Padic.lean`

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6 declarations, 0 contain sorry, 0 declared as axioms

The distributive Haar characters of the p-adics

This file computes ringHaarChar for ℤ_[p] and ℚ_[p].

This lets us know what volume (x • s) is in terms of ‖x‖ and volume s, when x is a p-adic/p-adic integer and s is a set of p-adics/p-adic integers.

Main declarations

ringHaarChar_padic: ringHaarChar is the usual p-adic norm on ℚ_[p]ˣ.
ringHaarChar_padicInt: ringHaarChar is constantly 1 on ℤ_[p]ˣ.
Padic.volume_padic_smul: volume (x • s) = ‖x‖₊ * volume s for all x : ℚ_[p] and s : Set ℚ_[p].
PadicInt.volume_padicInt_smul: volume (x • s) = ‖x‖₊ * volume s for all x : ℤ_[p] and s : Set ℤ_[p].

Imports (6)

Imported by 2 files

lemma MeasureTheory.ringHaarChar_padic_padicInt line 40 source
lemma MeasureTheory.ringHaarChar_padic The distributive Haar character of the action of ℚ_[p]ˣ on ℚ_[p] is the usual p-adic norm line 73 source
Source documentation
The distributive Haar character of the action of ℚ_[p]ˣ on ℚ_[p] is the usual p-adic norm.
This means that volume (x • s) = ‖x‖ * volume s for all x : ℚ_[p] and s : Set ℚ_[p]. See Padic.volume_padic_smul

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Units, Padic, Eq, NNReal, ContinuousMonoidHom, MeasureTheory.ringHaarChar, NNNorm.nnnorm
Proof uses: MonoidHom, MonoidHom.mk, OneHom.mk, of_eq_true, OfNat.ofNat, Eq.trans, True, congrFun', congrArg, nnnorm_one, eq_self, HMul.hMul, forall_congr, nnnorm_mul, implies_true, id, MulHom, ContinuousMonoidHom.toMonoidHom, MonoidHom.eq_of_eqOn_dense, Set.range and 18 more
110 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
lemma MeasureTheory.ringHaarChar_padic (x : ℚ_[p]ˣ) : ringHaarChar x = ‖(x : ℚ_[p])‖₊ := by
…
```
Mentions: MeasureTheory.ringHaarChar
lemma Padic.volume_padic_smul line 94 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Padic, Set, Eq, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, HSMul.hSMul, HMul.hMul, NNNorm.nnnorm
Proof uses: OfNat.ofNat, Ne, Or, eq_or_ne, of_eq_true, Eq.trans, True, congr, congrArg, MeasurableSpace, Zero, SMulWithZero, MeasureTheory.NullSingletonClass, Set.Finite.measure_zero, Set.Finite.subset, Set.finite_zero, Set.zero_smul_set_subset, congrFun', NNReal, nnnorm_zero and 16 more
111 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Padic.volume_padicInt_smul line 100 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, PadicInt, Set, Padic, Eq, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, HSMul.hSMul, HMul.hMul, NNNorm.nnnorm
Proof uses: Padic.volume_padic_smul, Membership.mem, Subring, PadicInt.subring
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma PadicInt.volume_padicInt_smul line 103 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, PadicInt, Set, Eq, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, HSMul.hSMul, HMul.hMul, NNNorm.nnnorm
Proof uses: Eq.mp, Padic, Set.image, Membership.mem, Subring, PadicInt.subring, congr, congrArg, Eq.trans, PadicInt.volume_coe, Padic.volume_padicInt_smul
57 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.ringHaarChar_padicInt The distributive Haar character of the action of ℤ_[p]ˣ on ℤ_[p] is the constant 1 line 112 source
Source documentation
The distributive Haar character of the action of ℤ_[p]ˣ on ℤ_[p] is the constant 1.
This means that volume (x • s) = ‖x‖ * volume s for all x : ℤ_[p] and s : Set ℤ_[p]. See PadicInt.volume_padicInt_smul

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Units, PadicInt, Eq, NNReal, ContinuousMonoidHom, MeasureTheory.ringHaarChar, OfNat.ofNat
Proof uses: MeasureTheory.ringHaarChar_eq_of_measure_smul_eq_mul, MeasureTheory.MeasureSpace.volume, Set.univ, of_eq_true, Ne, ENNReal, MeasureTheory.Measure, Set, Eq.trans, True, congrFun', congrArg, PadicInt.volume_univ, Not, False, not_false_eq_true, MeasureTheory.measure_ne_top, HSMul.hSMul, HMul.hMul, congr and 3 more
84 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma MeasureTheory.ringHaarChar_padicInt (x : ℤ_[p]ˣ) : ringHaarChar x = 1 :=
…
```
Mentions: MeasureTheory.ringHaarChar