`Carleson/LipschitzNorm.lean`

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8 declarations, 0 contain sorry, 0 declared as axioms

Inhomogeneous Lipschitz norm

This file defines the Lipschitz norm, which probably in some form should end up in Mathlib. Lemmas about this norm that are proven in Carleson are collected here.

TODO: Assess Mathlib-readiness, complete basic results, optimize imports.

Imports (1)

Carleson.Defs

Imported by 2 files

def iLipNNNorm The NNReal version of the inhomogeneous Lipschitz norm on a ball, iLipENorm line 36 source
Source documentation
The NNReal version of the inhomogeneous Lipschitz norm on a ball, iLipENorm.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NormedField, PseudoMetricSpace, Real, NNReal
Proof uses: ENNReal.toNNReal, iLipENorm
```
def iLipNNNorm (φ : X → 𝕜) (x₀ : X) (R : ℝ) : ℝ≥0 :=
…
```
lemma iLipENorm_le_add line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, NNReal, MetricSpace, NormedField, Membership.mem, Set, Metric.ball, LE.le, Norm.norm, Ne, HSub.hSub, HDiv.hDiv, HMul.hMul, Dist.dist, ENNReal, iLipENorm, HAdd.hAdd
Proof uses: add_le_add, iSup, ENorm.enorm, ENNReal.ofReal, EDist.edist, Eq.mpr, NNNorm.nnnorm, id, Eq, Eq.trans, forall_congr, implies_congr, Eq.refl, ENNReal.mul_le_of_le_div', Not, congrFun', congrArg, funext, iSup_congr_Prop, Iff.of_eq and 19 more
103 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
lemma iLipENorm_le line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, NNReal, MetricSpace, NormedField, Membership.mem, Set, Metric.ball, LE.le, Norm.norm, HMul.hMul, Inv.inv, OfNat.ofNat, Nat, Ne, HSub.hSub, HDiv.hDiv, Dist.dist, ENNReal, iLipENorm
Proof uses: LE.le.trans_eq, HAdd.hAdd, iLipENorm_le_add, of_eq_true, Eq, Eq.trans, True, congrFun', congrArg, congr, ENNReal.coe_inv, Not, False, not_false_eq_true, ENNReal.inv_two_add_inv_two, one_mul, eq_self
80 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma enorm_le_iLipENorm_of_mem line 69 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, MetricSpace, NormedField, Membership.mem, Set, Metric.ball, LE.le, ENNReal, ENorm.enorm, iLipENorm
Proof uses: le_trans, iSup, HAdd.hAdd, HMul.hMul, ENNReal.ofReal, Ne, HDiv.hDiv, HSub.hSub, EDist.edist, Eq.mpr, id, Eq, Eq.trans, forall_congr, implies_congr, Eq.refl, le_self_add
69 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma LipschitzOnWith.of_iLipENorm_ne_top line 75 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, MetricSpace, NormedField, Ne, ENNReal, iLipENorm, Top.top, LipschitzOnWith, HDiv.hDiv, NNReal, iLipNNNorm, Real.toNNReal, Metric.ball
Proof uses: Membership.mem, Set, LT.lt, OfNat.ofNat, LE.le.trans_lt, Dist.dist, dist_nonneg, Eq.mp, Eq, Or, LE.le, EDist.edist, HMul.hMul, eq_or_ne, of_eq_true, Eq.trans, True, congr, congrArg, WeakPseudoEMetricSpace.edist_self and 42 more
115 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
lemma continuous_of_iLipENorm_ne_top line 102 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, NormedField, Real, LE.le, Set, tsupport, Metric.ball, Ne, ENNReal, iLipENorm, Top.top, Continuous
Proof uses: ContinuousOn.continuous_of_tsupport_subset, LipschitzOnWith.continuousOn, HDiv.hDiv, NNReal, iLipNNNorm, Real.toNNReal, LipschitzOnWith.of_iLipENorm_ne_top, Metric.isOpen_ball
32 instances, coercions and auxiliary constants left out.
lemma norm_le_iLipNNNorm_of_mem line 110 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, MetricSpace, NormedField, Ne, ENNReal, iLipENorm, Top.top, Membership.mem, Set, Metric.ball, LE.le, Norm.norm, iLipNNNorm
Proof uses: Iff.mpr, ENNReal.toReal, ENorm.enorm, ENNReal.toReal_le_toReal, of_eq_true, Not, Eq, Eq.trans, False, True, congrArg, not_false_eq_true, enorm_le_iLipENorm_of_mem
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma norm_le_iLipNNNorm_of_subset line 114 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, MetricSpace, NormedField, Ne, ENNReal, iLipENorm, Top.top, LE.le, Set, Function.support, Metric.ball, Norm.norm, iLipNNNorm
Proof uses: dite, Membership.mem, Classical.propDecidable, norm_le_iLipNNNorm_of_mem, Not, of_eq_true, Eq.trans, OfNat.ofNat, True, congrFun', congrArg, Iff.mp, Eq, Function.notMem_support, norm_zero
34 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.