DiscriminantBounds.lean`

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17 declarations, 0 contain sorry, 0 declared as axioms

Discriminant bounds for number fields

Bounds of Minkowski type for the discriminant of a number field, derived from the comparison between root discriminants and small-degree explicit bounds.

Imports (7)

Imported by 1 file

FLT.lean

theorem strictMonoOn_Ici_nat_of_lt_succ upstream line 29 source
Source comment
upstream

no axioms what this means
Statement uses: Preorder, Nat, LE.le, LT.lt, HAdd.hAdd, OfNat.ofNat, StrictMonoOn, Set.ofPred
Proof uses: Membership.mem, Set, Nat.rel_of_forall_rel_succ_of_le_of_lt
13 instances, coercions and auxiliary constants left out.
```
theorem strictMonoOn_Ici_nat_of_lt_succ {α : Type*} [Preorder α] {f : ℕ → α} {x : ℕ}
    (h : ∀ n, x ≤ n → f n < f (n + 1)) :
    StrictMonoOn f {i | x ≤ i} :=
…
```
theorem strictMonoOn_Ioi_nat_of_lt_succ upstream line 35 source
Source comment
upstream

no axioms what this means
Statement uses: Preorder, Nat, LT.lt, HAdd.hAdd, OfNat.ofNat, StrictMonoOn, Set.Ioi
Proof uses: strictMonoOn_Ici_nat_of_lt_succ
11 instances, coercions and auxiliary constants left out.
```
theorem strictMonoOn_Ioi_nat_of_lt_succ {α : Type*} [Preorder α] {f : ℕ → α} {x : ℕ}
    (h : ∀ n, x < n → f n < f (n + 1)) :
    StrictMonoOn f (Set.Ioi x) :=
…
```
lemma constant_approx line 40 source
lemma log_approx line 48 source
lemma rootDiscrBound_strictMono_aux3 line 52 source
lemma rootDiscrBound_strictMono_aux2 line 78 source
lemma rootDiscrBound_strictMono_aux line 92 source
def rootDiscrBound Minkowski's lower bound for the root discriminant (that is, the nth root of the absolute value of the discriminant) of a totally complex number field of degree n line 110 source
Source documentation
Minkowski's lower bound for the root discriminant (that is, the nth root of the absolute value of the discriminant) of a totally complex number field of degree n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: HMul.hMul, HDiv.hDiv, Real.pi, OfNat.ofNat, HPow.hPow, Nat.factorial
16 instances, coercions and auxiliary constants left out.
```
noncomputable def rootDiscrBound (n : ℕ) : ℝ :=
…
```
lemma rootDiscrBound_strictMono line 113 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: StrictMono, Nat, Real, rootDiscrBound
Proof uses: StrictMono.const_mul, HDiv.hDiv, Real.pi, OfNat.ofNat, HPow.hPow, Nat.factorial, StrictMonoOn.Iic_union_Ici, Eq, Set, Set.Iic, Insert.insert, Singleton.singleton, Eq.mpr, Iff, LE.le, Or, id, Eq.trans, Membership.mem, forall_congr and 84 more
147 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma rootDiscrBound_four_lt line 125 source
lemma rootDiscrBound_five_gt line 135 source
lemma rootDiscrBound_thirteen_lt line 145 source
lemma rootDiscrBound_fourteen_gt line 160 source
theorem rootDiscrBound_lt_iff_lt_fourteen line 175 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Iff, LT.lt, Real, rootDiscrBound, HMul.hMul, HPow.hPow, OfNat.ofNat, HDiv.hDiv
Proof uses: Iff.intro, Iff.mp, StrictMono.lt_iff_lt, rootDiscrBound_strictMono, LT.lt.trans, LE.le, Decidable.byContradiction, Not, LE.le.trans_lt, Iff.mpr, StrictMono.le_iff_le
39 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
theorem rootDiscrBound_lt_iff_lt_five line 183 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Iff, LT.lt, Real, rootDiscrBound, OfScientific.ofScientific, Bool.true, OfNat.ofNat
Proof uses: Iff.intro, Iff.mp, StrictMono.lt_iff_lt, rootDiscrBound_strictMono, LT.lt.trans, LE.le, Decidable.byContradiction, Not, LE.le.trans_lt, Iff.mpr, StrictMono.le_iff_le
21 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
theorem le_fourteen_of_rootDiscrBound line 191 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, Real, rootDiscrBound, HMul.hMul, HPow.hPow, OfNat.ofNat, HDiv.hDiv, LE.le
Proof uses: LT.lt.le, Iff.mp, rootDiscrBound_lt_iff_lt_fourteen
29 instances, coercions and auxiliary constants left out.
theorem le_five_of_rootDiscrBound line 195 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LT.lt, Real, HMul.hMul, HDiv.hDiv, Real.pi, OfNat.ofNat, HPow.hPow, Nat.factorial, OfScientific.ofScientific, Bool.true, LE.le
Proof uses: LT.lt.le, Iff.mp, rootDiscrBound, rootDiscrBound_lt_iff_lt_five
41 instances, coercions and auxiliary constants left out.