Documentation Verification Report

Sqrt

📁 Source: Mathlib/Analysis/SpecialFunctions/Sqrt.lean

Statistics

Metric	Count
Definitions `sqPartialHomeomorph`	1
Theorems `contDiffAt_sqrt`, `deriv_sqrt_aux`, `hasDerivAt_sqrt`, `hasStrictDerivAt_sqrt`, `derivWithin_sqrt`, `deriv_sqrt`, `fderivWithin_sqrt`, `fderiv_sqrt`	8
Total	9

Real

Definitions

Name	Category	Theorems
`sqPartialHomeomorph` 📖	CompOp	—

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`contDiffAt_sqrt` 📖	mathematical	—	`ContDiffAt` `Real` `DenselyNormedField.toNontriviallyNormedField` `denselyNormedField` `normedAddCommGroup` `NormedField.toNormedSpace` `NontriviallyNormedField.toNormedField` `sqrt`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `Nat.instAtLeastTwoHAddOfNat` `deriv_sqrt_aux`
`deriv_sqrt_aux` 📖	mathematical	—	`HasStrictDerivAt` `Real` `DenselyNormedField.toNontriviallyNormedField` `denselyNormedField` `instAddCommGroup` `NormedSpace.toModule` `normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `instMul` `IsTopologicalSemiring.toContinuousMul` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `sqrt` `DivInvMonoid.toDiv` `instDivInvMonoid` `instOne` `instOfNatAtLeastTwo` `instNatCast` `Nat.instAtLeastTwoHAddOfNat` `ContDiffAt` `normedAddCommGroup`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `Nat.instAtLeastTwoHAddOfNat` `Ne.lt_or_gt` `sqrt_eq_zero_of_nonpos` `LT.lt.le` `MulZeroClass.mul_zero` `div_zero` `Filter.Eventually.mono` `gt_mem_nhds` `instOrderTopologyReal` `HasStrictDerivAt.congr_of_eventuallyEq` `hasStrictDerivAt_const` `Filter.EventuallyEq.symm` `ContDiffAt.congr_of_eventuallyEq` `contDiffAt_const` `pow_one` `NormMulClass.toNoZeroDivisors` `NormedDivisionRing.toNormMulClass` `two_ne_zero` `CharZero.NeZero.two` `IsStrictOrderedRing.toCharZero` `instIsStrictOrderedRing` `LT.lt.ne'` `sqrt_pos` `HasStrictDerivAt.congr_simp` `one_div` `mul_inv_rev` `IsTopologicalDivisionRing.toIsTopologicalRing` `NormedDivisionRing.to_isTopologicalDivisionRing` `OpenPartialHomeomorph.hasStrictDerivAt_symm` `hasStrictDerivAt_pow` `OpenPartialHomeomorph.contDiffAt_symm_deriv` `instCompleteSpace` `hasDerivAt_pow` `ContDiffAt.pow` `contDiffAt_id`
`hasDerivAt_sqrt` 📖	mathematical	—	`HasDerivAt` `Real` `DenselyNormedField.toNontriviallyNormedField` `denselyNormedField` `instAddCommGroup` `NormedSpace.toModule` `normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `instMul` `IsTopologicalSemiring.toContinuousMul` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `sqrt` `DivInvMonoid.toDiv` `instDivInvMonoid` `instOne` `instOfNatAtLeastTwo` `instNatCast` `Nat.instAtLeastTwoHAddOfNat`	—	`HasStrictDerivAt.hasDerivAt` `Nat.instAtLeastTwoHAddOfNat` `hasStrictDerivAt_sqrt`
`hasStrictDerivAt_sqrt` 📖	mathematical	—	`HasStrictDerivAt` `Real` `DenselyNormedField.toNontriviallyNormedField` `denselyNormedField` `instAddCommGroup` `NormedSpace.toModule` `normedField` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `normedCommRing` `NormedField.toNormedSpace` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `pseudoMetricSpace` `ContinuousMul.to_continuousSMul` `SeminormedRing.toPseudoMetricSpace` `SeminormedCommRing.toSeminormedRing` `NormedField.toNormedCommRing` `NontriviallyNormedField.toNormedField` `instMul` `IsTopologicalSemiring.toContinuousMul` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `sqrt` `DivInvMonoid.toDiv` `instDivInvMonoid` `instOne` `instOfNatAtLeastTwo` `instNatCast` `Nat.instAtLeastTwoHAddOfNat`	—	`ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `instIsTopologicalRingReal` `Nat.instAtLeastTwoHAddOfNat` `deriv_sqrt_aux`

(root)

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`derivWithin_sqrt` 📖	mathematical	`DifferentiableWithinAt` `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` `UniqueDiffWithinAt` `Real.semiring`	`derivWithin` `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` `Real.sqrt` `DivInvMonoid.toDiv` `Real.instDivInvMonoid` `Real.instMul` `instOfNatAtLeastTwo` `Real.instNatCast` `Nat.instAtLeastTwoHAddOfNat`	—	`HasDerivWithinAt.derivWithin` `Nat.instAtLeastTwoHAddOfNat` `HasDerivWithinAt.sqrt` `DifferentiableWithinAt.hasDerivWithinAt`
`deriv_sqrt` 📖	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`	`deriv` `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` `Real.sqrt` `DivInvMonoid.toDiv` `Real.instDivInvMonoid` `Real.instMul` `instOfNatAtLeastTwo` `Real.instNatCast` `Nat.instAtLeastTwoHAddOfNat`	—	`HasDerivAt.deriv` `Nat.instAtLeastTwoHAddOfNat` `HasDerivAt.sqrt` `DifferentiableAt.hasDerivAt`
`fderivWithin_sqrt` 📖	mathematical	`DifferentiableWithinAt` `Real` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `NormedAddCommGroup.toAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NormedAddCommGroup.toSeminormedAddCommGroup` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `Real.instAddCommGroup` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `Real.pseudoMetricSpace` `UniqueDiffWithinAt` `Real.semiring`	`fderivWithin` `Real` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `NormedAddCommGroup.toAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NormedAddCommGroup.toSeminormedAddCommGroup` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `Real.instAddCommGroup` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `Real.pseudoMetricSpace` `Real.sqrt` `ContinuousLinearMap` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `NormedField.toField` `NontriviallyNormedField.toNormedField` `RingHom.id` `Semiring.toNonAssocSemiring` `AddCommGroup.toAddCommMonoid` `ContinuousLinearMap.instSMul` `NonUnitalNonAssocSemiring.toDistribSMul` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `Algebra.to_smulCommClass` `Real.instCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `SeparatelyContinuousMul.to_continuousSMul` `Real.instMul` `IsSemitopologicalSemiring.toSeparatelyContinuousMul` `IsSemitopologicalRing.toIsSemitopologicalSemiring` `IsTopologicalRing.toIsSemitopologicalRing` `instIsTopologicalRingReal` `DivInvMonoid.toDiv` `Real.instDivInvMonoid` `Real.instOne` `instOfNatAtLeastTwo` `Real.instNatCast` `Nat.instAtLeastTwoHAddOfNat`	—	`HasFDerivWithinAt.fderivWithin` `Algebra.to_smulCommClass` `SeparatelyContinuousMul.to_continuousSMul` `IsSemitopologicalSemiring.toSeparatelyContinuousMul` `IsSemitopologicalRing.toIsSemitopologicalSemiring` `IsTopologicalRing.toIsSemitopologicalRing` `instIsTopologicalRingReal` `Nat.instAtLeastTwoHAddOfNat` `IsTopologicalAddGroup.toContinuousAdd` `SeminormedAddCommGroup.toIsTopologicalAddGroup` `IsBoundedSMul.continuousSMul` `NormedSpace.toIsBoundedSMul` `IsSemitopologicalSemiring.toContinuousAdd` `ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `TopologicalSpace.t2Space_of_metrizableSpace` `EMetricSpace.metrizableSpace` `HasFDerivWithinAt.sqrt` `DifferentiableWithinAt.hasFDerivWithinAt`
`fderiv_sqrt` 📖	mathematical	`DifferentiableAt` `Real` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `NormedAddCommGroup.toAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NormedAddCommGroup.toSeminormedAddCommGroup` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `Real.instAddCommGroup` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `Real.pseudoMetricSpace`	`fderiv` `Real` `DenselyNormedField.toNontriviallyNormedField` `Real.denselyNormedField` `NormedAddCommGroup.toAddCommGroup` `NormedSpace.toModule` `Real.normedField` `NormedAddCommGroup.toSeminormedAddCommGroup` `UniformSpace.toTopologicalSpace` `PseudoMetricSpace.toUniformSpace` `SeminormedAddCommGroup.toPseudoMetricSpace` `Real.instAddCommGroup` `NonUnitalSeminormedRing.toSeminormedAddCommGroup` `NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing` `SeminormedCommRing.toNonUnitalSeminormedCommRing` `NormedCommRing.toSeminormedCommRing` `Real.normedCommRing` `NormedField.toNormedSpace` `Real.pseudoMetricSpace` `Real.sqrt` `ContinuousLinearMap` `DivisionSemiring.toSemiring` `Semifield.toDivisionSemiring` `Field.toSemifield` `NormedField.toField` `NontriviallyNormedField.toNormedField` `RingHom.id` `Semiring.toNonAssocSemiring` `AddCommGroup.toAddCommMonoid` `ContinuousLinearMap.instSMul` `NonUnitalNonAssocSemiring.toDistribSMul` `NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring` `NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing` `NonUnitalCommRing.toNonUnitalNonAssocCommRing` `NonUnitalNormedCommRing.toNonUnitalCommRing` `NormedCommRing.toNonUnitalNormedCommRing` `Algebra.to_smulCommClass` `Real.instCommSemiring` `CommSemiring.toSemiring` `Algebra.id` `SeparatelyContinuousMul.to_continuousSMul` `Real.instMul` `IsSemitopologicalSemiring.toSeparatelyContinuousMul` `IsSemitopologicalRing.toIsSemitopologicalSemiring` `IsTopologicalRing.toIsSemitopologicalRing` `instIsTopologicalRingReal` `DivInvMonoid.toDiv` `Real.instDivInvMonoid` `Real.instOne` `instOfNatAtLeastTwo` `Real.instNatCast` `Nat.instAtLeastTwoHAddOfNat`	—	`HasFDerivAt.fderiv` `Algebra.to_smulCommClass` `SeparatelyContinuousMul.to_continuousSMul` `IsSemitopologicalSemiring.toSeparatelyContinuousMul` `IsSemitopologicalRing.toIsSemitopologicalSemiring` `IsTopologicalRing.toIsSemitopologicalRing` `instIsTopologicalRingReal` `Nat.instAtLeastTwoHAddOfNat` `IsTopologicalAddGroup.toContinuousAdd` `SeminormedAddCommGroup.toIsTopologicalAddGroup` `IsBoundedSMul.continuousSMul` `NormedSpace.toIsBoundedSMul` `IsSemitopologicalSemiring.toContinuousAdd` `ContinuousMul.to_continuousSMul` `IsTopologicalSemiring.toContinuousMul` `IsTopologicalRing.toIsTopologicalSemiring` `TopologicalSpace.t2Space_of_metrizableSpace` `EMetricSpace.metrizableSpace` `HasFDerivAt.sqrt` `DifferentiableAt.hasFDerivAt`

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