":0,"line":312}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))\n      a b →\n    ∀ (c : α),\n      Or\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c)\n        (@LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c b)","statement_sha256":"f43fc51f0283bb919144bd5e952ffa70347a37a532d3be91635667ba5be7e7e4","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e8f20dc9190fdb4a5aaa9aae5c162b7bac9a9f36c17f980c7baca1ff57baae1a","qualified_name":"LE.le.le_or_ge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":11480,"column":18,"line":318},"start":{"byte":11464,"column":2,"line":318}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))\n      b a →\n    ∀ (c : α),\n      Or\n        (@LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a)\n        (@LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b c)","statement_sha256":"aba781417e9d13f63af9efbbf123bf4e3e8b20a8702ec5205146af8c0332251d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-d98e2c58351cdba941920745736da50b66de9f87b53d1954c37b4374e26d4ff6","qualified_name":"LE.le.le_or_gt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":11364,"column":18,"line":315},"start":{"byte":11348,"column":2,"line":315}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))\n      b a →\n    ∀ (c : α),\n      Or\n        (@LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a)\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b c)","statement_sha256":"e8ddda12e042b43e7b37a5ba4bd8406979901f222bfa82635dac93ff96a80ddc","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f9ea42b6c0bc3d31828ecb54e4db37a05c220e082fab2e1a8be0fd6be0fb23d3","qualified_name":"LE.le.lt_iff_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7364,"column":81,"line":216},"start":{"byte":7261,"column":0,"line":215}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b →\n    Iff (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b) (@Ne.{u_2 + 1} α a b)","statement_sha256":"ce213b1040a58e40a019ad1c84be742a25ad1e29c1700396f064ff53db00cb15","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-97f4d35f20bf9fe245f788fc3910004090101a0bbd13dab81b430ddc9257d49b","qualified_name":"LE.le.lt_iff_ne'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7281,"column":20,"line":215},"start":{"byte":7263,"column":2,"line":215}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a →\n    Iff (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a) (@Ne.{u_2 + 1} α a b)","statement_sha256":"19ae568efe6caa1b0df4a2052ccff45ad2676d86ec6a11e40568483065847db9","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-9e5215229f652b36135fa12e408efd86bb19325460860f194137fdf068bc8d6f","qualified_name":"LE.le.lt_of_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7259,"column":53,"line":213},"start":{"byte":7206,"column":0,"line":213}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : PartialOrder.{u_1} α] {a b : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) a b →\n    @Ne.{u_1 + 1} α a b → @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) a b","statement_sha256":"7b9bfc81cf9737bdf16e0c628a09a7ebfb56074a13154dbf78d7a468d1a0aef7","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-35e652c0a150fe740ae9e3a24d47eddc82d3f6e1077ff56f4f48ddefae5393b1","qualified_name":"LE.le.lt_of_ne'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7225,"column":19,"line":213},"start":{"byte":7208,"column":2,"line":213}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : PartialOrder.{u_1} α] {a b : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) b a →\n    @Ne.{u_1 + 1} α a b → @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) b a","statement_sha256":"a25a1a19a102fe22be55dca212cb4023a0a467d8afabcd49dceffa866da551a3","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a58dedc638e8c4b052eca3d554bfcb8711ffb7f5c6eceb35ad3ef42c600f51ca","qualified_name":"LE.le.lt_of_not_ge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":4273,"column":59,"line":131},"start":{"byte":4214,"column":0,"line":131}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Preorder.{u_1} α] {a b : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) a b →\n    Not (@LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) b a) → @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) a b","statement_sha256":"792db058ebe4cbe4eb02301150f5f2b1c66ef28f3ab13fe3328c1b1facf2f08f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-0e1c1bf04d95844d5bd55275a5b3242bc5f14c63c67aba2c656dc827ae5c46af","qualified_name":"LE.le.lt_or_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":9291,"column":59,"line":264},"start":{"byte":9232,"column":0,"line":264}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : PartialOrder.{u_1} α] {a b : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) a b →\n    Or (@LT.lt.{u_1} α (@Preorder.toLT.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) a b) (@Eq.{u_1 + 1} α a b)","statement_sha256":"0ab123f30aff3eeb8c2f081b6624a80d42b49b99710ebd6792cd73fe955053e0","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-6edc4b87f82c683443b367566b8ebf5f1d8170b05740788e28e63e46c169bc28","qualified_name":"LE.le.lt_or_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":9251,"column":19,"line":264},"start":{"byte":9234,"column":2,"line":264}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : PartialOrder.{u_1} α] {a b : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) b a →\n    Or (@LT.lt.{u_1} α (@Preorder.toLT.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) b a) (@Eq.{u_1 + 1} α a b)","statement_sha256":"0f8f5bb640f5843a06bd76a03cdbdfb1ac3caa51935365a4c622e82224c6aa39","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ae6b0af6b585c91d835556be44300efa2f41756eb8d0b10327016529fca42c49","qualified_name":"LE.le.lt_or_eq_dec","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":9153,"column":77,"line":262},"start":{"byte":9076,"column":0,"line":262}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : PartialOrder.{u_1} α] {a b : α}\n  [@DecidableLE.{u_1} α (@Preorder.toLE.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst))],\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) a b →\n    Or (@LT.lt.{u_1} α (@Preorder.toLT.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) a b) (@Eq.{u_1 + 1} α a b)","statement_sha256":"334797c33cc76e41da5cbb9c73676b9f25104acfa325ead712f348011c9959d0","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-367ff1c4254a2675ed0badc62ae3afd342fd9db397bce355c6f9ffd8176df453","qualified_name":"LE.le.lt_or_eq_dec'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":9099,"column":23,"line":262},"start":{"byte":9078,"column":2,"line":262}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : PartialOrder.{u_1} α] {a b : α}\n  [@DecidableLE.{u_1} α (@Preorder.toLE.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst))],\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) b a →\n    Or (@LT.lt.{u_1} α (@Preorder.toLT.{u_1} α (@PartialOrder.toPreorder.{u_1} α inst)) b a) (@Eq.{u_1 + 1} α a b)","statement_sha256":"0800d43a90c7f52e3c9f3e6be1a3df1680d38262f069d3c0650da982d1e8903b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-19d598942b02375868a9e8a1222c187af9846f49f12b3751a6e56f619933bb74","qualified_name":"LE.le.lt_or_ge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":11251,"column":18,"line":312},"start":{"byte":11235,"column":2,"line":312}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))\n      b a →\n    ∀ (c : α),\n      Or\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a)\n        (@LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b c)","statement_sha256":"f9e119684954a12a687582ecf217b7eaeeb341f2c21c8b9e20593f162acf811f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-7677399c1bcf630e771e27ee7b322cad0bf0f257f3bf4eba552cd9edf866a9ff","qualified_name":"LE.le.not_lt_iff_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7471,"column":79,"line":219},"start":{"byte":7366,"column":0,"line":218}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b →\n    Iff (Not (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b))\n      (@Eq.{u_2 + 1} α a b)","statement_sha256":"394801d41056a1cc9a4306b73ca04412f7f47e0b52a4d09fdfa278a33f020668","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-3bb743ddbe57e1dc9893eb91b3e710954d451db8bf45bb4dba15bde41c97bd0a","qualified_name":"LE.le.not_lt_iff_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7390,"column":24,"line":218},"start":{"byte":7368,"column":2,"line":218}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a →\n    Iff (Not (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a))\n      (@Eq.{u_2 + 1} α a b)","statement_sha256":"bffb253240924803dfb1ea720c9e1cf083f0cc729c16508e61bbddbe67df2bbb","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-9b1bdeee8756d2a1151cc2931b3cc3e51273742254eec0fccf7fbd767d772ca2","qualified_name":"LE.le.trans","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":4044,"column":47,"line":126},"start":{"byte":3997,"column":0,"line":126}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Preorder.{u_1} α] {a b c : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) a b →\n    @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) b c → @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) a c","statement_sha256":"2d45d343e4d895626341d2677310777870d380fd631080f85bbee131edeaaecd","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-28518709fcbf002cea51da29116264cdb54cd46eac6a15d06440acfb50d661f6","qualified_name":"LE.le.trans'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":4013,"column":16,"line":126},"start":{"byte":3999,"column":2,"line":126}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Preorder.{u_1} α] {a b c : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) b a →\n    @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) c b → @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) c a","statement_sha256":"9531903c0b4ae83ed037be699e7bc05e5e6a24249a1026ffd1a0cab7e28af3a4","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-7839b4d52080875b1843b0edf29460140e9e053a053594ddb162f16ddf76c449","qualified_name":"LE.le.trans_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":2406,"column":59,"line":77},"start":{"byte":2347,"column":0,"line":77}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {a b c : α} [inst : LE.{u_1} α],\n  @LE.le.{u_1} α inst a b → @Eq.{u_1 + 1} α b c → @LE.le.{u_1} α inst a c","statement_sha256":"503364690189766b18936f8f1d813ad8a9258fbccb045d706f37e5c41fd82b13","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-067d66657e885d498a8981555ad411fc17cacaf5ae1de30667a06294c7df7c6d","qualified_name":"LE.le.trans_eq'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":2366,"column":19,"line":77},"start":{"byte":2349,"column":2,"line":77}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} {a b c : α} [inst : LE.{u_1} α],\n  @LE.le.{u_1} α inst b a → @Eq.{u_1 + 1} α b c → @LE.le.{u_1} α inst c a","statement_sha256":"b7c560012e996fe400c73020f00e3618369219f3c522354b3b74d7be1a193c0b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a0b2b0eb226bb982d0a5b4198d561452d314c50bdbd03cb23e5fb44e9e8287b6","qualified_name":"LE.le.trans_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":4152,"column":59,"line":128},"start":{"byte":4093,"column":0,"line":128}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Preorder.{u_1} α] {a b c : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) a b →\n    @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) b c → @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) a c","statement_sha256":"f36a92989b3d801bf0a1191c9e3368d869ba601bcd0316a1581c3a82ab090885","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-d583e9d4d04021db59a662414244195ec41b80b614dcf25a401827a4fbcc79fe","qualified_name":"LE.le.trans_lt'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":4112,"column":19,"line":128},"start":{"byte":4095,"column":2,"line":128}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Preorder.{u_1} α] {a b c : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) b a →\n    @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) c b → @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) c a","statement_sha256":"6139170e8c2b6f0dbc94e08b12da6ee5e8368637aeea352ba251441a1cf18151","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e2c92adda1db34baf2a34007979605d23ffdb36cfd7f9bd1e768ddcdc618d5c6","qualified_name":"LE.le.trans_strongLT","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":22351,"column":51,"line":608},"start":{"byte":22302,"column":2,"line":608}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : (i : ι) → Preorder.{u_4} (π i)] {a b c : (i : ι) → π i},\n  @LE.le.{max u_1 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a →\n    ∀ (c : α),\n      Or\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a)\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b 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(@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α A))) x y)\n        (@LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α B))) x y)) →\n    @Eq.{u_2 + 1} (LinearOrder.{u_2} α) A B","statement_sha256":"92e44e579b920edcb24f2afdbc0865e0f97de08bb83339297602f9c1a08b232c","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-6a4072e3899b105da952b8bb0e020a85728984859a7e0b75df8232cf1dd570a4","qualified_name":"LinearOrder.ext_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":18867,"column":73,"line":508},"start":{"byte":18678,"column":0,"line":506}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {A B : LinearOrder.{u_2} α},\n  (∀ (x y : α),\n      Iff\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α 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@Function.Injective.{u_2 + 1, u_3 + 1} α β f →\n              (∀ (x y : α),\n                  @Eq.{u_3 + 1} β (f (@Max.max.{u_2} α inst_1 x y))\n                    (@Max.max.{u_3} β (@LinearOrder.toMax.{u_3} β inst) (f x) (f y))) →\n                (∀ (x y : α),\n                    @Eq.{u_3 + 1} β (f (@Min.min.{u_2} α inst_2 x y))\n                      (@Min.min.{u_3} β (@LinearOrder.toMin.{u_3} β inst) (f x) (f y))) →\n                  LinearOrder.{u_2} 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PartialOrder.{u_2} α] {a b : α},\n  @Ne.{u_2 + 1} α a b →\n    @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b →\n      @LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a 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@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a","statement_sha256":"8b16ee94a9732e4650c0880e45be8f725621eb820753ed0536087994d1135712","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-20befcd863887b7b18f9a203a799735e5ef5a126693ee6c3633396e686d3dc89","qualified_name":"Ne.lt_or_gt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":11820,"column":18,"line":332},"start":{"byte":11727,"column":0,"line":330}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  @Ne.{u_2 + 1} α a b →\n    Or\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a b)\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b 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1, u_2 + 1} α α (@Order.Preimage.{u_2, u_3} α β f s)","statement_sha256":"986c0d42c195a20ba65a98d7fb676aebfd80eaa3d9d1fb05e88608f517e69846","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-969ab8871c8c24a9cf31a1cd4caa580306383b766693fa40988e7935972e316e","qualified_name":"PUnit.instDenselyOrdered","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":42883,"column":30,"line":1089},"start":{"byte":42818,"column":0,"line":1088}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"@DenselyOrdered.{u_5} PUnit.{u_5 + 1}\n  (@Preorder.toLT.{u_5} PUnit.{u_5 + 1}\n    (@PartialOrder.toPreorder.{u_5} PUnit.{u_5 + 1}\n      (@LinearOrder.toPartialOrder.{u_5} PUnit.{u_5 + 1} PUnit.instLinearOrder.{u_5})))","statement_sha256":"3f1798d80a663fea1978a38904857c164ef71fcd5c231e107d752a999d03d1d7","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-911e50004609d97d2b5a85c4951788731d4de117132ae5b11e4b629b04cbc985","qualified_name":"PUnit.instLinearOrder","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":42645,"column":63,"line":1074},"start":{"byte":42582,"column":0,"line":1074}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"LinearOrder.{u_5} PUnit.{u_5 + 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(@LinearOrder.toPartialOrder.{u_5} PUnit.{u_5 + 1} PUnit.instLinearOrder.{u_5})))\n    a b","statement_sha256":"b03794052460162123ec7206cb2ebb2df9ebee596a6e3646c25253abdf1975ac","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-3dcbf89b3c9bfee2c09afca9075332908fc667dd850cc433765b868b9ae08b8d","qualified_name":"PUnit.max_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":42698,"column":5,"line":1078},"start":{"byte":42647,"column":0,"line":1076}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (a b : PUnit.{u_5 + 1}),\n  @Eq.{u_5 + 1} PUnit.{u_5 + 1}\n    (@Max.max.{u_5} PUnit.{u_5 + 1} (@LinearOrder.toMax.{u_5} PUnit.{u_5 + 1} PUnit.instLinearOrder.{u_5}) a b)\n    PUnit.unit.{u_5 + 1}","statement_sha256":"d2e5598ca5dbaf010ae07d230f89fb58f306aabd546205c9c15244d2f7b7996f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-32a39a4e58716eb1c9b2d7857fe0845481ee8ab031d83ed993716f5db6ff13f8","qualified_name":"PUnit.min_eq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":42656,"column":9,"line":1076},"start":{"byte":42649,"column":2,"line":1076}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (a b : PUnit.{u_5 + 1}),\n  @Eq.{u_5 + 1} PUnit.{u_5 + 1}\n    (@Min.min.{u_5} PUnit.{u_5 + 1} (@LinearOrder.toMin.{u_5} PUnit.{u_5 + 1} PUnit.instLinearOrder.{u_5}) a b)\n    PUnit.unit.{u_5 + 1}","statement_sha256":"23bc00a4d914d5dbbc37cee0c4682c8012760dfad4e6d98cd606ede9898f8629","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-aaa0fffac135723a0eb56849a59c8a49ad69df0447a299391a1cf4be81fd80ec","qualified_name":"PUnit.not_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":42816,"column":11,"line":1086},"start":{"byte":42761,"column":0,"line":1084}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ (a b : PUnit.{u_5 + 1}),\n  Not\n    (@LT.lt.{u_5} PUnit.{u_5 + 1}\n      (@Preorder.toLT.{u_5} PUnit.{u_5 + 1}\n        (@PartialOrder.toPreorder.{u_5} PUnit.{u_5 + 1}\n          (@LinearOrder.toPartialOrder.{u_5} PUnit.{u_5 + 1} PUnit.instLinearOrder.{u_5})))\n      a b)","statement_sha256":"cb36618eb7af76d9c2826125124ac69d6b4048864b203d8773ca07790a63fdaf","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-578547ab625879de27610411b285d32f095ea0dda3e80a7c9a28d7483d1a45c3","qualified_name":"PartialOrder.ext","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":18331,"column":36,"line":496},"start":{"byte":18176,"column":0,"line":494}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {A B : PartialOrder.{u_2} α},\n  (∀ (x y : α),\n      Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α A)) x y)\n        (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α B)) x y)) →\n    @Eq.{u_2 + 1} (PartialOrder.{u_2} α) A B","statement_sha256":"bfc8934ca08d3ec6c06b24c1951ec823173ea03373fe463423112ed524abbf05","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-eda42c01ebe5a5258b101e51b5ae0cfacd9d243aef75fbfb1873fdf1eb38a100","qualified_name":"PartialOrder.ext_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":18521,"column":70,"line":500},"start":{"byte":18333,"column":0,"line":498}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {A B : PartialOrder.{u_2} α},\n  (∀ (x y : α),\n      Iff (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α A)) x y)\n        (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α B)) x y)) →\n    @Eq.{u_2 + 1} (PartialOrder.{u_2} α) A B","statement_sha256":"cf65478bea4806826d97ba08aa1c398e6eed2c6e6ad491d50bc95dffb07ac203","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-9afe6a592b0330d23bd372823cb3bdddd0555aaa949aab5c3787c0915ae90004","qualified_name":"PartialOrder.lift","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":27432,"column":49,"line":739},"start":{"byte":26927,"column":0,"line":730}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u_2} →\n  {β : Type u_3} →\n    [PartialOrder.{u_3} β] → (f : α → β) → @Function.Injective.{u_2 + 1, u_3 + 1} α β f → PartialOrder.{u_2} 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[inst : (i : ι) → Compl.{u_4} (π i)] (x : (i : ι) → π i),\n  @Eq.{max (u_1 + 1) (u_4 + 1)} ((i : ι) → π i)\n    (@Compl.compl.{max u_1 u_4} ((i : ι) → π i) (@Pi.instCompl.{u_1, u_4} ι π inst) x) fun i =>\n    @Compl.compl.{u_4} (π i) (inst i) (x 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@Eq.{max (u_1 + 1) (u_4 + 1)} ((i : ι) → π i)\n    (@HImp.himp.{max u_1 u_4} ((i : ι) → π i) (@Pi.instHImp.{u_1, u_4} ι π inst) y x) fun i =>\n    @HImp.himp.{u_4} (π i) (inst i) (y i) (x 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((i : ι) → π i)\n    (@HNot.hnot.{max u_1 u_4} ((i : ι) → π i) (@Pi.instHNot.{u_1, u_4} ι π inst) x) fun i =>\n    @HNot.hnot.{u_4} (π i) (inst i) (x i)","statement_sha256":"f25883398f11a3edc6ac6706852694c111627d48567b523ed52d8f96544bf169","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-a5f99cc60542d1c2bf2cc40c8834ef279c6f2b5fee341815c4f145e34ffabae6","qualified_name":"Pi.instCompl","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":19060,"column":28,"line":518},"start":{"byte":18943,"column":0,"line":516}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{ι : Type u_1} → {π : ι → Type u_4} → [(i : ι) → Compl.{u_4} (π i)] → Compl.{max u_1 u_4} ((i : ι) → π i)","statement_sha256":"925e7f6726344a723424b51b712a2225757fc6c0673e77055c74a8e56c82e0c2","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-baeed87850a647944991c69a8cda9b694161bbc718c128c3816d255a05b35102","qualified_name":"Pi.instHImp","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23506,"column":18,"line":642},"start":{"byte":23490,"column":2,"line":642}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{ι : Type u_1} → {π : ι → Type u_4} → [(i : ι) → HImp.{u_4} (π i)] → HImp.{max u_1 u_4} ((i : ι) → π i)","statement_sha256":"a8a8a307dd7d356b1ffa7b1054ac56736b570bd96384122d200b4964d1c464cd","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-8b4e566352aa07133f7be938130cb63610c06268c17ce7dc689d9c8cc744b47d","qualified_name":"Pi.instHNot","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":18961,"column":18,"line":516},"start":{"byte":18945,"column":2,"line":516}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{ι : Type u_1} → {π : ι → Type u_4} → [(i : ι) → HNot.{u_4} (π i)] → HNot.{max u_1 u_4} ((i : ι) → π i)","statement_sha256":"f0f5dbcdf98407c7735c79ad324370023dd338cff2fd36c1cc40fb7f6eb23317","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-ae17e26ccf22552d2ea284c38c54d182971a57e61bf0cc2c1dc68e81480e4d51","qualified_name":"Pi.instSDiff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23608,"column":31,"line":644},"start":{"byte":23488,"column":0,"line":642}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{ι : Type u_1} → {π : ι → Type u_4} → [(i : ι) → SDiff.{u_4} (π i)] → SDiff.{max u_1 u_4} ((i : ι) → π i)","statement_sha256":"2478ae2d769aa2ae6cf275b15cca8a5be270bcde5fa8e55bf50583827ffd953e","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-0353476583393a49d7960bf51be072e70e366bf38b6f4a6318d0ec5d60e483e2","qualified_name":"Pi.lt_def","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":20542,"column":48,"line":555},"start":{"byte":20364,"column":0,"line":552}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : (i : ι) → Preorder.{u_4} (π i)] {x y : (i : ι) → π i},\n  Iff\n    (@LT.lt.{max u_1 u_4} ((i : ι) → π i)\n      (@Preorder.toLT.{max u_1 u_4} ((i : ι) → π i) (@Pi.preorder.{u_1, u_4} ι π inst)) x y)\n    (And\n      (@LE.le.{max u_1 u_4} ((i : ι) → π i) (@Pi.hasLe.{u_1, u_4} ι π fun i => @Preorder.toLE.{u_4} (π i) (inst i)) x y)\n      (∃ i, @LT.lt.{u_4} (π i) (@Preorder.toLT.{u_4} (π i) (inst i)) (x i) (y i)))","statement_sha256":"6ff8f886b9ea71938c12967545989925539375a25adc425508fa445ce9157704","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-72875474abe0b0d02b4373418f135635abcd4120f4f9a31864a30d537f4682f2","qualified_name":"Pi.partialOrder","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":20727,"column":74,"line":559},"start":{"byte":20544,"column":0,"line":557}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{ι : Type u_1} → {π : ι → Type u_4} → [(i : ι) → PartialOrder.{u_4} (π i)] → PartialOrder.{max u_1 u_4} ((i : ι) → π i)","statement_sha256":"3937c9611846f080233caa037aefe6ab7ddea2a89e4c9dfb2b23fecd77b5f53e","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-2f4040c48fc6598e4cc762538620887297edd3d5fae69e9c3746b269950ec2be","qualified_name":"Pi.preorder","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":20362,"column":66,"line":550},"start":{"byte":20137,"column":0,"line":547}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{ι : Type u_1} → {π : ι → Type u_4} → [(i : ι) → Preorder.{u_4} (π i)] → Preorder.{max u_1 u_4} ((i : ι) → π i)","statement_sha256":"6e48fcf1f22039f09900ce7041cbbda3e7abb01638db5b5a6829c396b5879c07","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-c3d421d0cad1b72df9d9e9311bded9c4a16a862565351452caa6304ef4ac3c41","qualified_name":"Pi.sdiff_apply","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23903,"column":5,"line":654},"start":{"byte":23756,"column":0,"line":651}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : (i : ι) → SDiff.{u_4} (π i)] (x y : (i : ι) → π i) (i : ι),\n  @Eq.{u_4 + 1} (π i) (@SDiff.sdiff.{max u_1 u_4} ((i : ι) → π i) (@Pi.instSDiff.{u_1, u_4} ι π inst) x y i)\n    (@SDiff.sdiff.{u_4} (π i) (inst i) (x i) (y i))","statement_sha256":"a88be44b4bc57537aa11c6182ac46579a7d3c3cdc95c5b3f0b0a125f96292a60","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-545704fe5fa28687c04bd1438e7c6ada1be908a8c1d73cc8de10fc6bd5ee038d","qualified_name":"Pi.sdiff_def","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23754,"column":5,"line":649},"start":{"byte":23610,"column":0,"line":646}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : (i : ι) → SDiff.{u_4} (π i)] (x y : (i : ι) → π i),\n  @Eq.{max (u_1 + 1) (u_4 + 1)} ((i : ι) → π i)\n    (@SDiff.sdiff.{max u_1 u_4} ((i : ι) → π i) (@Pi.instSDiff.{u_1, u_4} ι π inst) x y) fun i =>\n    @SDiff.sdiff.{u_4} (π i) (inst i) (x i) (y i)","statement_sha256":"ff72affbdea1eae466e87ddd9ca36f1b884f93746e6060e312cb48dcb667ef28","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f7d8306bd430add25a629358dec544df19e142da8034a0b15dbaf99e5aa3b804","qualified_name":"Preorder.ext","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":18174,"column":22,"line":492},"start":{"byte":18029,"column":0,"line":490}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {A B : Preorder.{u_2} α},\n  (∀ (x y : α), Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α A) x y) (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α B) x y)) →\n    @Eq.{u_2 + 1} (Preorder.{u_2} α) A B","statement_sha256":"37d4b9e6e761c9419385cfbe96b33ff8997bdffc6aa43958fa1d4ea747df8363","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-d0bab17e13ae10c0086615057ba807297b83fc02bd94c5fd808fa8a2e89d8435","qualified_name":"Preorder.lift","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":26925,"column":41,"line":728},"start":{"byte":26483,"column":0,"line":720}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u_2} → {β : Type u_3} → [Preorder.{u_3} β] → (α → β) → Preorder.{u_2} 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α)","statement_sha256":"317c7802e406105e2322d8da2c753b3182cafe680d46a5701284955d222aca04","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-eb076b93d84283970be61a4a1bfb9645024f6db38a141d0102be2ecc6751e278","qualified_name":"Preorder.toLE_injective_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":16260,"column":5,"line":446},"start":{"byte":16257,"column":2,"line":446}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {a₁ a₂ : Preorder.{u_2} α},\n  Iff (@Eq.{u_2 + 1} (Preorder.{u_2} α) a₁ a₂)\n    (@Eq.{u_2 + 1} (LE.{u_2} α) (@Preorder.toLE.{u_2} α a₁) (@Preorder.toLE.{u_2} α a₂))","statement_sha256":"c6c07c69b25cf17b283c6a06758ebaeca34810c10046c104c3ddcd6827710088","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a0ce9ed04187037abb8063d18b58d2f305a245ea6140c42affb3cd9303f935e0","qualified_name":"Prod.GCongr.mk_le_mk","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":34838,"column":111,"line":886},"start":{"byte":34703,"column":0,"line":885}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : LE.{u_2} α] [inst_1 : LE.{u_3} β] {a₁ a₂ : α} {b₁ b₂ : β},\n  @LE.le.{u_2} α inst a₁ a₂ →\n    @LE.le.{u_3} β inst_1 b₁ b₂ →\n      @LE.le.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instLE_mathlib.{u_2, u_3} α β inst inst_1)\n        (@Prod.mk.{u_2, u_3} α β a₁ b₁) (@Prod.mk.{u_2, u_3} α β a₂ b₂)","statement_sha256":"33e7b003d48c8447cadf790044e314fae22131b91cd56be105b64eca80ad195f","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-5b71f32ea2dd47ea78466cb57563cea6f2efe200840681d6c0fea1996de1ff59","qualified_name":"Prod.instDecidableLE","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":34507,"column":60,"line":879},"start":{"byte":34329,"column":0,"line":877}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{α : Type u_2} →\n  {β : Type u_3} →\n    [inst : LE.{u_2} α] →\n      [inst_1 : LE.{u_3} β] →\n        {x y : Prod.{u_2, u_3} α β} →\n          [Decidable (@LE.le.{u_2} α inst (@Prod.fst.{u_2, u_3} α β x) (@Prod.fst.{u_2, u_3} α β y))] →\n            [Decidable (@LE.le.{u_3} β inst_1 (@Prod.snd.{u_2, u_3} α β x) (@Prod.snd.{u_2, u_3} α β y))] →\n              Decidable\n                (@LE.le.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instLE_mathlib.{u_2, u_3} α β inst inst_1) x 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(@Preorder.toLT.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instPreorder.{u_2, u_3} α β inst inst_1)) x y)\n    (Or\n      (And (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α inst) (@Prod.fst.{u_2, u_3} α β x) (@Prod.fst.{u_2, u_3} α β y))\n        (@LE.le.{u_3} β (@Preorder.toLE.{u_3} β inst_1) (@Prod.snd.{u_2, u_3} α β x) (@Prod.snd.{u_2, u_3} α β y)))\n      (And (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) (@Prod.fst.{u_2, u_3} α β x) (@Prod.fst.{u_2, u_3} α β y))\n        (@LT.lt.{u_3} β (@Preorder.toLT.{u_3} β inst_1) (@Prod.snd.{u_2, u_3} α β x) (@Prod.snd.{u_2, u_3} α β 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y","statement_sha256":"03ba72b5af4db167bd62183fecee115af6f33ec9f49cefdab0ab28c844a04a69","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-b9a3d1778efc0bbf013469e5748b70222b8d0ac07a68ab6e107d89cf4e00cec1","qualified_name":"Prod.lt_of_lt_of_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":36991,"column":101,"line":942},"start":{"byte":36874,"column":0,"line":941}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_3} β] {x y : Prod.{u_2, u_3} α β},\n  @LT.lt.{u_2} α (@Preorder.toLT.{u_2} α inst) (@Prod.fst.{u_2, 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y","statement_sha256":"5f48ffb12f2116edf85ae1713e4cca5ab2af424523ccf9802f91354efb97cfb4","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-8f70c39d8ace89ae8a42251f9f871125c7a89ae5da4549b930d480ce1ac23bee","qualified_name":"Prod.mk_le_mk","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":34701,"column":112,"line":883},"start":{"byte":34589,"column":0,"line":883}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : LE.{u_2} α] [inst_1 : LE.{u_3} β] {a₁ a₂ : α} {b₁ b₂ : β},\n  Iff\n    (@LE.le.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instLE_mathlib.{u_2, 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(@Prod.instLE_mathlib.{u_2, u_3} α β (@Preorder.toLE.{u_2} α inst) (@Preorder.toLE.{u_3} β inst_1))\n      (@Prod.mk.{u_2, u_3} α β a b₁) (@Prod.mk.{u_2, u_3} α β a b₂))\n    (@LE.le.{u_3} β (@Preorder.toLE.{u_3} β inst_1) b₁ 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x)","statement_sha256":"d899ac0046ac2aac9093decea0c28c24b3265276000f677eb03843e940967f60","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ddf75e09858e62cc4a6927116dcb557c5c6f4b63f96f8c57caa300979368c37a","qualified_name":"Prod.mk_lt_mk","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":36872,"column":8,"line":939},"start":{"byte":36724,"column":0,"line":937}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_3} β] {a₁ a₂ : α} {b₁ b₂ : β},\n  Iff\n    (@LT.lt.{max u_2 u_3} (Prod.{u_2, u_3} α β)\n      (@Preorder.toLT.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instPreorder.{u_2, u_3} α β inst inst_1))\n      (@Prod.mk.{u_2, u_3} α β a₁ b₁) (@Prod.mk.{u_2, u_3} α β a₂ b₂))\n    (Or\n      (And (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α inst) a₁ a₂) (@LE.le.{u_3} β (@Preorder.toLE.{u_3} β inst_1) b₁ b₂))\n      (And (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) a₁ a₂) (@LT.lt.{u_3} β (@Preorder.toLT.{u_3} β inst_1) b₁ b₂)))","statement_sha256":"7f0f95415ce55c54f20645371ed01ceb8472b5cf6d35f17b457396bf00e8d00d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-acc9c52a4375642b9692102025f89ae2adeefdca88190d494e916169dbce858e","qualified_name":"Prod.mk_lt_mk_iff_left","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":36055,"column":61,"line":921},"start":{"byte":35901,"column":0,"line":919}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_3} β] {a₁ a₂ : α} {b : β},\n  Iff\n    (@LT.lt.{max u_2 u_3} (Prod.{u_2, u_3} α β)\n      (@Preorder.toLT.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instPreorder.{u_2, u_3} α β inst inst_1))\n      (@Prod.mk.{u_2, u_3} α β a₁ b) (@Prod.mk.{u_2, u_3} α β a₂ b))\n    (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α inst) a₁ a₂)","statement_sha256":"e838c996782e4bf44b5a83dba3e93f13e4e04b008113cf85e6132a143f161138","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-b8173dfb5724a073443d34daceed2f3e20c449b757e2cda3b27d69631387d3d1","qualified_name":"Prod.mk_lt_mk_iff_right","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":36214,"column":63,"line":925},"start":{"byte":36057,"column":0,"line":923}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_3} β] {a : α} {b₁ b₂ : β},\n  Iff\n    (@LT.lt.{max u_2 u_3} (Prod.{u_2, u_3} α β)\n      (@Preorder.toLT.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instPreorder.{u_2, u_3} α β inst inst_1))\n      (@Prod.mk.{u_2, u_3} α β a b₁) (@Prod.mk.{u_2, u_3} α β a b₂))\n    (@LT.lt.{u_3} β (@Preorder.toLT.{u_3} β inst_1) b₁ b₂)","statement_sha256":"4c4c3dfb97672c4a9394869c6ac0d3e840c41079503e89367a9bf4b38a93684a","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-9ebbec2ad2eeaed420bd0634f39bb7399b33d1918ccd748f4d29226f7886f6d5","qualified_name":"Prod.mk_lt_mk_of_le_of_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":37396,"column":18,"line":953},"start":{"byte":37255,"column":0,"line":951}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_3} β] {a₁ a₂ : α} {b₁ b₂ : β},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) a₁ a₂ →\n    @LT.lt.{u_3} β (@Preorder.toLT.{u_3} β inst_1) b₁ b₂ →\n      @LT.lt.{max u_2 u_3} (Prod.{u_2, u_3} α β)\n        (@Preorder.toLT.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instPreorder.{u_2, u_3} α β inst inst_1))\n        (@Prod.mk.{u_2, u_3} α β a₁ b₁) (@Prod.mk.{u_2, u_3} α β a₂ 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@LE.le.{u_3} β (@Preorder.toLE.{u_3} β inst_1) b₁ b₂ →\n      @LT.lt.{max u_2 u_3} (Prod.{u_2, u_3} α β)\n        (@Preorder.toLT.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instPreorder.{u_2, u_3} α β inst inst_1))\n        (@Prod.mk.{u_2, u_3} α β a₁ b₁) (@Prod.mk.{u_2, u_3} α β a₂ b₂)","statement_sha256":"9f21bced96a99527f2f85c9b3e0a98b0b4781fdad6ccb8dbc2e277408655f9a3","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-53253c621f707e9834c4203dcb3b77f0acec77f6fa54d36371a7b7b02867d983","qualified_name":"Prod.mk_lt_swap","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":35572,"column":35,"line":908},"start":{"byte":35539,"column":2,"line":908}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_3} β] {a : α} {b : β}\n  {x : Prod.{u_2, u_3} α β},\n  Iff\n    (@LT.lt.{max u_2 u_3} (Prod.{u_3, u_2} 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x)","statement_sha256":"6a2056d2d05cff9385410bfa85af6bdec956b73fbc93ea5e8928f661c691f506","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-0275d2035fe7cdded02b0b7bf4cb806893989d8a1672f19de29bbb096f4b4faa","qualified_name":"Prod.swap_le_mk","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":35028,"column":65,"line":891},"start":{"byte":34926,"column":0,"line":890}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : LE.{u_2} α] [inst_1 : LE.{u_3} β] {x : Prod.{u_2, u_3} α β} {a : α} {b : β},\n  Iff\n    (@LE.le.{max u_2 u_3} (Prod.{u_3, u_2} β α) 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b))","statement_sha256":"6639ff0f4fcdafa1842eb40dca56273e6ae42ca5ca727ca0b5742edb313561e5","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-3e67ddb287e7792239d7fcd1f4550b34e8e0cfa8d7e0323aae4659123e43e5b5","qualified_name":"Prod.swap_le_swap","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":34924,"column":84,"line":888},"start":{"byte":34840,"column":0,"line":888}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : LE.{u_2} α] [inst_1 : LE.{u_3} β] {x y : Prod.{u_2, u_3} α β},\n  Iff\n    (@LE.le.{max u_2 u_3} (Prod.{u_3, u_2} β α) (@Prod.instLE_mathlib.{u_3, u_2} β α inst_1 inst)\n      (@Prod.swap.{u_2, u_3} α β x) (@Prod.swap.{u_2, u_3} α β y))\n    (@LE.le.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instLE_mathlib.{u_2, u_3} α β inst inst_1) x y)","statement_sha256":"1abe18659966c49d9c5915c9b9c3fdf0a02d5e338a5c07fa0ca0b7e616702fd4","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a6f504d5eb4b32f47f77566cb505f7d95da53a6507ec13825e1f89fcb39e9f15","qualified_name":"Prod.swap_lt_mk","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":35657,"column":83,"line":909},"start":{"byte":35537,"column":0,"line":908}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_3} β] {a : α} {b : β}\n  {x : Prod.{u_2, u_3} α β},\n  Iff\n    (@LT.lt.{max u_2 u_3} (Prod.{u_3, u_2} β α)\n      (@Preorder.toLT.{max u_2 u_3} (Prod.{u_3, u_2} β α) (@Prod.instPreorder.{u_3, u_2} β α inst_1 inst))\n      (@Prod.swap.{u_2, u_3} α β x) (@Prod.mk.{u_3, u_2} β α b a))\n    (@LT.lt.{max u_2 u_3} (Prod.{u_2, u_3} α β)\n      (@Preorder.toLT.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instPreorder.{u_2, u_3} α β inst inst_1)) x\n      (@Prod.mk.{u_2, u_3} α β a b))","statement_sha256":"e5e6fafac511ba39920e12b7fbb5919258f7a9acd8baad3f2803860da66afc9b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-bf025e2bc07d34301c4bfd8779aac3b21fb2c6fd81094947c507b8b046c349f4","qualified_name":"Prod.swap_lt_swap","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":35535,"column":49,"line":906},"start":{"byte":35412,"column":0,"line":904}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_3} β] {x y : Prod.{u_2, u_3} α β},\n  Iff\n    (@LT.lt.{max u_2 u_3} (Prod.{u_3, u_2} β α)\n      (@Preorder.toLT.{max u_2 u_3} (Prod.{u_3, u_2} β α) (@Prod.instPreorder.{u_3, u_2} β α inst_1 inst))\n      (@Prod.swap.{u_2, u_3} α β x) (@Prod.swap.{u_2, u_3} α β y))\n    (@LT.lt.{max u_2 u_3} (Prod.{u_2, u_3} α β)\n      (@Preorder.toLT.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instPreorder.{u_2, u_3} α β inst inst_1)) x y)","statement_sha256":"14a91863cfe0c550a277c7dc1f0c47a7bc0eb3eb42e7d6c4b6fa42543d6c8c9e","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-d23b934e40d410e027146cb2c9a997a1dbf1a811fe26181a868e952fcd267d49","qualified_name":"Prop.instCompl","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":18941,"column":11,"line":514},"start":{"byte":18890,"column":0,"line":513}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"Compl.{0} 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Prop","statement_sha256":"5b022c59d5b3ff29e08b25525403fb1add261e9ceb299472910509b9fbfd03e8","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-c2ab92f5f1e88cc61ea7f207c178f5ba063cf1084ea30e1db713687a9b1f8028","qualified_name":"Std.Irrefl.compl","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":19454,"column":22,"line":531},"start":{"byte":19347,"column":0,"line":530}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} (r : α → α → Prop) [@Std.Irrefl.{u_2 + 1} α r],\n  @Std.Refl.{u_2 + 1} α\n    (@Compl.compl.{u_2} (α → α → Prop)\n      (@Pi.instCompl.{u_2, u_2} α (fun a => α → Prop) fun i =>\n        @Pi.instCompl.{u_2, 0} α (fun a => Prop) fun i => «Prop».instCompl)\n      r)","statement_sha256":"484e59f21100782edd2851d2b406e455e7c2754c4fc7d458949cfad3420c7d98","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-bfe5522ba4e312159f6b1860df19250c095e753f74a0bc25e5e0b5fd5472c82b","qualified_name":"Std.Refl.compl","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":19579,"column":40,"line":534},"start":{"byte":19456,"column":0,"line":533}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} (r : α → α → Prop) [@Std.Refl.{u_2 + 1} α r],\n  @Std.Irrefl.{u_2 + 1} α\n    (@Compl.compl.{u_2} (α → α → Prop)\n      (@Pi.instCompl.{u_2, u_2} α (fun a => α → Prop) fun i =>\n        @Pi.instCompl.{u_2, 0} α (fun a => Prop) fun i => «Prop».instCompl)\n      r)","statement_sha256":"a0d48fb301e927dd988efac8e611bde4fbc96f48118b4be802bf6bfe8f7b1655","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-25cd9ab36bb05c493a1d16b6f9c2ed031ce4457bceb5ab489d16b871ee9240a7","qualified_name":"StrongLT","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":21667,"column":18,"line":585},"start":{"byte":21467,"column":0,"line":582}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"{ι : Type u_1} → {π : ι → Type u_4} → [(i : ι) → LT.{u_4} (π i)] → ((i : ι) → π i) → ((i : ι) → π i) → 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@LE.le.{max u_1 u_4} ((i : ι) → π i) (@Pi.hasLe.{u_1, u_4} ι π fun i => @Preorder.toLE.{u_4} (π i) (inst i)) a b","statement_sha256":"5d595623ca00e001d385f6cb24a078d52ffc38f2c39e553300ae13190d999fc2","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-8f6e2a22ccdd501889da9e625419b4d37dde593216be540b28d2ca9714edbb6b","qualified_name":"StrongLT.lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":22298,"column":51,"line":606},"start":{"byte":22247,"column":0,"line":606}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : (i : ι) → Preorder.{u_4} (π i)] {a b : (i : ι) → π i}\n  [Nonempty.{u_1 + 1} ι],\n  @StrongLT.{u_1, u_4} ι π (fun i => @Preorder.toLT.{u_4} (π i) (inst i)) a b →\n    @LT.lt.{max u_1 u_4} ((i : ι) → π i)\n      (@Preorder.toLT.{max u_1 u_4} ((i : ι) → π i) (@Pi.preorder.{u_1, u_4} ι π inst)) a b","statement_sha256":"14e549f2e992321ba3ca06adfea1d8ac07e54a52a90a64e459f5c7361a3aa848","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-03be3cf77d14c489d57da1f859ad3c8fe329192cf04e5ef94d71cef1afbcb8d8","qualified_name":"StrongLT.trans_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":22406,"column":53,"line":609},"start":{"byte":22300,"column":0,"line":608}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : (i : ι) → Preorder.{u_4} (π i)] {a b c : (i : ι) → π i},\n  @StrongLT.{u_1, u_4} ι π (fun i => @Preorder.toLT.{u_4} (π i) (inst i)) a b →\n 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{α : Type u_2} [inst : PartialOrder.{u_2} α] {f : α → α → α},\n  @Std.Commutative.{u_2 + 1} α f →\n    (∀ (a b c : α),\n        @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) (f (f a b) c) (f a (f b c))) →\n      @Std.Associative.{u_2 + 1} α f","statement_sha256":"be89199acd5dc47810c14e9dbb64875cd1efa93486d198a4ac156771860e0a14","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-f5ec324ab7b44719f003799cc13c326bb38c7e6edd5bfd43ccb3acf74c524c3d","qualified_name":"commutative_of_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":10684,"column":45,"line":294},"start":{"byte":10414,"column":0,"line":291}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : PartialOrder.{u_2} α] {f : β → β → α},\n  (∀ (a b : β), @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) (f a b) (f b a)) →\n    ∀ (a b : β), @Eq.{u_2 + 1} α (f a b) (f b a)","statement_sha256":"06bce08d43da3b01105145f70f705b3fd088d9550ffcd41a4278aa3ccbe5874b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-76bcf144017c09e375e4088a4cf38972ac236d887d61c1f4b2862743072d78f5","qualified_name":"compare_of_injective_eq_compareOfLessAndEq","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":27982,"column":10,"line":751},"start":{"byte":27434,"column":0,"line":741}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} (a b : α) [inst : LinearOrder.{u_3} β] [inst_1 : DecidableEq.{u_2 + 1} α] (f : α → β)\n  (inj : @Function.Injective.{u_2 + 1, u_3 + 1} α β f)\n  [inst_2 :\n    Decidable\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α\n          (@PartialOrder.toPreorder.{u_2} α\n            (@PartialOrder.lift.{u_2, u_3} α β (@LinearOrder.toPartialOrder.{u_3} β inst) f inj)))\n        a b)],\n  @Eq.{1} Ordering (@Ord.compare.{u_3} β (@LinearOrder.toOrd.{u_3} β inst) (f a) (f b))\n    (@compareOfLessAndEq.{u_2} α a b\n      (@Preorder.toLT.{u_2} α\n        (@PartialOrder.toPreorder.{u_2} α\n          (@PartialOrder.lift.{u_2, u_3} α β (@LinearOrder.toPartialOrder.{u_3} β inst) f inj)))\n      inst_2 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x2","statement_sha256":"aecbcedf1a789a4adf3b3902aa41766816c5ff0701cbdb667e145964bf3ba50b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-54f2b6f8d86b3b99fa843954cee0f4bf0f7342e40022b6ee4e09875c9225debf","qualified_name":"compl_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":19680,"column":99,"line":536},"start":{"byte":19581,"column":0,"line":536}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α],\n  @Eq.{u_2 + 1} (α → α → Prop)\n    (@Compl.compl.{u_2} (α → α → Prop)\n      (@Pi.instCompl.{u_2, u_2} α (fun x1 => α → Prop) fun i =>\n       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x2","statement_sha256":"ef8feb6077d8f296e88ed1e8fd55d934f6763dad0818b5359c3b6d0f1724df2e","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4e6721a175cf94b3bc650f8100bb5e6e363d6521829c7923a32844bea2ea8df9","qualified_name":"dense_or_discrete","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":41427,"column":74,"line":1043},"start":{"byte":41048,"column":0,"line":1038}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] (a₁ a₂ : α),\n  Or\n    (∃ a,\n      And\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₁ a)\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a a₂))\n    (And\n      (∀ (a : α),\n        @LT.lt.{u_2} α\n            (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₁\n            a →\n          @LE.le.{u_2} α\n            (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₂ a)\n      (∀ (a : α),\n        @LT.lt.{u_2} α\n            (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a\n            a₂ →\n          @LE.le.{u_2} α\n            (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a\n            a₁))","statement_sha256":"d63c26267272f777abb8b87a504c2bc321ea0d94209e3b18de2590009f0762ea","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-38958a832c1d35969820eccd0a0c749e665df787509b9afe182a97d8738df1d4","qualified_name":"dense_or_discrete'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":41076,"column":28,"line":1038},"start":{"byte":41050,"column":2,"line":1038}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] (a₁ a₂ : α),\n  Or\n    (∃ a,\n      And\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a a₁)\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₂ a))\n    (And\n      (∀ (a : α),\n        @LT.lt.{u_2} α\n            (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a\n            a₁ →\n          @LE.le.{u_2} α\n            (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a a₂)\n      (∀ (a : α),\n        @LT.lt.{u_2} α\n            (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₂\n            a →\n          @LE.le.{u_2} α\n            (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₁\n            a))","statement_sha256":"92069c042a199705c61d535d504cf8fb4db73e6c9d410c2321c3114253245c05","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a4c8986ecd6f42959060097b11cd2177ae05af12a897731f6a8d399c4c066198","qualified_name":"eq_iff_eq_of_lt_iff_lt_of_gt_iff_gt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":14019,"column":33,"line":387},"start":{"byte":13832,"column":0,"line":384}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {x y x' y' : α},\n  Iff\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) x y)\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) x' y') →\n    Iff\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) y x)\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) y'\n          x') →\n      Iff (@Eq.{u_2 + 1} α x y) (@Eq.{u_2 + 1} α x' y')","statement_sha256":"9693bb4d08f9ee37c03292aa80dfa6a06fb5a5d6a46d2894f985deb5d2708e56","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-6e12b86ca80cb926d503f7b0a8d8807bb292d5bfdaded8f4bf574410e9ced562","qualified_name":"eq_iff_ge_not_gt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":8656,"column":26,"line":250},"start":{"byte":8632,"column":2,"line":250}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  Iff (@Eq.{u_2 + 1} α a b)\n    (And (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a)\n 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 (And (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b)\n      (Not (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b)))","statement_sha256":"8080eee1e20687b69475bad02c0fe240bfbbbafd6ea5786b3b8c4521339b0849","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-2873e3b69fc5f25fdbc686d50d22997e51731d01b94b49190a54b060bea8f126","qualified_name":"eq_of_forall_ge_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":10296,"column":29,"line":287},"start":{"byte":10269,"column":2,"line":287}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a c)\n        (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b c)) →\n    @Eq.{u_2 + 1} α a b","statement_sha256":"d2684d090e2e9da62cab152cc2e1a7781673cd27649274d5084ca1cc80c6d0f3","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4d5641e377e778e7963f46cf6cba1206953a50e00b45613a6f6fa1cbe8633564","qualified_name":"eq_of_forall_gt_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13677,"column":29,"line":380},"start":{"byte":13650,"column":2,"line":380}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      Iff\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c)\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b c)) →\n    @Eq.{u_2 + 1} α a b","statement_sha256":"6e8c15763c26df6418e9724e891da07507feee496a7ee0bcbe5dcecca01ac52f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-fb77d1bb7524c78773ce82c2f2064e187a9c5a62bc55d3ab5bc984e4da41a2d6","qualified_name":"eq_of_forall_le_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":10412,"column":44,"line":289},"start":{"byte":10267,"column":0,"line":287}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) c a)\n        (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) c b)) →\n    @Eq.{u_2 + 1} α a b","statement_sha256":"9e0ab9c32f032c2e7d3363be125b4ca3e6984428c68010e7e44576496a63c066","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-80a0c565febad4db1e200087434ba3ef700f6c85ed85e0b4191f76ac8534ea47","qualified_name":"eq_of_forall_lt_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13830,"column":83,"line":382},"start":{"byte":13648,"column":0,"line":380}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      Iff\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a)\n        (@LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c b)) →\n    @Eq.{u_2 + 1} α a b","statement_sha256":"1a8a26294a6637d8e5b5885739878cf9f76e06080e0cbd90d8bc2c89046a62fe","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4eb56a841be88ce2b378d46abf43d4111b45ffb82eaf8c6ab97db6670788ca4f","qualified_name":"eq_of_le_of_forall_gt_imp_ge_of_dense","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":40846,"column":47,"line":1032},"start":{"byte":40801,"column":2,"line":1032}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α]\n  [@DenselyOrdered.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))]\n  {a₁ a₂ : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))\n      a₁ a₂ →\n    (∀ (a : α),\n        @LT.lt.{u_2} α\n            (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a\n            a₂ →\n          @LE.le.{u_2} α\n            (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a\n            a₁) →\n      @Eq.{u_2 + 1} α a₁ a₂","statement_sha256":"58efdca3db66a9180c48bffa366976a5050b4b9173e20edbeaee6c1a60d126d2","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-7c0b53b0c05df5240f35528359380d37d0b53f9119fd98289ce4e5c11cbfefe5","qualified_name":"eq_of_le_of_forall_lt_imp_le_of_dense","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":41029,"column":57,"line":1034},"start":{"byte":40799,"column":0,"line":1032}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α]\n  [@DenselyOrdered.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))]\n  {a₁ a₂ : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))\n      a₂ a₁ →\n    (∀ (a : α),\n        @LT.lt.{u_2} α\n            (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₂\n            a →\n          @LE.le.{u_2} α\n            (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₁\n            a) →\n      @Eq.{u_2 + 1} α a₁ a₂","statement_sha256":"4f42ec83cdfedc9a1dc83a48e2951bf47cda4dd1f8f6751f96c9670ac3f79e4d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-427726e972890dde84312f6142b02143049748fecad201a442582f4025c3b523","qualified_name":"eq_of_le_of_not_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":9488,"column":104,"line":268},"start":{"byte":9353,"column":0,"line":267}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b →\n    Not (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b) → @Eq.{u_2 + 1} α a b","statement_sha256":"46553fc521689ab1a01086de6ad66f2521ee78dec654677b1c78deb614be3451","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-d2d6e5de8794d3e321e677b0c5409500d4c668240ca18fb000a84fe7913f9166","qualified_name":"eq_of_le_of_not_lt'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":9382,"column":29,"line":267},"start":{"byte":9355,"column":2,"line":267}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a →\n    Not (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a) → @Eq.{u_2 + 1} α a b","statement_sha256":"d683fd2708271e459ab459b3bbe6c920ee3e17e89dcee325352beb1dc1cc57e4","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-6bc1b3d5a56827d22492dc521bc6a3a61bbbb16c704fcec9ac3eb0aab2e6cf90","qualified_name":"eq_or_eq_or_eq_of_forall_not_lt_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":42029,"column":113,"line":1054},"start":{"byte":41429,"column":0,"line":1045}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α],\n  (∀ ⦃x y z : α⦄,\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) x y →\n        @LT.lt.{u_2} α\n            (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) y z →\n          False) →\n    ∀ (x y z : α), Or (@Eq.{u_2 + 1} α x y) (Or (@Eq.{u_2 + 1} α y z) (@Eq.{u_2 + 1} α x z))","statement_sha256":"0dc0fcb22863db56c168c740fafd33b79c04d18a991690f62edbfa926a9a5b40","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-fbac49f031215715c917bd79fe338579374c7d7b22f3e8d75f5001ecfea5889f","qualified_name":"eq_or_lt_of_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":9074,"column":81,"line":260},"start":{"byte":8966,"column":0,"line":259}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b →\n    Or (@Eq.{u_2 + 1} α a b) (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b)","statement_sha256":"a612a0b160113e40095a7b9401124842779e25e90ae2bd22a842b8386eeae862","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e62f5e5a73db80a153a3d5aa2e0b17d8a5361b6240023b997a306e41aaae00b5","qualified_name":"eq_or_lt_of_le'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":8991,"column":25,"line":259},"start":{"byte":8968,"column":2,"line":259}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a →\n    Or (@Eq.{u_2 + 1} α a b) (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a)","statement_sha256":"cf31096192f6b389ee60b4abda8ec254b5997d495dd436bf05e8422428c4ba2a","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-7b914d5d80daf122c970eba310ae18faf23686ddb8761ea5e565779256af6c53","qualified_name":"exists_between","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":38937,"column":26,"line":988},"start":{"byte":38766,"column":0,"line":986}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LT.{u_2} α] [@DenselyOrdered.{u_2} α inst] {a₁ a₂ : α},\n  @LT.lt.{u_2} α inst a₁ a₂ → ∃ a, And (@LT.lt.{u_2} α inst a₁ a) (@LT.lt.{u_2} α inst a a₂)","statement_sha256":"60145ae0be513f2628b85b4e1fd3c24b14c6c250b7933a4cf76eafaa21d0400b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e6a19c0e534023e450340b202a4b8a70fac5e3bb4bdf6b5dd7ac4ab2632fcf2c","qualified_name":"exists_between'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":38791,"column":25,"line":986},"start":{"byte":38768,"column":2,"line":986}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LT.{u_2} α] 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α] {p q : α → Prop},\n  (∃ i,\n      ∀ (j : α),\n        @GE.ge.{u_2} α\n            (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) j i →\n          p j) →\n    (∃ i,\n        ∀ (j : α),\n          @GE.ge.{u_2} α\n              (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) j\n              i →\n            q j) →\n      ∃ i,\n        ∀ (j : α),\n          @GE.ge.{u_2} α\n              (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) j\n              i →\n            And (p j) (q 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(@LinearOrder.toPartialOrder.{u_2} α inst))) c a)\n      (@LE.le.{u_2} α\n        (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c b)","statement_sha256":"e54df5bf3234697f3904721621671ff269ca9c463b7a029340d99e6a72d2185f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-5bb48a4da0e0db070a52703e408270d5c83ef68d735ae087e5d123315a8a9fff","qualified_name":"forall_ge_iff_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":5185,"column":26,"line":156},"start":{"byte":5161,"column":2,"line":156}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : Preorder.{u_2} α] {a b : α},\n  Iff (∀ ⦃c : α⦄, @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) a c → @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) b c)\n    (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) b a)","statement_sha256":"a4949c6c402600b7b5d1d564ae9b0ee0c23ed2c9c2da3396a66a99614f973e1a","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e84ff4031201180c73ea6afb9d80c4ebfc705162eb32187f73d190295bf13aab","qualified_name":"forall_ge_imp_ne_iff_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13507,"column":33,"line":376},"start":{"byte":13476,"column":2,"line":376}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  Iff\n    (∀ (c : α),\n      @LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c →\n        @Ne.{u_2 + 1} α c b)\n    (@LT.lt.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a)","statement_sha256":"b947c259f16cce7c4255cebcbde9bc366ae6f44a64499da449f9a92973339922","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-dbd0b71b181242a8e99403c5adf697e98c88db9103044a79de2e323fde2cf3c7","qualified_name":"forall_gt_iff_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":12880,"column":26,"line":360},"start":{"byte":12856,"column":2,"line":360}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  Iff\n    (∀ ⦃c : α⦄,\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c →\n        @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b c)\n    (@LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a)","statement_sha256":"ed638cfbf5c12449910b2b7200281e397209066815f06a600dae20d32e0750df","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-8dc5a375f3da9728078913316c739076767e36630f46ace4ca4694681b26ea9c","qualified_name":"forall_gt_imp_ge_iff_le_of_dense","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":40741,"column":78,"line":1029},"start":{"byte":40524,"column":0,"line":1027}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α]\n  [@DenselyOrdered.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))]\n  {a₁ a₂ : α},\n  Iff\n    (∀ (a : α),\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₂ a →\n        @LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₁ a)\n    (@LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₁ a₂)","statement_sha256":"627d2e3402e9070555fcfaf4e9abe558981777f688c62f10ea77eec37eeab068","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-6f8f080cf5cb2a90b623751f4e22bcce44d5eed7efdf0fa295d41fa12283e3f4","qualified_name":"forall_gt_imp_ne_iff_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13333,"column":33,"line":372},"start":{"byte":13302,"column":2,"line":372}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  Iff\n    (∀ (c : α),\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c →\n        @Ne.{u_2 + 1} α c b)\n    (@LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a)","statement_sha256":"12d4f6914f7ab1363318855de17bb6804a1d214917267a4c38dcf74c52c7eee1","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ae3525464138144a763f44eda1f949da87c8d3f3de25b040c34e71e2110824f5","qualified_name":"forall_le_iff_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":5319,"column":55,"line":158},"start":{"byte":5159,"column":0,"line":156}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : Preorder.{u_2} α] {a b : α},\n  Iff (∀ ⦃c : α⦄, @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) c a → @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) c b)\n    (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) a b)","statement_sha256":"fd403e173b889ed3909afce0ba3da1e657007d3ea2956449f1af90186889aab5","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a2720e7f12d7c6981c8c5995332186a66bcf4717948e47efdb5983e8b1aa85ad","qualified_name":"forall_le_imp_ne_iff_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13646,"column":67,"line":378},"start":{"byte":13474,"column":0,"line":376}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  Iff\n    (∀ (c : α),\n      @LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a →\n        @Ne.{u_2 + 1} α c b)\n    (@LT.lt.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a b)","statement_sha256":"d51eeef08edcff9662d703701195b6570b248be31ae81b255d8ecc268621dc72","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-de6e9ed1d4d937419845f4e627f292a5007a61f4b8833b7a72a4e333f67e9c3f","qualified_name":"forall_lt_iff_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13016,"column":61,"line":362},"start":{"byte":12854,"column":0,"line":360}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  Iff\n    (∀ ⦃c : α⦄,\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a →\n        @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c b)\n    (@LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a b)","statement_sha256":"24d9be80516f6b7048dc6e37f4de6053ed693c97a97b4d3ba89b153eef5375a4","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-2f59c2ebd784904ce3c40a1839188b56bc4adaa08bf3fd664702615214a23ed4","qualified_name":"forall_lt_imp_le_iff_le_of_dense","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":40566,"column":42,"line":1027},"start":{"byte":40526,"column":2,"line":1027}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α]\n  [@DenselyOrdered.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))]\n  {a₁ a₂ : α},\n  Iff\n    (∀ (a : α),\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a a₂ →\n        @LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a a₁)\n    (@LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₂ a₁)","statement_sha256":"cfebd7734e8fe15d8d72a702afbdca802ce281e5864aa4727412fbdf37798f96","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-70cd5b2205692c4275c881b2efcf80b2c8d3167c14ff18255120b0cc60f7ba7e","qualified_name":"forall_lt_imp_ne_iff_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13472,"column":67,"line":374},"start":{"byte":13300,"column":0,"line":372}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  Iff\n    (∀ (c : α),\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a →\n        @Ne.{u_2 + 1} α c b)\n    (@LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a 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@GE.ge.{u_2} α (@Preorder.toLE.{u_2} α inst) a b → @GE.ge.{u_2} α (@Preorder.toLE.{u_2} α inst) c d","statement_sha256":"9623af053f98a6b11cb5a05e403ceb61bd05bf77d602cd0257671abda7e5395b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-33e153dc1e9d1a25511d3c31a99be91736b5d94fac65e197df2be3988c875f43","qualified_name":"ge_imp_gt_of_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":6467,"column":76,"line":183},"start":{"byte":6339,"column":0,"line":182}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : Preorder.{u_2} α] {a b c : α},\n  @LT.lt.{u_2} α 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c : α},\n  @LT.lt.{u_2} α (@Preorder.toLT.{u_2} α inst) b a →\n    @GE.ge.{u_2} α (@Preorder.toLE.{u_2} α inst) c a → @GT.gt.{u_2} α (@Preorder.toLT.{u_2} α inst) c b","statement_sha256":"f038bcdb0bfd0f6b6bf1e20dd232c186e736e046597ac6fdb3f2618dadfdbb8b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-547d6644424bff084cd5c9a76fe466726e0ca988383359f53173b00d70df5087","qualified_name":"ge_trans'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":3918,"column":19,"line":124},"start":{"byte":3901,"column":2,"line":124}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Preorder.{u_1} α] {a b c : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) a b →\n    @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) b c → @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) a 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@GT.gt.{u_2} α (@Preorder.toLT.{u_2} α inst) a b → @GT.gt.{u_2} α (@Preorder.toLT.{u_2} α inst) c d","statement_sha256":"c2e760c56bdf413e764d5dd8fdd54d1501bc942b366b319c884e9e93c075f438","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ba88742320e68233b7d289bb6c5d09eab355a7f6e2fd060e0d44bbdd0e9541f7","qualified_name":"gt_trans'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":3967,"column":19,"line":125},"start":{"byte":3950,"column":2,"line":125}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Preorder.{u_1} α] {a b c : α},\n  @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) a b →\n    @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) b c → @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) a c","statement_sha256":"c97a6cd2f006f0023564eaaef27daff4c876e12b3607cae70cf0e5d8cbaff8d6","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-3985c3f1eec9ee581abe5f8b4baa42ece3c20f50a43290f27b1e3bda2929c150","qualified_name":"instDenselyOrderedForall","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":40151,"column":87,"line":1016},"start":{"byte":39685,"column":0,"line":1006}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : (i : ι) → Preorder.{u_4} (π i)]\n  [∀ (i : ι), @DenselyOrdered.{u_4} (π i) (@Preorder.toLT.{u_4} (π i) (inst i))],\n  @DenselyOrdered.{max u_1 u_4} ((i : ι) → π i)\n    (@Preorder.toLT.{max u_1 u_4} ((i : ι) → π i) (@Pi.preorder.{u_1, u_4} ι π inst))","statement_sha256":"25ea9f52c1a16a1cbafe1dd9e4d2f0854b4d74536d2f6370a8ecce6e4000798a","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-9815d7298d6f086ec7343eb40d02ec01ccdc86cea0ab917ece5152202c6a9326","qualified_name":"instDenselyOrderedProd","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":39683,"column":75,"line":1004},"start":{"byte":39226,"column":0,"line":997}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_3} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_3} β]\n  [@DenselyOrdered.{u_2} α (@Preorder.toLT.{u_2} α inst)] [@DenselyOrdered.{u_3} β (@Preorder.toLT.{u_3} β inst_1)],\n  @DenselyOrdered.{max u_3 u_2} (Prod.{u_2, u_3} α β)\n    (@Preorder.toLT.{max u_2 u_3} (Prod.{u_2, u_3} α β) (@Prod.instPreorder.{u_2, u_3} α β inst inst_1))","statement_sha256":"3679207dee246994bd526b04ba95c1cd3f32df2e2ae5f10d5b79a861e7dbd6b3","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-2cc36caa7a9e2276a72dcec0d8569445e0706e566405cc4552582887c330aac7","qualified_name":"instLinearOrderEmpty","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":42490,"column":47,"line":1067},"start":{"byte":42443,"column":0,"line":1067}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"LinearOrder.{0} 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1}","statement_sha256":"11416ee22afdd464c3edb5a89212d1ffadfcaeaae3c50af1cb79d8c809430d40","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-d49767e702c9a7c37227dda0a2df80fcd555b63d0fb3cc413b6d81abaefd1935","qualified_name":"le_iff_eq_or_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":8130,"column":86,"line":238},"start":{"byte":8016,"column":0,"line":237}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b)\n    (Or (@Eq.{u_2 + 1} α a b) (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b))","statement_sha256":"8c6d33b707b45eedbb824b16bcbad73cd828b4cd4e5c5bd575572d0c14872dff","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-5fc8cfc96161c7c35e6dd7bd4fde323b720b4286ad59cc4d98088081ff8a6d7c","qualified_name":"le_iff_eq_or_lt'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":8042,"column":26,"line":237},"start":{"byte":8018,"column":2,"line":237}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a)\n    (Or (@Eq.{u_2 + 1} α a b) (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a))","statement_sha256":"2ea240df47a0c7a6fb5a3a3bb33430462d3c464522f9590b077b7b191d1a64ca","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-b2308e8836007c2f319bc5c7bdd859ae0e94e6041d7d2e22b552c4b52bf950af","qualified_name":"le_iff_le_iff_lt_iff_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":15886,"column":92,"line":437},"start":{"byte":15634,"column":0,"line":434}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_5} [inst : LinearOrder.{u_2} α] [inst_1 : LinearOrder.{u_5} β] {a b : α} {c d : β},\n  Iff\n    (Iff\n      (@LE.le.{u_2} α\n        (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a b)\n      (@LE.le.{u_5} β\n        (@Preorder.toLE.{u_5} β (@PartialOrder.toPreorder.{u_5} β (@LinearOrder.toPartialOrder.{u_5} β inst_1))) c d))\n    (Iff\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a)\n      (@LT.lt.{u_5} β\n        (@Preorder.toLT.{u_5} β (@PartialOrder.toPreorder.{u_5} β (@LinearOrder.toPartialOrder.{u_5} β inst_1))) d c))","statement_sha256":"ac919e1eb695901b39bd3da2bbb943b38b39b3e2016e250a7a8c385ae598df93","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-fb9496e0fedb50252319b52c9a21bd64b7649277462284f3766a1ebaf3201ca3","qualified_name":"le_imp_eq_iff_le_imp_ge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7815,"column":35,"line":230},"start":{"byte":7635,"column":0,"line":227}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b → @Eq.{u_2 + 1} α a b)\n    (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b →\n      @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a)","statement_sha256":"810ecfdc627501a37b898877f15c7a955556449afefe2b6f860f82b79d5f2e45","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-1cebfe09c9f31f7424d1ba7d7e5dd44e0a9505041e1b7d746b64ea8b502bcd34","qualified_name":"le_imp_eq_iff_le_imp_ge'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":7669,"column":34,"line":227},"start":{"byte":7637,"column":2,"line":227}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a → @Eq.{u_2 + 1} α a b)\n    (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a →\n      @LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b)","statement_sha256":"5ed839fdf0ebdb879a448aaf146bf3ed536cf2e41eea997c4101e6ca538bdcb5","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-018cb40571159983331cc95d31ddf0b28d94063f65e7c27e74ea39ce5cb50eab","qualified_name":"le_imp_le_iff_lt_imp_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":15158,"column":54,"line":423},"start":{"byte":14948,"column":0,"line":420}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_5} [inst : LinearOrder.{u_2} α] [inst_1 : LinearOrder.{u_5} β] {a b : α} {c d : β},\n  Iff\n    (@LE.le.{u_2} α\n        (@Preorder.toLE.{u_2} α 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a)","statement_sha256":"a34abe4c4a6c3cdfe2ac51b2b1e80c698a40e0dbaf99f344e0ed9cd678874f85","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a99983f98e009cbd390013169043beb049e0bded24a76ef486961214759f968e","qualified_name":"le_imp_le_of_le_of_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":5559,"column":41,"line":163},"start":{"byte":5321,"column":0,"line":160}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : Preorder.{u_2} α] {a b c d : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) c a →\n    @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) b d →\n      @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) a b → @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) c d","statement_sha256":"bf827b6fb5a2bc41e82c7e32b37ed1bea52ffe9d1fc77f891447025d23ff25bd","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-c4994fc8c03245e8b2512968643b9d9461f3558234db8689d8cc61c86118c072","qualified_name":"le_of_forall_ge","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":5075,"column":25,"line":153},"start":{"byte":5052,"column":2,"line":153}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : Preorder.{u_2} α] {a b : α},\n  (∀ (c : α), @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) a c → @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) b c) →\n    @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) b a","statement_sha256":"d481209fc01ac3d1f19c3cc5a74ca90cdd9c2bd116a660604286e37f5769ec82","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-c12f42a642d8c86cca6181cec4c40de8bc26974885c23a9b21facc808e6dd00b","qualified_name":"le_of_forall_gt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":12740,"column":25,"line":356},"start":{"byte":12717,"column":2,"line":356}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c →\n        @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b c) →\n    @LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a","statement_sha256":"1dad963b0b128b20deca3057131b8c5662d3feb2c1808f415900aad01267ed2b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ff799edb267b87ec1627946823ee09bf33eac4f544c489b0b2c56f8ba79fcfc3","qualified_name":"le_of_forall_gt_imp_ge_of_dense","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":40522,"column":69,"line":1025},"start":{"byte":40237,"column":0,"line":1021}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α]\n  [@DenselyOrdered.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))]\n  {a₁ a₂ : α},\n  (∀ (a : α),\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₂ a →\n        @LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₁ a) →\n    @LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a₁ a₂","statement_sha256":"57d49a95575660981da12064379c298b649e3dd4dbc0fcb94b0b6f6e014a16bc","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-32975609d31b722027f05af18a0b34dea8ff1b817de6c0f3bcecc83a183427ce","qualified_name":"le_of_forall_gt_imp_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13050,"column":32,"line":364},"start":{"byte":13020,"column":2,"line":364}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c →\n        @Ne.{u_2 + 1} α c b) →\n    @LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a","statement_sha256":"d8851aa28ee1d6be7498c1c78902fee5970a5b959af11d7fbad49110cc33310b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-dc6865f883d9e675e17c04f1f1288937e67d3eda8c024afc7dc04b454c28bb2e","qualified_name":"le_of_forall_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":5157,"column":80,"line":154},"start":{"byte":5050,"column":0,"line":153}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : Preorder.{u_2} α] {a b : α},\n  (∀ (c : α), @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) c a → @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) c b) →\n    @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) a b","statement_sha256":"679293d8e48f5984b8a27933509e71c1fc383221cbeea337ada9248b8a0ed66b","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-d01a255a924dead389ead27ca67826a0e528d9f9a5ae888b1d84b49206de12dd","qualified_name":"le_of_forall_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":12852,"column":44,"line":358},"start":{"byte":12715,"column":0,"line":356}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a →\n        @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c b) →\n    @LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a b","statement_sha256":"20127ee8aa0ea5c65e8b20dd98176c8de7faac43fe16286332d68b1671b01d7d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-29877d9fe9b3edc2f56b7883f45fde112144d40ee2b4f3aae941bc362e0eadfe","qualified_name":"le_of_forall_lt_imp_le_of_dense","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":40278,"column":41,"line":1021},"start":{"byte":40239,"column":2,"line":1021}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α]\n  [@DenselyOrdered.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α 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a₁","statement_sha256":"3bbc638b5a871f7bc799df8bd008740217cb598c08ff94b08b360b8bb9374a95","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-7101325e6d578083477691e4100046e472b5358c6721b2341b31ffec42475bd2","qualified_name":"le_of_forall_lt_imp_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13157,"column":36,"line":366},"start":{"byte":13018,"column":0,"line":364}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      @LT.lt.{u_2} α\n          (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a →\n        @Ne.{u_2 + 1} α c b) →\n    @LE.le.{u_2} α\n      (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a b","statement_sha256":"5bbc52abb7dd798fe7e02ac916a0fda5b10f5e10037711bb8a05ef085b0f763e","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-08c64bfeaab95ed5dbc1d7fd5233a32c9b4cb5061187afe5b0c308501a8759a3","qualified_name":"le_of_strongLT","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":21862,"column":68,"line":593},"start":{"byte":21778,"column":0,"line":592}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : (i : ι) → Preorder.{u_4} (π i)] {a b : (i : ι) → π i},\n  @StrongLT.{u_1, u_4} ι π (fun i => @Preorder.toLT.{u_4} (π i) (inst i)) a b →\n    @LE.le.{max u_1 u_4} ((i : ι) → π i) (@Pi.hasLe.{u_1, u_4} ι π fun i => @Preorder.toLE.{u_4} (π i) (inst i)) a b","statement_sha256":"5d595623ca00e001d385f6cb24a078d52ffc38f2c39e553300ae13190d999fc2","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-23185abc9589c84be66f30291ae35344ccdc80e19e70ff4f5b79169b4dadea5c","qualified_name":"le_of_subsingleton","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":4855,"column":82,"line":147},"start":{"byte":4751,"column":0,"line":146}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : Preorder.{u_2} α] {a b : α} [Subsingleton.{u_2 + 1} α],\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) a b","statement_sha256":"89738dae463a913185a6a9418a8384f13308236148e7bd0dd85c48b805bb14f7","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-cce8d3bd952354ff3ad0ad2ec323715ca15db3e63dde06b4aafa25ca1a33bfb5","qualified_name":"le_trans'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":3947,"column":48,"line":124},"start":{"byte":3899,"column":0,"line":124}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Preorder.{u_1} α] {a b c : α},\n  @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) b a →\n    @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) c b → @LE.le.{u_1} α (@Preorder.toLE.{u_1} α inst) c a","statement_sha256":"9531903c0b4ae83ed037be699e7bc05e5e6a24249a1026ffd1a0cab7e28af3a4","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-768413f89a4825b0ee38961c94053a140359e156c17881ce98641d4bf9fb1bee","qualified_name":"le_update_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":22713,"column":52,"line":619},"start":{"byte":22528,"column":0,"line":617}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : DecidableEq.{u_1 + 1} ι] [inst_1 : (i : ι) → Preorder.{u_4} (π i)]\n  {x y : (i : ι) → π i} {i : ι} {a : π i},\n  Iff\n    (@LE.le.{max u_1 u_4} ((i : ι) → π i) (@Pi.hasLe.{u_1, u_4} ι π fun i => @Preorder.toLE.{u_4} (π i) (inst_1 i)) x\n      (@Function.update.{u_1 + 1, u_4 + 1} ι π inst y i a))\n    (And (@LE.le.{u_4} (π i) (@Preorder.toLE.{u_4} (π i) (inst_1 i)) (x i) a)\n      (∀ (j : ι), @Ne.{u_1 + 1} ι j i → @LE.le.{u_4} (π j) (@Preorder.toLE.{u_4} (π j) (inst_1 j)) (x j) (y j)))","statement_sha256":"cb20a1763ff29c1c8f7ae063519cdd9a769e95663a593a9ea4359048d6f147fc","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-fb408933897f8de3a2ae8007953185fe3923a169bd8b273627c5bfb3f2028a90","qualified_name":"le_update_self_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23338,"column":88,"line":635},"start":{"byte":23205,"column":0,"line":634}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : DecidableEq.{u_1 + 1} ι] [inst_1 : (i : ι) → Preorder.{u_4} (π i)]\n  {x : (i : ι) → π i} {i : ι} {a : π i},\n  Iff\n    (@LE.le.{max u_1 u_4} ((i : ι) → π i) (@Pi.hasLe.{u_1, u_4} ι π fun i => @Preorder.toLE.{u_4} (π i) (inst_1 i)) x\n      (@Function.update.{u_1 + 1, u_4 + 1} ι π inst x i a))\n    (@LE.le.{u_4} (π i) (@Preorder.toLE.{u_4} (π i) (inst_1 i)) (x i) 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(@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) a b) (@Ne.{u_2 + 1} α a b))","statement_sha256":"4daf27bd75a4c6df6535bb92c258f413e139db446d6a36b853b52c9d96194e97","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-eab4043810a51c20200bdee3759572b03ae22561bfac4fe3c37180f0ed8fefb6","qualified_name":"lt_iff_le_and_ne'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":8159,"column":27,"line":240},"start":{"byte":8134,"column":2,"line":240}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : PartialOrder.{u_2} α] {a b : α},\n  Iff (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a)\n    (And (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α inst)) b a) (@Ne.{u_2 + 1} α a b))","statement_sha256":"9da9a0fb4aa133de31212fad8955a4e83c58d710782214e35ccbbacfd4f11d75","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-5b21c18ad75cb30553eef25dae365742b9cf73b6c1545984a55b05e6cdd07662","qualified_name":"lt_iff_lt_of_le_iff_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":15632,"column":101,"line":432},"start":{"byte":15425,"column":0,"line":430}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_5} [inst : LinearOrder.{u_2} α] [inst_1 : LinearOrder.{u_5} β] {a b : α} {c d : β},\n  Iff\n      (@LE.le.{u_2} α\n        (@Preorder.toLE.{u_2} α 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c)","statement_sha256":"7325fd09f22b4b16d296fb4a3576261f770b49e39b64a1953ead395ec3a8a37f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ec69418715fda5de0c5d365315ddd43f24bd87316e30eb728741bba8bce4ae83","qualified_name":"lt_iff_lt_of_le_iff_le'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":15423,"column":84,"line":428},"start":{"byte":15160,"column":0,"line":425}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_5} [inst : Preorder.{u_2} α] [inst_1 : Preorder.{u_5} β] {a b : α} {c d : β},\n  Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) a b) (@LE.le.{u_5} β (@Preorder.toLE.{u_5} β inst_1) c d) →\n    Iff (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) b a) (@LE.le.{u_5} β (@Preorder.toLE.{u_5} β inst_1) d c) →\n      Iff (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α inst) b a) (@LT.lt.{u_5} β (@Preorder.toLT.{u_5} β inst_1) d c)","statement_sha256":"cf9d5e5453e9e85252e21a01eeacbe5efb70938e8507dca2c57a343ad6b4d39a","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-64934a12d1cc1d2e080983f09c5f7406b872a327a161dc2a6400db95dffd5bcd","qualified_name":"lt_imp_lt_of_le_imp_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":14946,"column":41,"line":418},"start":{"byte":14749,"column":0,"line":415}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} {β : Type u_5} [inst : LinearOrder.{u_2} α] [inst_1 : Preorder.{u_5} β] {a b : α} {c d : β},\n  (@LE.le.{u_2} α (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst)))\n        a b →\n      @LE.le.{u_5} β (@Preorder.toLE.{u_5} β inst_1) c d) →\n    @LT.lt.{u_5} β (@Preorder.toLT.{u_5} β inst_1) d c →\n      @LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a","statement_sha256":"e92676a60ad1b8e3705f91de27a6dd7d47a097a155ed01cedfbd51035f8e0595","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-668cef4ed7817b89edc198f2cd876e8fe868b016cb4a3155d080f659bcdf708d","qualified_name":"lt_imp_lt_of_le_of_le","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":5799,"column":47,"line":168},"start":{"byte":5561,"column":0,"line":165}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : Preorder.{u_2} α] {a b c d : α},\n  @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) c a →\n    @LE.le.{u_2} α (@Preorder.toLE.{u_2} α inst) b d →\n      @LT.lt.{u_2} α (@Preorder.toLT.{u_2} α inst) a b → @LT.lt.{u_2} α (@Preorder.toLT.{u_2} α inst) c d","statement_sha256":"392d434220bc9bdfd9c5996cb94ac327cf15d4abfe43a8ef8fcbc125c30d8957","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-d4d9ec631740eab1a98771e7b0857b64b111d8ad880ee4ece309be025fa680ac","qualified_name":"lt_of_forall_ge_imp_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13191,"column":32,"line":368},"start":{"byte":13161,"column":2,"line":368}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      @LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a c →\n        @Ne.{u_2 + 1} α c b) →\n    @LT.lt.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a","statement_sha256":"7da11d1a219b43bc5ccba272b7f384ec36145862b493c5c324de1a235e36031c","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-3c743fdd56968199887037ae830bb7af4bc230b285d0ec41d331e70f49865adf","qualified_name":"lt_of_forall_le_imp_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":13298,"column":36,"line":370},"start":{"byte":13159,"column":0,"line":368}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  (∀ (c : α),\n      @LE.le.{u_2} α\n          (@Preorder.toLE.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) c a →\n        @Ne.{u_2 + 1} α c b) →\n    @LT.lt.{u_2} α\n      (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a b","statement_sha256":"380c331fbca839f237823cfea41ae6bac6ed45ecce999e3e6ea3006b86ade096","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4d9ab7a3d27bab7fa53d4e1c19840455d79cb1ad32fadfe280a43a23d1af1a26","qualified_name":"lt_of_strongLT","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":22014,"column":56,"line":598},"start":{"byte":21864,"column":0,"line":595}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : (i : ι) → Preorder.{u_4} (π i)] {a b : (i : ι) → π i}\n  [Nonempty.{u_1 + 1} ι],\n  @StrongLT.{u_1, u_4} ι π (fun i => @Preorder.toLT.{u_4} (π i) (inst i)) a b →\n    @LT.lt.{max u_1 u_4} ((i : ι) → π i)\n      (@Preorder.toLT.{max u_1 u_4} ((i : ι) → π i) (@Pi.preorder.{u_1, u_4} ι π inst)) a b","statement_sha256":"14e549f2e992321ba3ca06adfea1d8ac07e54a52a90a64e459f5c7361a3aa848","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-fb07c21ffdbaad61d62a21eac2380a601b73d5092135d94b07eb48c0def0f461","qualified_name":"lt_or_gt_iff_ne'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":11913,"column":26,"line":335},"start":{"byte":11889,"column":2,"line":335}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  Iff\n    (Or\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a)\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a b))\n    (@Ne.{u_2 + 1} α a b)","statement_sha256":"ac29ecf6662cf9d61ba8882ce0c567e5de628d82c6033ce655475f51b7e81aa7","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-c06d45df6b523032678bf68e4f8e3f80d57f0a5bc916bda4b7b535dbaf24da82","qualified_name":"lt_or_lt_iff_ne","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":12000,"column":22,"line":337},"start":{"byte":11822,"column":0,"line":334}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] {a b : α},\n  Iff\n    (Or\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) a b)\n      (@LT.lt.{u_2} α\n        (@Preorder.toLT.{u_2} α (@PartialOrder.toPreorder.{u_2} α (@LinearOrder.toPartialOrder.{u_2} α inst))) b a))\n    (@Ne.{u_2 + 1} α a b)","statement_sha256":"802f12967e28e1f7ba52c43df2f605322e347f2bb240b16631f12cd8ab705133","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-433cc24043c8f5e06eaaaa39f39b11975c9c272dbb0d66a5e238d055f23a9a95","qualified_name":"lt_self_iff_false","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":3897,"column":83,"line":122},"start":{"byte":3806,"column":0,"line":121}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : Preorder.{u_2} α] (x : α), Iff (@LT.lt.{u_2} α (@Preorder.toLT.{u_2} α inst) x x) False","statement_sha256":"cf85dd19ad2e58febcd70f644fcbaa1fd82b526697a48a68e2873cb25465da33","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4c0f66b85ca8a2c53acb183ed770963cf0491c179582bc0164a528b91296b445","qualified_name":"lt_trans'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":3996,"column":48,"line":125},"start":{"byte":3948,"column":0,"line":125}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_1} [inst : Preorder.{u_1} α] {a b c : α},\n  @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) b a →\n    @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) c b → @LT.lt.{u_1} α (@Preorder.toLT.{u_1} α inst) c a","statement_sha256":"21a06770221a56c9f6b8a4b88995dcf02fc6de1dcca1eb778441695dc2706149","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-d44d88a291e15f1f76bca52a05328e26fdaecb95a988f2772e0ea11059f37d56","qualified_name":"lt_update_self_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23472,"column":87,"line":638},"start":{"byte":23340,"column":0,"line":637}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : DecidableEq.{u_1 + 1} ι] [inst_1 : (i : ι) → Preorder.{u_4} (π i)]\n  {x : (i : ι) → π i} {i : ι} {a : π i},\n  Iff\n    (@LT.lt.{max u_1 u_4} ((i : ι) → π i)\n      (@Preorder.toLT.{max u_1 u_4} ((i : ι) → π i) (@Pi.preorder.{u_1, u_4} ι π inst_1)) x\n      (@Function.update.{u_1 + 1, u_4 + 1} ι π inst x i a))\n    (@LT.lt.{u_4} (π i) (@Preorder.toLT.{u_4} (π i) (inst_1 i)) (x i) a)","statement_sha256":"0e31d4043d9f498247d944076012c96b9c708e698ac67b089e52a96ce5a7448f","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-221c8abf115e38d2a0bb6c4bb032c751e11ae2a98f80f70fbb59bc7c94c08c7a","qualified_name":"max_def_lt","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":14691,"column":57,"line":408},"start":{"byte":14542,"column":0,"line":406}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {α : Type u_2} [inst : LinearOrder.{u_2} α] (a b : α),\n  @Eq.{u_2 + 1} α (@Max.max.{u_2} α (@LinearOrder.toMax.{u_2} α inst) a b)\n    (@ite.{u_2 + 1} α\n      (@LT.lt.{u_2} α\n        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j)))","statement_sha256":"3ac8c5a7336469efacc05507629aa4f85ca9ab273f303d4250cd11305e5b890c","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-ac01d9c44669d3b852dedd9ff56ccf6e81d68dc76f6b3a51cb1458347275ff0b","qualified_name":"update_le_self_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23248,"column":43,"line":634},"start":{"byte":23207,"column":2,"line":634}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : DecidableEq.{u_1 + 1} ι] [inst_1 : (i : ι) → Preorder.{u_4} (π i)]\n  {x : (i : ι) → π i} {i : ι} {a : π i},\n  Iff\n    (@LE.le.{max 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(@LE.le.{max u_1 u_4} ((a : ι) → π a) (@Pi.hasLe.{u_1, u_4} ι π fun i => @Preorder.toLE.{u_4} (π i) (inst_1 i))\n      (@Function.update.{u_1 + 1, u_4 + 1} ι π inst x i a) (@Function.update.{u_1 + 1, u_4 + 1} ι π inst y i b))\n    (And (@LE.le.{u_4} (π i) (@Preorder.toLE.{u_4} (π i) (inst_1 i)) a b)\n      (∀ (j : ι), @Ne.{u_1 + 1} ι j i → @LE.le.{u_4} (π j) (@Preorder.toLE.{u_4} (π j) (inst_1 j)) (x j) (y j)))","statement_sha256":"739df0f18b946227f5b147830ca1c2a2d8dda43f289769c76ad8a6d946c2b981","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-a2c603d2d72a8b8f230a97e22ffe2f896a438c84f7ce076bea52ef56247d96c8","qualified_name":"update_le_update_iff'","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23038,"column":29,"line":628},"start":{"byte":22907,"column":0,"line":626}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : DecidableEq.{u_1 + 1} ι] [inst_1 : (i : ι) → Preorder.{u_4} (π i)]\n  {x : (i : ι) → π i} {i : ι} {a b : π i},\n  Iff\n    (@LE.le.{max u_1 u_4} ((a : ι) → π a) (@Pi.hasLe.{u_1, u_4} ι π fun i => @Preorder.toLE.{u_4} (π i) (inst_1 i))\n      (@Function.update.{u_1 + 1, u_4 + 1} ι π inst x i a) (@Function.update.{u_1 + 1, u_4 + 1} ι π inst x i b))\n    (@LE.le.{u_4} (π i) (@Preorder.toLE.{u_4} (π i) (inst_1 i)) a b)","statement_sha256":"d19df0aa54593b16f23027e02747725109419852b6bacf03418f8b16fb59f8ad","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-4ffd4301136b1938c09a62589021c2eabd4fb90d6d617e7e1d1ff491796b442c","qualified_name":"update_lt_self_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23383,"column":43,"line":637},"start":{"byte":23342,"column":2,"line":637}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : DecidableEq.{u_1 + 1} ι] [inst_1 : (i : ι) → Preorder.{u_4} (π i)]\n  {x : (i : ι) → π i} {i : ι} {a : π i},\n  Iff\n    (@LT.lt.{max u_1 u_4} ((i : ι) → π i)\n      (@Preorder.toLT.{max u_1 u_4} ((i : ι) → π i) (@Pi.preorder.{u_1, u_4} ι π inst_1))\n      (@Function.update.{u_1 + 1, u_4 + 1} ι π inst x i a) x)\n    (@LT.lt.{u_4} (π i) (@Preorder.toLT.{u_4} (π i) (inst_1 i)) a (x i))","statement_sha256":"6b23a944629ea60cd038e6cc526dfd8c5df7406e173555a4e200b9f71bedb47d","verification_status":"NOT_CHECKED"},{"authority":"NONE","authorization":"NONE","current_applicability":"NOT_CHECKED","extensions":[],"freshness":"PINNED_HISTORICAL_SOURCE","kind":"theorem","object_contract_version":1,"object_id":"object-e445606784457f63ffa8db7444e710301f0136f98bda6478efa20df1ffa7fdc4","qualified_name":"update_lt_update_iff","source":{"commit":"d13f23b723b8a846827a245b89c10fc7d3f11612","module":"Mathlib.Order.Basic","path":"Mathlib/Order/Basic.lean","range":{"end":{"byte":23203,"column":69,"line":632},"start":{"byte":23040,"column":0,"line":630}},"repository":"https://github.com/leanprover-community/mathlib4","tree":"efa132418a56d0e38c56cc412a06df22178dac5c"},"source_status":"SOURCE_INDEXED","statement":"∀ {ι : Type u_1} {π : ι → Type u_4} [inst : DecidableEq.{u_1 + 1} ι] [inst_1 : (i : ι) → Preorder.{u_4} (π i)]\n  {x : (i : ι) → π i} {i : ι} {a b : π i},\n  Iff\n    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