Documentation Verification Report

Darboux

📁 Source: Mathlib/Analysis/Calculus/Darboux.lean

Statistics

Metric	Count
Definitions	0
Theorems `image_deriv`, `image_derivWithin`, `image_hasDerivWithinAt`, `image_deriv`, `image_derivWithin`, `image_hasDerivWithinAt`, `exists_hasDerivWithinAt_eq_of_ge_of_le`, `exists_hasDerivWithinAt_eq_of_gt_of_lt`, `exists_hasDerivWithinAt_eq_of_le_of_ge`, `exists_hasDerivWithinAt_eq_of_lt_of_gt`, `hasDerivWithinAt_forall_lt_or_forall_gt_of_forall_ne`	11
Total	11

Convex

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`image_deriv` 📖	mathematical	`Convex` `Real` `Real.semiring` `Real.partialOrder` `Real.instAddCommMonoid` `Algebra.toSMul` `Real.instCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `DifferentiableAt` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace`	`Set.image` `deriv`	—	`Set.OrdConnected.convex` `Real.instIsOrderedAddMonoid` `IsOrderedModule.toPosSMulMono` `IsStrictOrderedModule.toIsOrderedModule` `IsStrictOrderedRing.toIsStrictOrderedModule` `Real.instIsStrictOrderedRing` `Set.OrdConnected.image_deriv` `ordConnected`
`image_derivWithin` 📖	mathematical	`Convex` `Real` `Real.semiring` `Real.partialOrder` `Real.instAddCommMonoid` `Algebra.toSMul` `Real.instCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `DifferentiableOn` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace`	`Set.image` `derivWithin`	—	`Set.OrdConnected.convex` `Real.instIsOrderedAddMonoid` `IsOrderedModule.toPosSMulMono` `IsStrictOrderedModule.toIsOrderedModule` `IsStrictOrderedRing.toIsStrictOrderedModule` `Real.instIsStrictOrderedRing` `Set.OrdConnected.image_derivWithin` `ordConnected`
`image_hasDerivWithinAt` 📖	mathematical	`Convex` `Real` `Real.semiring` `Real.partialOrder` `Real.instAddCommMonoid` `Algebra.toSMul` `Real.instCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `HasDerivWithinAt` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `MulZeroClass.toMul` `MulZeroOneClass.toMulZeroClass` `MonoidWithZero.toMulZeroOneClass` `Semiring.toMonoidWithZero` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `NormedField.toField` `IsTopologicalSemiring.toContinuousMul` `NonUnitalSemiring.toNonUnitalNonAssocSemiring` `Semiring.toNonUnitalSemiring` `IsTopologicalRing.toIsTopologicalSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `instIsTopologicalRingReal`	`Set.image`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `Set.OrdConnected.convex` `Real.instIsOrderedAddMonoid` `IsOrderedModule.toPosSMulMono` `IsStrictOrderedModule.toIsOrderedModule` `IsStrictOrderedRing.toIsStrictOrderedModule` `Real.instIsStrictOrderedRing` `Set.OrdConnected.image_hasDerivWithinAt` `ordConnected`

Set.OrdConnected

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`image_deriv` 📖	mathematical	`DifferentiableAt` `Real` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace`	`Set.OrdConnected` `Real.instPreorder` `Set.image` `deriv`	—	`image_hasDerivWithinAt` `HasDerivAt.hasDerivWithinAt` `DifferentiableAt.hasDerivAt`
`image_derivWithin` 📖	mathematical	`DifferentiableOn` `Real` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace`	`Set.OrdConnected` `Real.instPreorder` `Set.image` `derivWithin`	—	`image_hasDerivWithinAt` `DifferentiableWithinAt.hasDerivWithinAt`
`image_hasDerivWithinAt` 📖	mathematical	`HasDerivWithinAt` `Real` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `MulZeroClass.toMul` `MulZeroOneClass.toMulZeroClass` `MonoidWithZero.toMulZeroOneClass` `Semiring.toMonoidWithZero` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `NormedField.toField` `IsTopologicalSemiring.toContinuousMul` `NonUnitalSemiring.toNonUnitalNonAssocSemiring` `Semiring.toNonUnitalSemiring` `IsTopologicalRing.toIsTopologicalSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `instIsTopologicalRingReal`	`Set.OrdConnected` `Real.instPreorder` `Set.image`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `Set.ordConnected_of_Ioo` `le_total` `out` `exists_hasDerivWithinAt_eq_of_gt_of_lt` `HasDerivWithinAt.mono` `Set.Ioo_subset_Icc_self` `exists_hasDerivWithinAt_eq_of_lt_of_gt`

(root)

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`exists_hasDerivWithinAt_eq_of_ge_of_le` 📖	mathematical	`Real` `Real.instLE` `HasDerivWithinAt` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `MulZeroClass.toMul` `MulZeroOneClass.toMulZeroClass` `MonoidWithZero.toMulZeroOneClass` `Semiring.toMonoidWithZero` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `NormedField.toField` `IsTopologicalSemiring.toContinuousMul` `NonUnitalSemiring.toNonUnitalNonAssocSemiring` `Semiring.toNonUnitalSemiring` `IsTopologicalRing.toIsTopologicalSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `instIsTopologicalRingReal` `Set.Icc` `Real.instPreorder`	`Set` `Set.instMembership` `Set.image`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `Set.OrdConnected.out` `Set.OrdConnected.image_hasDerivWithinAt` `Set.ordConnected_Icc` `Set.mem_image_of_mem` `Set.left_mem_Icc` `Set.right_mem_Icc`
`exists_hasDerivWithinAt_eq_of_gt_of_lt` 📖	mathematical	`Real` `Real.instLE` `HasDerivWithinAt` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `MulZeroClass.toMul` `MulZeroOneClass.toMulZeroClass` `MonoidWithZero.toMulZeroOneClass` `Semiring.toMonoidWithZero` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `NormedField.toField` `IsTopologicalSemiring.toContinuousMul` `NonUnitalSemiring.toNonUnitalNonAssocSemiring` `Semiring.toNonUnitalSemiring` `IsTopologicalRing.toIsTopologicalSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `instIsTopologicalRingReal` `Set.Icc` `Real.instPreorder` `Real.instLT`	`Set` `Set.instMembership` `Set.image` `Set.Ioo`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `LE.le.eq_or_lt` `lt_asymm` `IsBoundedSMul.continuousSMul` `NormedSpace.toIsBoundedSMul` `HasDerivWithinAt.congr_simp` `mul_one` `HasDerivWithinAt.sub` `HasDerivWithinAt.const_mul` `hasDerivWithinAt_id` `IsCompact.exists_isMinOn` `instClosedIicTopology` `HasSolidNorm.orderClosedTopology` `instHasSolidNormReal` `Real.instIsOrderedAddMonoid` `CompactIccSpace.isCompact_Icc` `ConditionallyCompleteLinearOrder.toCompactIccSpace` `instOrderTopologyReal` `Set.nonempty_Icc` `HasDerivWithinAt.continuousWithinAt` `sub_nonneg` `covariant_swap_add_of_covariant_add` `IsOrderedAddMonoid.toAddLeftMono` `nonneg_of_mul_nonneg_right` `IsStrictOrderedRing.toPosMulStrictMono` `Real.instIsStrictOrderedRing` `sub_mem_posTangentConeAt_of_segment_subset` `Set.Subset.rfl` `segment_eq_Icc` `IsLocalMinOn.hasFDerivWithinAt_nonneg` `IsMinOn.localize` `Set.left_mem_Icc` `sub_pos` `IsRightCancelAdd.addRightStrictMono_of_addRightMono` `AddRightCancelSemigroup.toIsRightCancelAdd` `not_le_of_gt` `LE.le.eq_or_lt'` `sub_nonpos` `nonpos_of_mul_nonneg_right` `IsStrictOrderedRing.toMulPosStrictMono` `segment_symm` `subset_refl` `Set.instReflSubset` `Set.right_mem_Icc` `sub_lt_zero` `sub_eq_zero` `mem_interior_iff_mem_nhds` `interior_Icc` `LinearOrderedSemiField.toDenselyOrdered` `PosMulReflectLE.toPosMulReflectLT` `PosMulStrictMono.toPosMulReflectLE` `instNoMinOrderOfNontrivial` `Real.instIsOrderedRing` `Real.instNontrivial` `instNoMaxOrderOfNontrivial` `IsLocalMin.hasDerivAt_eq_zero` `IsMinOn.isLocalMin` `HasDerivWithinAt.hasDerivAt`
`exists_hasDerivWithinAt_eq_of_le_of_ge` 📖	mathematical	`Real` `Real.instLE` `HasDerivWithinAt` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `MulZeroClass.toMul` `MulZeroOneClass.toMulZeroClass` `MonoidWithZero.toMulZeroOneClass` `Semiring.toMonoidWithZero` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `NormedField.toField` `IsTopologicalSemiring.toContinuousMul` `NonUnitalSemiring.toNonUnitalNonAssocSemiring` `Semiring.toNonUnitalSemiring` `IsTopologicalRing.toIsTopologicalSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `instIsTopologicalRingReal` `Set.Icc` `Real.instPreorder`	`Set` `Set.instMembership` `Set.image`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `Set.OrdConnected.out` `Set.OrdConnected.image_hasDerivWithinAt` `Set.ordConnected_Icc` `Set.mem_image_of_mem` `Set.left_mem_Icc` `Set.right_mem_Icc`
`exists_hasDerivWithinAt_eq_of_lt_of_gt` 📖	mathematical	`Real` `Real.instLE` `HasDerivWithinAt` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `MulZeroClass.toMul` `MulZeroOneClass.toMulZeroClass` `MonoidWithZero.toMulZeroOneClass` `Semiring.toMonoidWithZero` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `NormedField.toField` `IsTopologicalSemiring.toContinuousMul` `NonUnitalSemiring.toNonUnitalNonAssocSemiring` `Semiring.toNonUnitalSemiring` `IsTopologicalRing.toIsTopologicalSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `instIsTopologicalRingReal` `Set.Icc` `Real.instPreorder` `Real.instLT`	`Set` `Set.instMembership` `Set.image` `Set.Ioo`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `exists_hasDerivWithinAt_eq_of_gt_of_lt` `HasDerivWithinAt.neg` `neg_lt_neg` `Real.instIsOrderedAddMonoid` `neg_injective`
`hasDerivWithinAt_forall_lt_or_forall_gt_of_forall_ne` 📖	mathematical	`Convex` `Real` `Real.semiring` `Real.partialOrder` `Real.instAddCommMonoid` `Algebra.toSMul` `Real.instCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `HasDerivWithinAt` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `Real.instAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `Real.pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `MulZeroClass.toMul` `MulZeroOneClass.toMulZeroClass` `MonoidWithZero.toMulZeroOneClass` `Semiring.toMonoidWithZero` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `NormedField.toField` `IsTopologicalSemiring.toContinuousMul` `NonUnitalSemiring.toNonUnitalNonAssocSemiring` `Semiring.toNonUnitalSemiring` `IsTopologicalRing.toIsTopologicalSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `instIsTopologicalRingReal`	`Real.instLT`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `Mathlib.Tactic.Contrapose.contrapose₁` `Mathlib.Tactic.Push.not_forall_eq` `Set.OrdConnected.out` `Set.OrdConnected.image_hasDerivWithinAt` `Convex.ordConnected` `Real.instIsStrictOrderedRing` `Set.mem_image_of_mem`

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