Trace — 2607.20422v1

$ /bin/bash -lc "pwd && rg --files -g 'AGENTS.md' -g 'paper.txt' -g 'paper.json' -g 'paper.pdf' -g 'lakefile.*' -g 'lean-toolchain' | sort | sed -n '1,120p'"
/work
paper.json
paper.pdf
paper.txt
repro/DownstreamTest/lakefile.toml
repro/lakefile.lean
repro/lean-toolchain
repro/scripts/SideSkimmer/lakefile.lean

$ /bin/bash -lc "wc -l paper.txt && sed -n '1,220p' paper.txt"
802 paper.txt
THE SHARP EXPONENT FOR THE MINIMAL DISTANCE PROBLEM
COSMIN POHOATA
Abstract. We show that for every fixed ε > 0, there exist arbitrarily large families of
point–line pairs ( x1 , ℓ1 ), . . . , ( xn , ℓn ) in [0, 1]2 , with xi ∈ ℓi for all i, and such that
dist( xi , ℓ j ) ≥ n−2/3−ε

( i ̸ = j ).

Combined with a previous result of Cohen, the author, and Zakharov, this solves the
minimal distance problem.

arXiv:2607.20422v1 [math.CO] 22 Jul 2026

1. Introduction
Choose points x1 , . . . , xn ∈ [0, 1]2 and, for each i = 1, . . . , n, let ℓi be a line through xi .
The minimal distance problem asks how far every point can be kept from every line to
which it is not assigned. More precisely, we define
∆PL (n) := sup min dist( xi , ℓ j ),
i̸= j

(1)

where the supremum is over all configurations of points x1 , . . . , xn ∈ [0, 1]2 and lines
ℓ1 , . . . , ℓn ⊂ R2 such that xi ∈ ℓi . For convenience, let us call these point-line incidence
configurations of size n. The quantity ∆PL (n) can therefore be regarded as the largest
scale at which one can realize n prescribed point–line incidences while avoiding every
nontrivial incidence. In [7], Cohen, the author, and Zakharov introduced the problem
of determining the asymptotics of ∆PL (n) as n grows, in connection with the Heilbronn
triangle problem.
Trivially, taking xi = (i/n, 0) and ℓi = xi + (0, 1)R, for i = 1, . . . n, gives a point-line
incidence configuration with dist( xi , ℓ j ) ≥ 1/n for all i ̸= j. Thus, ∆PL (n) ≥ 1/n. In the
other direction, it is not difficult to see that in any point-line incidence configuration of
size n, we must have dist( xi , ℓ j ) ≤ | xi − x j | ≲ n−1/2 for some i ̸= j. Hence ∆PL (n) ≲ n−1/2 .
In [7], Cohen-Pohoata-Zakharov improved this upper bound to
∆PL (n) ≤ n−2/3+o(1) ,

(2)

and used this estimate to establish the latest record for the Heilbronn triangle problem.
For more history and context, see this recent survey by Zakharov [24].
The connection itself between the Heilbro

$ /bin/bash -lc 'rg -n "''^(Theorem|THEOREM|Definition|Proposition|Lemma|Corollary|Conjecture)|main theorem|minimal distance|Theorem 1|sharp" paper.txt'
10:minimal distance problem.
16:The minimal distance problem asks how far every point can be kept from every line to
41:The connection itself between the Heilbronn triangle problem and the minimal distance
72:several new connections between the minimal distance problem and various other topics
81:circle-method proof was quantitative. This estimate was subsequently sharpened by Pintz–
87:In [11, Theorem 1.15 and Proposition 2.3], Hunter–Pohoata–Verstraëte–Zhang proved
89:minimal distance problem:
140:Theorem 1.1. For every ε > 0, there exists n0 (ε) such that, for every integer n ≥ n0 (ε), there are
155:Combining Theorem 1.1 with (2), it thus follows that the correct exponent for the
156:minimal distance problem is 2/3.
157:Corollary 1.2. As n → ∞,
159:The proof of Theorem 1.1 will build upon the ideas from [11]. Roughly speaking, the
177:(4) would readily convert into the exponent γ = 4/5 + o (1) for the minimal distance
185:The proof of Theorem 1.1 bypasses this barrier and the Ruzsa limit altogether, by
191:Curiously, the high degree number fields enter the proof of Theorem 1.1 in a very
206:number-field construction and the proof of Theorem 1.1.
287:Corollary 2.2. If A ⊂ F is square-difference-free, then the pairs {( p, L p ) : p ∈ P ( A, Y )} form
323:Proposition 2.3. The set Pq ( A) has q| A| elements, every p∗ lies in [0, 1]2 , and p∗ ∈ ℓ∗p . For
374:Proposition 2.4. The number of incidences between Pq and Lq is
429:Thus the unrestricted parabola model is simultaneously a sharp incidence configuration
490:Proposition 3.2. The set AK ( R) is square-difference-free. Moreover, for every R, M ≥ 1,
543:Proposition 2.3.
544:Proposition 4.1. Let A ⊂ 2OK ∩ BK ( R) be square-difference-free, and let Y ⊂ 2OK ∩ BK ( M ),
663:Corollary 4.2. Let M ≥ 1, and take
683:We are now ready to derive Theorem 1.1.
686:Proof of Theorem 1.1. Fix ε > 0. Choose an odd prime p so large that 9p−
705:n, which yields ∆PL (n) ≥ n−2/3−ε . This completes the proof of Theorem 1.1.
708:We do not

$ /bin/bash -lc 'git status --short && find . -maxdepth 2 -name AGENTS.md -print'

$ /bin/bash -lc "sed -n '460,710p' paper.txt"
ker(TrK/Q ) ∩ {z2 : z ∈ K } = {0}.

(20)

Geometrically, every nonzero square lies in the open half-space TrK/Q > 0, whose boundary is the trace-zero hyperplane.

THE SHARP EXPONENT FOR THE MINIMAL DISTANCE PROBLEM

7

For X ≥ 0, let us now define the symmetric Minkowski box of radius X
BK ( X ) := { a ∈ OK : |σj ( a)| ≤ X for 1 ≤ j ≤ d}.

(21)

One of the main new ideas in this construction is to consider the trace-zero lattice
Λ0K := { a ∈ 2OK : TrK/Q ( a) = 0}.

(22)

The additive group 2OK has rank d, and the trace map is nonzero because TrK/Q (2) = 2d.
Hence Λ0K has rank d − 1 (it lives in the trace-zero hyperplane x1 + · · · + xd = 0).
For two parameters R, M ≥ 1 (to be balanced appropriately later), let us denote
AK ( R) := Λ0K ∩ BK ( R),

YK ( M ) := 2OK ∩ BK ( M ).

(23)

We record two important facts about AK ( R) and YK ( M ).
Proposition 3.2. The set AK ( R) is square-difference-free. Moreover, for every R, M ≥ 1,

| AK ( R)| ≍K Rd−1 ,

|YK ( M)| ≍K Md .

(24)

Proof. Let’s first check the square-difference-freeness. Suppose that a − a′ = z2 with
a, a′ ∈ AK ( R) and z ∈ OK . Then
0 = TrK/Q ( a − a′ ) = TrK/Q (z2 ) = qK (z).
By the positive definiteness from Observation 3.1, one has z = 0, and therefore a = a′ .
The size estimates for AK ( R) and YK ( M ) immediately follow from the following standard fact about lattices:
Fact 3.3. Let Λ ⊂ Rm be a lattice of rank r ≥ 1. Then

|Λ ∩ [− T, T ]m | ≍Λ T r
for every T ≥ 1.
Therefore, | AK ( R)| ≍K Rd−1 comes from the fact that the lattice Λ0K has rank d − 1,
whereas |YK ( M )| ≍K Md follows from the fact that 2OK has full rank d.
□
The fixed factor 2 in (22) only clears the possible half-integral first coordinate in (13). If
a, y ∈ 2OK , then
a + y2
x=
∈ OK .
2
Consequently, for two points p = ( x, y) and p′ = ( x ′ , y′ ) arising from A × Y, the quantity
a − a ′ − ( y ′ − y )2
2
is also an algebraic integer. Passing from OK to the finite-index sublattice 2OK has no
effect on any exponent.
D ( p, p′ ) = x − x 

$ /bin/bash -lc "sed -n '206,460p' paper.txt"
number-field construction and the proof of Theorem 1.1.

COSMIN POHOATA

4

2. The Hunter-Pohoata-Verstraëte-Zhang construction
With an eye on the number field story that will follow, let us first isolate some basic
facts that hold in greater generality.
2.1. The induced-matching identity. Let F be any field of characteristic different from
2, and let us Q( x, y) := 2x − y2 as a polynomial in F [ x, y]. In particular, the level set
Q( x, y) = a is a parabola in F2 . For a point p = ( x, y) ∈ F2 , let

L p := p + F (y, 1) = {( x + ty, y + t) : t ∈ F }.

(7)

This is the tangent line to the parabola Q = Q( p) at p. For two points p = ( x, y) and
p′ = ( x ′ , y′ ), let us also single out the following recurring quantity:
D ( p, p′ ) := x − x ′ + y(y′ − y).

(8)

We next record the key identity mentioned above, along with some important consequences.


Observation 2.1. For p = ( x, y) ∈ F2 and t ∈ F, Q p + t(y, 1) = Q( p) − t2 . Moreover, for
p′ = ( x ′ , y′ ), we have that Q( p) − Q( p′ ) = 2D ( p, p′ ) + (y′ − y)2 , and
p′ ∈ L p

D ( p, p′ ) = 0.

⇐⇒

(9)

In particular, Observation 2.1 gives us the following important implication:
p′ ∈ L p

=⇒

Q ( p ) − Q ( p ′ ) = ( y ′ − y )2 .

(10)

For the sake of completeness, we include the easy proof as well.
Proof. The parametrisation in (7) gives
Q( x + ty, y + t) = 2( x + ty) − (y + t)2 = 2x − y2 − t2 ,
which gives the first equality. Expanding 2D ( p, p′ ) + (y′ − y)2 gives
2x − 2x ′ + 2y(y′ − y) + (y′ − y)2 = 2x − y2 − (2x ′ − y′2 ) = Q( p) − Q( p′ ).
Finally, a defining equation for L p is
Λ p ( X, Y ) := X − yY + (y2 − x ) = 0.

(11)

Its value at p′ is Λ p ( p′ ) = x ′ − yy′ + y2 − x = − D ( p, p′ ), so it vanishes exactly when
p′ ∈ L p .
□
A useful take-away is that the quantity D ( p, p′ ) is precisely the negative of the value
at p′ of the normalized affine equation (11) for L p (so it detects incidences). For any
A, Y ⊂ F, let us next define the set

P ( A, Y ) := {( x, y) ∈ F2 : y ∈ Y, Q( x, y) ∈ A}.

(12)

Bec

$ /bin/bash -lc "sed -n '730,770p' paper.txt"
translating, it follows that sQ ( X ) = s(2⌊ X ⌋ + 1). See also the closely related version from
[16] for general number fields.
For a totally real field K of degree d ≥ 2, the trace-zero lattice used in our proof of
Theorem 1.1 gives sK ( X ) ≳K X d−1 . Conversely, Z · 1 is a primitive direct summand of
the additive group OK , so one may partition BK ( X ) into OK ( X d−1 ) fibers parallel to this
copy of Z. The intersection of A with each such fiber, after translation, is an ordinary
square-difference-free set contained in an interval of length OK ( X ). The result of [9]
therefore implies
p


X d−1 ≲K sK ( X ) ≲K X d exp −c log X
for an absolute constant c > 0. It would be interesting to narrow this gap for large d.
Acknowledgments
The author would like to acknowledge the important role of OpenAI’s GPT-5.6 Sol Pro
in preparing the manuscript and in developing the proof of Theorem 1.1. The author’s
original idea was to use a high-degree number field analogue of the Hunter–Pohoata–
Verstraëte–Zhang construction to get from ∆PL (n) ≳ n−0.81075... to the Ruzsa endpoint
∆PL (n) ≥ n−4/5−o(1) . Naturally, the point was to leverage a better behaved version of the
Ruzsa construction (5) inside the rings of integers of such number fields. The decisive new
idea of using the codimension one, square-difference-free, trace-zero lattice Λ0K ⊂ 2OK
instead of a Ruzsa-like set, in order to subsequently upgrade the exponent 4/5 to the
sharp exponent 2/3, is entirely due to GPT-5.6 Sol Pro.
References
[1] N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. Shankar, J. Tsimerman, V. Wang, and M. Matchett
Wood, Remarks on the disproof of the unit distance conjecture, arXiv:2605.20695, 2026.
[2] R. Beigel and W. Gasarch, Square-difference-free sets of size Ω(n0.7334... ), arXiv:0804.4892, 2008.
[3] T. F. Bloom, Sum-product, unit distances, and number fields, Blog post, May 31, 2026. https://www.erdosp
roblems.com/forum/thread/blog:6.
[4] T. F. Bloom and J. Maynard, A new upper bo

$ /bin/bash -lc "sed -n '1,220p' lakefile.lean && sed -n '1,80p' lean-toolchain"
import Lake

open Lake DSL

/-!
## Mathlib dependencies on upstream projects
-/

require "leanprover-community" / "batteries" @ git "main"
require "leanprover-community" / "Qq" @ git "master"

require "leanprover-community" / "aesop" @ git "master"
require "leanprover-community" / "proofwidgets" @ git "main"
  with NameMap.empty.insert `errorOnBuild
    "ProofWidgets failed to reuse pre-built JS code. \
    Please report this issue on the Lean Zulip."
require "leanprover-community" / "importGraph" @ git "main"
require "leanprover-community" / "LeanSearchClient" @ git "main"
require "leanprover-community" / "plausible" @ git "main"


/-!
## Options for building mathlib
-/

/-- These options are used as `leanOptions`, prefixed by `` `weak``, so that
`lake build` uses them, as well as `Archive` and `Counterexamples`. -/
abbrev mathlibOnlyLinters : Array LeanOption := #[
  ⟨`linter.mathlibStandardSet, true⟩,
  -- Explicitly enable the header linter, since the standard set is defined in `Mathlib.Init`
  -- but we want to run this linter in files imported by `Mathlib.Init`.
  ⟨`linter.style.header, true⟩,
  ⟨`linter.checkInitImports, true⟩,
  ⟨`linter.allScriptsDocumented, true⟩,
  ⟨`linter.pythonStyle, true⟩,
  ⟨`linter.style.longFile, .ofNat 1500⟩,
  -- ⟨`linter.nightlyRegressionSet, true⟩,
  -- `latest_import.yml` uses this comment: if you edit it, make sure that the workflow still works
]

/-- These options are passed as `leanOptions` to building mathlib, as well as the
`Archive` and `Counterexamples`. (`tests` omits the first two options.) -/
abbrev mathlibLeanOptions := #[
    ⟨`pp.unicode.fun, true⟩, -- pretty-prints `fun a ↦ b`
    ⟨`autoImplicit, false⟩,
    ⟨`maxSynthPendingDepth, .ofNat 3⟩,
  ] ++ -- options that are used in `lake build`
    mathlibOnlyLinters.map fun s ↦ { s with name := `weak ++ s.name }

package mathlib where
  testDriver := "MathlibTest"
  lintDriver := "batteries/runLinter"
  lintDriverArgs := #["Mathlib"]
  -- A version of Mathlib only su

$ /bin/bash -lc "rg -n \"TotallyReal|totally real|NumberField.*Trace|trace.*positive|Algebra.trace|Algebra.norm|norm_eq_prod_embeddings|prod.*embedding|Minkowski\" Mathlib/NumberTheory Mathlib/FieldTheory | sed -n '1,260p'"
Mathlib/FieldTheory/KummerExtension.lean:129:    apply ha q hq (dvd_mul_of_dvd_right hqn p) (Algebra.norm _ b)
Mathlib/FieldTheory/KummerPolynomial.lean:112:  have key : (Algebra.norm K (AdjoinRoot.root g)) ^ p = a ^ g.natDegree := by
Mathlib/FieldTheory/KummerPolynomial.lean:115:    rw [← map_pow, this, ← AdjoinRoot.algebraMap_eq, Algebra.norm_algebraMap,
Mathlib/FieldTheory/LinearDisjoint.lean:746:    Algebra.trace A E (algebraMap B E x) = algebraMap F A (Algebra.trace F B x) := by
Mathlib/FieldTheory/LinearDisjoint.lean:756:    Algebra.norm A (algebraMap B E x) = algebraMap F A (Algebra.norm F x) := by
Mathlib/FieldTheory/Finite/GaloisField.lean:225:    algebraMap K K' (Algebra.norm K x) = x ^ ((Nat.card K' - 1) / (Nat.card K - 1)) := by
Mathlib/FieldTheory/Finite/GaloisField.lean:229:  simp_rw [← Fintype.card_eq_nat_card, Algebra.norm_eq_prod_automorphisms,
Mathlib/FieldTheory/Finite/GaloisField.lean:236:theorem unitsMap_norm_surjective : Function.Surjective (Units.map <| Algebra.norm K (S := K')) :=
Mathlib/FieldTheory/Finite/GaloisField.lean:251:theorem norm_surjective : Function.Surjective (Algebra.norm K (S := K')) := fun k ↦ by
Mathlib/FieldTheory/Finite/GaloisField.lean:253:  · exact ⟨0, Algebra.norm_zero ..⟩
Mathlib/NumberTheory/Cyclotomic/Discriminant.lean:101:    replace H := congr_arg (Algebra.norm K) H
Mathlib/NumberTheory/Cyclotomic/Discriminant.lean:107:      Algebra.norm_algebraMap, finrank L hirr, ← succ_eq_add_one,
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean:284:  rw [hζ.eq_neg_one_of_two_right, show -1 = algebraMap K L (-1) by simp, Algebra.norm_algebraMap]
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean:296:    rw [hζ, show 1 = algebraMap K L 1 by simp, Algebra.norm_algebraMap, one_pow]
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean:309:  rw [← (algebraMap K L).map_one, Algebra.norm_algebraMap, one_pow, map_pow, ← one_pow n] at hz
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean:341:  rw [norm_eq_prod_embeddings]
Mathlib/Num

$ /bin/bash -lc "rg -n \"Nat\\.squarefree|Squarefree.*prod|Prime.*not_isSquare|not.*IsSquare.*prime|irrational_sqrt\" Mathlib | sed -n '1,240p'"
Mathlib/NumberTheory/Real/GoldenRatio.lean:121:  have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
Mathlib/NumberTheory/Real/GoldenRatio.lean:129:  have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
Mathlib/NumberTheory/Real/Irrational.lean:97:theorem irrational_sqrt_of_multiplicity_odd (m : ℤ) (hm : 0 < m) (p : ℕ) [hp : Fact p.Prime]
Mathlib/NumberTheory/Real/Irrational.lean:106:theorem irrational_sqrt_ratCast_iff_of_nonneg {q : ℚ} (hq : 0 ≤ q) :
Mathlib/NumberTheory/Real/Irrational.lean:116:theorem irrational_sqrt_ratCast_iff {q : ℚ} :
Mathlib/NumberTheory/Real/Irrational.lean:119:  · simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq]
Mathlib/NumberTheory/Real/Irrational.lean:123:theorem irrational_sqrt_intCast_iff_of_nonneg {z : ℤ} (hz : 0 ≤ z) :
Mathlib/NumberTheory/Real/Irrational.lean:125:  rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),
Mathlib/NumberTheory/Real/Irrational.lean:128:theorem irrational_sqrt_intCast_iff {z : ℤ} :
Mathlib/NumberTheory/Real/Irrational.lean:130:  rw [← Rat.cast_intCast, irrational_sqrt_ratCast_iff, Rat.isSquare_intCast_iff,
Mathlib/NumberTheory/Real/Irrational.lean:133:theorem irrational_sqrt_natCast_iff {n : ℕ} : Irrational (√n) ↔ ¬IsSquare n := by
Mathlib/NumberTheory/Real/Irrational.lean:134:  rw [← Rat.isSquare_natCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg n.cast_nonneg,
Mathlib/NumberTheory/Real/Irrational.lean:137:theorem irrational_sqrt_ofNat_iff {n : ℕ} [n.AtLeastTwo] :
Mathlib/NumberTheory/Real/Irrational.lean:139:  irrational_sqrt_natCast_iff
Mathlib/NumberTheory/Real/Irrational.lean:141:theorem Nat.Prime.irrational_sqrt {p : ℕ} (hp : Nat.Prime p) : Irrational (√p) :=
Mathlib/NumberTheory/Real/Irrational.lean:142:  irrational_sqrt_natCast_iff.mpr hp.not_isSquare
Mathlib/NumberTheory/Real/Irrational.lean:145:theorem irrational_sqrt_two : Irrational (√2) := by
Mathlib/NumberTheory/Real/Irrational.lean:146:  simpa using Nat.prime_two.

$ /bin/bash -lc "rg -n \"LinearIndependent.*sqrt|linearIndependent.*sqrt|sqrt.*LinearIndependent|squarefree.*sqrt|Multiquadratic|multi.?quadratic|QuadraticExtension|IsSquare.*Rat|irrational.*sqrt|sqrt.*irrational\" Mathlib | sed -n '1,240p'"
Mathlib/NumberTheory/Real/GoldenRatio.lean:121:  have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
Mathlib/NumberTheory/Real/GoldenRatio.lean:129:  have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
Mathlib/NumberTheory/Real/Irrational.lean:97:theorem irrational_sqrt_of_multiplicity_odd (m : ℤ) (hm : 0 < m) (p : ℕ) [hp : Fact p.Prime]
Mathlib/NumberTheory/Real/Irrational.lean:106:theorem irrational_sqrt_ratCast_iff_of_nonneg {q : ℚ} (hq : 0 ≤ q) :
Mathlib/NumberTheory/Real/Irrational.lean:116:theorem irrational_sqrt_ratCast_iff {q : ℚ} :
Mathlib/NumberTheory/Real/Irrational.lean:119:  · simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq]
Mathlib/NumberTheory/Real/Irrational.lean:123:theorem irrational_sqrt_intCast_iff_of_nonneg {z : ℤ} (hz : 0 ≤ z) :
Mathlib/NumberTheory/Real/Irrational.lean:125:  rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),
Mathlib/NumberTheory/Real/Irrational.lean:128:theorem irrational_sqrt_intCast_iff {z : ℤ} :
Mathlib/NumberTheory/Real/Irrational.lean:130:  rw [← Rat.cast_intCast, irrational_sqrt_ratCast_iff, Rat.isSquare_intCast_iff,
Mathlib/NumberTheory/Real/Irrational.lean:133:theorem irrational_sqrt_natCast_iff {n : ℕ} : Irrational (√n) ↔ ¬IsSquare n := by
Mathlib/NumberTheory/Real/Irrational.lean:134:  rw [← Rat.isSquare_natCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg n.cast_nonneg,
Mathlib/NumberTheory/Real/Irrational.lean:137:theorem irrational_sqrt_ofNat_iff {n : ℕ} [n.AtLeastTwo] :
Mathlib/NumberTheory/Real/Irrational.lean:139:  irrational_sqrt_natCast_iff
Mathlib/NumberTheory/Real/Irrational.lean:141:theorem Nat.Prime.irrational_sqrt {p : ℕ} (hp : Nat.Prime p) : Irrational (√p) :=
Mathlib/NumberTheory/Real/Irrational.lean:142:  irrational_sqrt_natCast_iff.mpr hp.not_isSquare
Mathlib/NumberTheory/Real/Irrational.lean:145:theorem irrational_sqrt_two : Irrational (√2) := by
Mathlib/NumberTheory/Real/Irrational.lean:146:  simpa using Nat.prime_two.

$ /bin/bash -lc "sed -n '1,240p' Mathlib/NumberTheory/NumberField/Cyclotomic/Embeddings.lean && sed -n '1,230p' Mathlib/NumberTheory/NumberField/CMField.lean"
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module

public import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
public import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex

/-!
# Cyclotomic extensions of `ℚ` are totally complex number fields.

We prove that cyclotomic extensions of `ℚ` are totally complex, meaning that
`NrRealPlaces K = 0` if `IsCyclotomicExtension {n} ℚ K` and `2 < n`.

## Main results
* `nrRealPlaces_eq_zero`: If `K` is an `n`-th cyclotomic extension of `ℚ`, where `2 < n`,
  then there are no real places of `K`.
-/

public section

universe u

namespace IsCyclotomicExtension.Rat

open NumberField InfinitePlace Module Complex Nat Polynomial

variable {n : ℕ} [NeZero n] (K : Type u) [Field K] [CharZero K]

/-- If `K` is an `n`-th cyclotomic extension of `ℚ`, where `2 < n`, then there are no real places
of `K`. -/
theorem nrRealPlaces_eq_zero [IsCyclotomicExtension {n} ℚ K] (hn : 2 < n) :
    haveI := IsCyclotomicExtension.numberField {n} ℚ K
    nrRealPlaces K = 0 := by
  have := IsCyclotomicExtension.numberField {n} ℚ K
  apply (IsCyclotomicExtension.zeta_spec n ℚ K).nrRealPlaces_eq_zero_of_two_lt hn

theorem isTotallyComplex [IsCyclotomicExtension {n} ℚ K] (hn : 2 < n) :
    IsTotallyComplex K := by
  have := IsCyclotomicExtension.numberField {n} ℚ K
  exact nrRealPlaces_eq_zero_iff.mp <| nrRealPlaces_eq_zero K hn

variable (n)

/-- If `K` is an `n`-th cyclotomic extension of `ℚ`, then there are `φ n / n` complex places
of `K`. Note that this uses `1 / 2 = 0` in the cases `n = 1, 2`. -/
theorem nrComplexPlaces_eq_totient_div_two [h : IsCyclotomicExtension {n} ℚ K] :
    haveI := IsCyclotomicExtension.numberField {n} ℚ K
    nrComplexPlaces K = φ n / 2 := by
  have := IsCyclotomicExtension.numberField {n} ℚ K
  by_cases hn : 2 < n
  · obtain ⟨k, hk : φ n = k + k⟩ := totient_even hn
    have key := card_add_two_mul_car

$ /bin/bash -lc "rg -n \"maximalRealSubfield.*finrank|finrank.*maximalRealSubfield|maximalRealSubfield.*degree|IsTotallyReal.*Cyclotomic|cyclotomic.*totally\" Mathlib/NumberTheory | sed -n '1,260p'"
Mathlib/NumberTheory/NumberField/Cyclotomic/Embeddings.lean:14:We prove that cyclotomic extensions of `ℚ` are totally complex, meaning that

$ /bin/bash -lc "rg -n \"exists.*NumberField|NumberField.*exists|finrank.*CyclotomicField|CyclotomicField.*finrank|totient.*finrank|IsCyclotomicExtension.*finrank|finrank.*totient\" Mathlib/NumberTheory Mathlib/FieldTheory | sed -n '1,300p'"
Mathlib/NumberTheory/NumberField/ExistsRamified.lean:31:lemma NumberField.exists_not_isUnramifiedIn (H : Module.finrank ℚ K ≠ 1) :
Mathlib/NumberTheory/NumberField/ExistsRamified.lean:41:lemma NumberField.exists_not_isUnramifiedAt_int (H : Module.finrank ℚ K ≠ 1) :
Mathlib/NumberTheory/NumberField/ExistsRamified.lean:43:  obtain ⟨p, hp1, hp2⟩ := NumberField.exists_not_isUnramifiedIn (𝒪 := 𝒪) H
Mathlib/NumberTheory/NumberField/ExistsRamified.lean:54:  obtain ⟨P, _, H⟩ := NumberField.exists_not_isUnramifiedAt_int (𝒪 := 𝒪) H
Mathlib/NumberTheory/NumberField/ExistsRamified.lean:73:lemma NumberField.exists_not_isUnramifiedAt_int_of_isGalois [IsGalois ℚ K]
Mathlib/NumberTheory/NumberField/ExistsRamified.lean:83:  obtain ⟨P, _, hP'⟩ := NumberField.exists_not_isUnramifiedAt_int (𝒪 := 𝒪) H.ne'
Mathlib/NumberTheory/NumberField/CMField.lean:39:* `NumberField.IsCM.ofIsCMExtension`: Assume that there exists `F` such that `K/F` is a
Mathlib/NumberTheory/NumberField/Cyclotomic/Ideal.lean:128:  rw [← Nat.totient_prime_pow_succ hp.out, ← finrank _ K,
Mathlib/NumberTheory/NumberField/Cyclotomic/Ideal.lean:383:    rw [finrank m, finrank (p ^ (k + 1)), finrank n, hn, mul_comm, Nat.totient_mul]
Mathlib/NumberTheory/NumberField/Cyclotomic/Embeddings.lean:56:    rw [nrRealPlaces_eq_zero K hn, zero_add, IsCyclotomicExtension.finrank (n := n) K
Mathlib/NumberTheory/NumberField/Cyclotomic/Embeddings.lean:68:    rw [IsCyclotomicExtension.finrank K (cyclotomic.irreducible_rat (NeZero.pos n)), this]
Mathlib/NumberTheory/NumberField/Cyclotomic/PID.lean:36:  rw [discr_prime 3 K, IsCyclotomicExtension.finrank (n := 3) K
Mathlib/NumberTheory/NumberField/Cyclotomic/PID.lean:51:  rw [discr_prime 5 K, IsCyclotomicExtension.finrank (n := 5) K
Mathlib/NumberTheory/NumberField/Cyclotomic/Basic.lean:45:theorem finrank [NeZero k] [IsCyclotomicExtension {k} ℚ K] : Module.finrank ℚ K = k.totient :=
Mathlib/NumberTheory/NumberField/Cyclotomic/Basic.lean:46:  IsCyclotomicExtension.finrank K <| Polynomial.cyclo

$ /bin/bash -lc "rg -n \"integralBasis|Basis.*RingOfIntegers|RingOfIntegers.*Basis|trace_eq.*finrank|trace_one|Algebra.trace.*one|trace.*basis\" Mathlib/NumberTheory/NumberField Mathlib/LinearAlgebra | sed -n '1,300p'"
Mathlib/NumberTheory/NumberField/FractionalIdeal.lean:23:  field `K`, the absolute value of the determinant of the base change from `integralBasis` to
Mathlib/NumberTheory/NumberField/FractionalIdeal.lean:95:  rw [finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank,
Mathlib/NumberTheory/NumberField/FractionalIdeal.lean:104:/-- The absolute value of the determinant of the base change from `integralBasis` to
Mathlib/NumberTheory/NumberField/FractionalIdeal.lean:108:    |(integralBasis K).det ((basisOfFractionalIdeal K I).reindex e.symm)| =
Mathlib/NumberTheory/NumberField/EquivReindex.lean:35:    rw [Embeddings.card, ← finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank]
Mathlib/NumberTheory/NumberField/EquivReindex.lean:45:      (integralBasis K ∘ (equivReindex K)) (RingHom.equivRatAlgHom K ℂ) := by
Mathlib/NumberTheory/NumberField/EquivReindex.lean:58:      (Algebra.discr_not_zero_of_basis ℚ (integralBasis K))
Mathlib/NumberTheory/NumberField/EquivReindex.lean:59:  rw [← Algebra.discr_reindex ℚ (integralBasis K) (equivReindex K).symm]
Mathlib/NumberTheory/NumberField/EquivReindex.lean:61:    (integralBasis K ∘ (equivReindex K)) (RingHom.equivRatAlgHom K ℂ)).symm
Mathlib/NumberTheory/NumberField/EquivReindex.lean:70:      (fun i ↦ (((integralBasis K).reindex (equivReindex K).symm).repr α i : ℂ)) := by
Mathlib/NumberTheory/NumberField/EquivReindex.lean:73:  simp only [canonicalEmbedding.integralBasis_repr_apply, mulVec, dotProduct,
Mathlib/NumberTheory/NumberField/EquivReindex.lean:80:      ((integralBasis K).reindex (equivReindex K).symm).repr α i := fun i => by
Mathlib/NumberTheory/NumberField/Discriminant/Defs.lean:38:theorem coe_discr : (discr K : ℚ) = Algebra.discr ℚ (integralBasis K) :=
Mathlib/NumberTheory/NumberField/Discriminant/Defs.lean:43:  exact Algebra.discr_not_zero_of_basis ℚ (integralBasis K)
Mathlib/NumberTheory/NumberField/Discriminant/Defs.lean:47:  let b₀ := Basis.reindex (RingOfIntegers.basis K) (Basis.indexEquiv (RingOfIntegers.basis K) b)
M

$ /bin/bash -lc "rg -n \"dist.*affine.*line|infDist.*line|dist.*AffineSubspace|AffineSubspace.*dist|distance.*line|point.*line.*dist|dist_eq.*inner|infDist.*hyperplane\" Mathlib/Geometry Mathlib/Analysis Mathlib/Topology | sed -n '1,300p'"
Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean:201:/-- For two points on the same vertical line, the distance is equal to the distance between the
Mathlib/Analysis/InnerProductSpace/Affine.lean:49:  rw [dist_eq_norm_vsub V, inner_eq_norm_sq_left_iff, vsub_sub_vsub_cancel_left,
Mathlib/Analysis/InnerProductSpace/Affine.lean:58:  rw [dist_eq_norm_vsub V, inner_eq_norm_sq_left_iff, vsub_sub_vsub_cancel_right,
Mathlib/Analysis/InnerProductSpace/Affine.lean:66:/-- Squared distance between two points on lines from a common origin,
Mathlib/Analysis/InnerProductSpace/Affine.lean:80:/-- Squared distance from `p` to a point on the line from `a` to `b`,
Mathlib/Geometry/Euclidean/Circumcenter.lean:54:theorem existsUnique_dist_eq_of_insert {s : AffineSubspace ℝ P}
Mathlib/Geometry/Euclidean/SignedDist.lean:21:* `AffineSubspace.signedInfDist` is the signed distance between an affine subspace and a point.
Mathlib/Geometry/Euclidean/SignedDist.lean:23:  In the case of a triangle, these distances are trilinear coordinates.
Mathlib/Geometry/Euclidean/SignedDist.lean:174:  rw [signedDist_apply_apply, dist_eq_norm_vsub', NormedSpace.normalize, real_inner_smul_left,
Mathlib/Geometry/Euclidean/SignedDist.lean:185:  rw [signedDist_apply_apply, dist_eq_norm_vsub', NormedSpace.normalize, real_inner_smul_left]
Mathlib/Geometry/Euclidean/SignedDist.lean:343:  rw [signedInfDist, AffineSubspace.abs_signedInfDist_eq_dist_of_mem_affineSpan_insert,
Mathlib/Geometry/Euclidean/Sphere/Tangent.lean:100:lemma IsTangentAt.dist_sq_eq_of_mem {s : Sphere P} {p q : P} {as : AffineSubspace ℝ P}
Mathlib/Geometry/Euclidean/Sphere/Tangent.lean:130:lemma IsTangentAt.radius_lt_dist_center {s : Sphere P} {as : AffineSubspace ℝ P} {p q : P}
Mathlib/Geometry/Euclidean/Sphere/Tangent.lean:165:lemma IsTangent.radius_le_dist_center {s : Sphere P} {as : AffineSubspace ℝ P} (h : s.IsTangent as)
Mathlib/Geometry/Euclidean/Sphere/Tangent.lean:172:lemma IsTangent.notMem_of_dist_lt {s : Sphere P} {as : AffineSubspace ℝ

$ /bin/bash -lc "rg -n \"Nat.*eventually.*pow|atTop.*rpow|exists.*pow.*le|Real.rpow.*eventually|isLittleO.*rpow|tendsto.*rpow.*atTop|eventually.*n \\"'^" Mathlib/Analysis Mathlib/Topology Mathlib/NumberTheory | sed -n '"'1,280p'"
Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean:447:  tendsto_one_add_div_rpow_exp t |>.comp tendsto_natCast_atTop_atTop |>.congr (by simp)
Mathlib/Topology/Algebra/Valued/WithZeroMulInt.lean:50:lemma exists_pow_lt_of_le_exp_neg_one [Valued R ℤᵐ⁰] {x : R} (hx : v x ≤ exp (-1)) (γ : ℤᵐ⁰ˣ) :
Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean:102:      ((tendsto_rpow_atTop (sub_pos.2 hlt)).comp tendsto_natCast_atTop_atTop).eventually
Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean:271:  rcases ((eventually_gt_atTop 0).and_frequently (h.frequently_lt_rpow_neg hp)).exists with
Mathlib/NumberTheory/Transcendental/Liouville/Basic.lean:149:  -- Use the key lemma `exists_one_le_pow_mul_dist`: we are left to show that ...
Mathlib/Topology/Algebra/Order/Field.lean:84:A version for positive real powers exists as `tendsto_rpow_neg_atTop`. -/
Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean:232:    refine tendsto_const_nhds.div_atTop <| (tendsto_rpow_atTop (by linarith)).comp ?_
Mathlib/NumberTheory/Ostrowski.lean:56:  exact Tendsto.comp tendsto_rpow_div tendsto_natCast_atTop_atTop
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean:594:  obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure
Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean:318:    simpa using hv'.trans (isLittleO_exp_neg_mul_rpow_atTop hv _).isBigO
Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean:322:    simpa using hv'.trans (isLittleO_exp_neg_mul_rpow_atTop hv _).isBigO
Mathlib/NumberTheory/LSeries/Injectivity.lean:106:        (tendsto_rpow_atTop_of_base_lt_one _ (neg_one_lt_zero.trans_le H₀') H₂).ofReal.const_mul _
Mathlib/NumberTheory/Chebyshev.lean:738:      apply isLittleO_log_rpow_rpow_atTop _ (by norm_num)
Mathlib/NumberTheory/LSeries/SumCoeff.lean:66:  · refine (IsBigO.mul_atTop_rpow_natCast_of_isBigO_rpow _ _ _ ?_ hO h₂).congr_right (by simp)
Mathlib/NumberTheory/LSeries/SumCoeff.lean:68:  · refine h₄.isBigO.of_c

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
CyclotomicField.{w} (n : ℕ) (K : Type w) [Field K] : Type w
SharpDistanceScratch.lean:10:7: error(lean.unknownIdentifier): Unknown constant `CyclotomicField.instIsCyclotomicExtension`
SharpDistanceScratch.lean:11:7: error(lean.unknownIdentifier): Unknown constant `IsCyclotomicExtension.Rat.finrank`
IsCyclotomicExtension.Rat.isCMField.{u_1} (K : Type u_1) [Field K] [CharZero K] {S : Set ℕ} (hS : ∃ n ∈ S, 2 < n)
  [IsCyclotomicExtension S ℚ K] : NumberField.IsCMField K
NumberField.maximalRealSubfield.{u_2} (K : Type u_2) [Field K] : Subfield K
NumberField.IsCMField.isQuadraticExtension.{u_1} (K : Type u_1) [Field K] [CharZero K] [NumberField.IsCMField K] :
  Algebra.IsQuadraticExtension (↥(NumberField.maximalRealSubfield K)) K
NumberField.IsTotallyReal.finrank.{u_2} (K : Type u_2) [Field K] [NumberField K] [h : NumberField.IsTotallyReal K] :
  Module.finrank ℚ K = NumberField.InfinitePlace.nrRealPlaces K
Module.finrank_mul_finrank.{u, v, w} (F : Type u) (K : Type v) (A : Type w) [Semiring F] [Semiring K] [AddCommMonoid A]
  [Module F K] [Module K A] [Module F A] [IsScalarTower F K A] [StrongRankCondition F] [StrongRankCondition K]
  [Module.Free F K] [Module.Free K A] : Module.finrank F K * Module.finrank K A = Module.finrank F A
NumberField.RingOfIntegers.basis.{u_1} (K : Type u_1) [Field K] [NumberField K] :
  Module.Basis (Module.Free.ChooseBasisIndex ℤ (NumberField.RingOfIntegers K)) ℤ (NumberField.RingOfIntegers K)
NumberField.integralBasis.{u_1} (K : Type u_1) [Field K] [NumberField K] :
  Module.Basis (Module.Free.ChooseBasisIndex ℤ (NumberField.RingOfIntegers K)) ℚ K
SharpDistanceScratch.lean:19:7: error(lean.unknownIdentifier): Unknown constant `Algebra.trace_one`
Algebra.trace.{u_1, u_2} (R : Type u_1) (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] : S →ₗ[R] R
Algebra.norm.{u_1, u_2} (R : Type u_1) {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] : S →* R
Algebra.norm_eq_prod_embeddings.{u_6, u_7, u_9} {L : Type u_6} (K : Type u_7) [Field K] [Field

$ /bin/bash -lc "sed -n '1,85p' Mathlib/NumberTheory/NumberField/Cyclotomic/Basic.lean && rg -n \"isCMField|trace_one|dist_orthogonalProjection_eq_infDist|def line|lineMap\" Mathlib/NumberTheory/NumberField/CMField.lean Mathlib/LinearAlgebra Mathlib/Geometry/Euclidean/Projection.lean Mathlib/Geometry/Euclidean/Basic.lean | sed -n '1,260p'"
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
module

public import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
public import Mathlib.NumberTheory.Cyclotomic.Discriminant
public import Mathlib.NumberTheory.NumberField.Cyclotomic.Embeddings
public import Mathlib.NumberTheory.NumberField.Discriminant.Different
public import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
public import Mathlib.RingTheory.Prime

/-!
# Ring of integers of cyclotomic fields

We gather results about cyclotomic extensions of `ℚ`. In particular, we compute the ring of
integers of a cyclotomic extension of `ℚ`.

## Main results
* `IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton`: if `K` is a cyclotomic
  extension of `ℚ`, then `adjoin ℤ {ζ}` is the integral closure of `ℤ` in `K`.
* `IsCyclotomicExtension.Rat.cyclotomicRing_isIntegralClosure`: the integral
  closure of `ℤ` inside `CyclotomicField n ℚ` is `CyclotomicRing n ℤ ℚ`.
* `IsCyclotomicExtension.Rat.discr` and related results: the absolute discriminant
  of cyclotomic fields.
-/

@[expose] public section

universe u

open Algebra IsCyclotomicExtension Polynomial NumberField

open scoped Cyclotomic Nat

variable {p k n : ℕ} {K : Type u} [Field K] {ζ : K} [hp : Fact p.Prime]

namespace IsCyclotomicExtension.Rat

variable [CharZero K]

variable (k K) in
theorem finrank [NeZero k] [IsCyclotomicExtension {k} ℚ K] : Module.finrank ℚ K = k.totient :=
  IsCyclotomicExtension.finrank K <| Polynomial.cyclotomic.irreducible_rat (NeZero.pos _)

/-- The discriminant of the power basis given by `ζ - 1`. -/
theorem discr_prime_pow_ne_two' [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ (p ^ (k + 1))) (hk : p ^ (k + 1) ≠ 2) :
    discr ℚ (hζ.subOnePowerBasis ℚ).basis =
      (-1) ^ ((p ^ (k + 1)).totient / 2) * p ^ (p ^ k * ((p - 1) * (k + 1) - 1)) := by
  rw [← discr_prime_pow_ne_two hζ (cyclotomic.irr

$ /bin/bash -lc "sed -n '305,335p' Mathlib/Geometry/Euclidean/Projection.lean; sed -n '180,200p' Mathlib/LinearAlgebra/Trace.lean; rg -n \"CyclotomicField.*instance|instance.*CyclotomicField|IsCyclotomicExtension.*CyclotomicField\" Mathlib/NumberTheory/Cyclotomic Mathlib/NumberTheory/NumberField/Cyclotomic | sed -n '1,180p'"
lemma dist_orthogonalProjection_eq_dist_iff_eq_of_mem {s : AffineSubspace 𝕜 P}
    [s.direction.HasOrthogonalProjection] {p₁ p₂ : P} (hp₂ : p₂ ∈ s) :
    haveI : Nonempty s := ⟨p₂, hp₂⟩
    dist p₁ (orthogonalProjection s p₁) = dist p₁ p₂ ↔ orthogonalProjection s p₁ = p₂ := by
  have : Nonempty s := ⟨p₂, hp₂⟩
  constructor
  · intro h
    rwa [← sq_eq_sq₀ dist_nonneg dist_nonneg, pow_two, pow_two, dist_comm _ p₂,
      dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq _ hp₂,
      right_eq_add, mul_eq_zero, dist_eq_zero, or_self, eq_comm] at h
  · intro h
    nth_rw 4 [← h]

/-- The distance between a point and its orthogonal projection to a subspace equals the distance
to that subspace as given by `Metric.infDist`. This is not a `simp` lemma since the simplest form
depends on the context (if any calculations are to be done with the distance, the version with
the orthogonal projection gives access to more lemmas about orthogonal projections that may be
useful). -/
lemma dist_orthogonalProjection_eq_infDist (s : AffineSubspace 𝕜 P) [Nonempty s]
    [s.direction.HasOrthogonalProjection] (p : P) :
    dist p (orthogonalProjection s p) = Metric.infDist p s := by
  refine le_antisymm ?_ (Metric.infDist_le_dist_of_mem (orthogonalProjection_mem _))
  rw [Metric.infDist_eq_iInf]
  refine le_ciInf fun x ↦ le_of_sq_le_sq ?_ dist_nonneg
  rw [dist_comm _ (x : P)]
  simp_rw [pow_two,
    dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq p x.property]
  simp [mul_self_nonneg]

/-- The nonnegative distance between a point and its orthogonal projection to a subspace equals
the distance to that subspace as given by `Metric.infNndist`. This is not a `simp` lemma since

/-- When `M` is finite free, the trace of a linear map corresponds to the contraction pairing under
the isomorphism `End(M) ≃ M* ⊗ M`. -/
theorem trace_eq_contract' :
    LinearMap.trace R M = contractLeft R M ∘ₗ (dualTensorHomEquiv R M M).symm.toLinearMap := by
  rw [dual

$ /bin/bash -lc "rg -n \"IsTotallyReal.*embedding|isReal.*embedding|embedding_of_isReal|exists.*real.*embedding|RatAlgHom.*Real|K →\\+\\* ℝ\" Mathlib/NumberTheory/NumberField | sed -n '1,300p'"
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:34:  is real, then this extends the embedding `v.embedding_of_isReal : K →+* ℝ` to
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:71:theorem isometry_embedding_of_isReal (hv : v.IsReal) :
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:72:    Isometry ((v.embedding_of_isReal hv).comp (WithAbs.equiv v.1).toRingHom) :=
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:74:    simpa using! v.norm_embedding_of_isReal hv (WithAbs.equiv v.1 x)
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:229:/-- The embedding `K →+* ℝ` associated to a real infinite place extended to `v.Completion →+* ℝ`. -/
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:231:  (v.isometry_embedding_of_isReal hv).extensionHom.comp (equiv v).toRingHom
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:240:    extensionEmbeddingOfIsReal hv x = embedding_of_isReal hv (WithAbs.equiv v.1 x) :=
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:241:  (v.isometry_embedding_of_isReal hv).extensionHom_coe _
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:250:  (v.isometry_embedding_of_isReal hv).completion_extension.comp (isometry_toCompletion v)
Mathlib/NumberTheory/NumberField/Completion/InfinitePlace.lean:411:  over := (comap_eq w v ▸ comap_embedding_of_isReal _ (comap_eq w v ▸ h)).symm
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean:66:    embedding_of_isReal_apply, Subtype.forall, ← forall₂_or_left, ← not_isReal_iff_isComplex, em,
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean:159:    embedding_of_isReal_apply, norm_embedding_eq, Subtype.forall]
Mathlib/NumberTheory/NumberField/InfinitePlace/Embeddings.lean:219:def IsReal.embedding {φ : K →+* ℂ} (hφ : IsReal φ) : K →+* ℝ where
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean:34:  then `φ_w : K →+* ℝ` and, if `w` is complex, `φ_w` i

$ /bin/bash -lc "sed -n '80,115p' Mathlib/NumberTheory/NumberField/House.lean; sed -n '370,405p' Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean; rg -n \"house.*embedding|embedding.*le_house|norm_le_house|abs.*house\" Mathlib/NumberTheory/NumberField/House.lean | sed -n '1,180p'"
use w.embedding
  rwa [InfinitePlace.norm_embedding_eq]

lemma norm_embedding_le_house (α : K) (σ : K →+* ℂ) : ‖σ α‖ ≤ house α := by
  rw [house_eq_sup']
  exact Finset.le_sup' (f := (‖· α‖₊)) (Finset.mem_univ σ)

set_option backward.isDefEq.respectTransparency.types false in
lemma one_le_house_of_isIntegral {α : K} (hα : IsIntegral ℤ α) (hα0 : α ≠ 0) :
    1 ≤ house α := by
  have ⟨σ, hσ⟩ : ∃ σ : K →+* ℂ, 1 ≤ ‖σ α‖ := by
    apply exists_conjugate_one_le_norm (K := K) (α := ⟨α, hα⟩)
    simpa [RingOfIntegers.ext_iff]
  apply hσ.trans (norm_embedding_le_house α σ)

lemma norm_norm_le_norm_mul_house_pow (α : K) (σ : K →+* ℂ) :
    ‖Algebra.norm ℚ α‖ ≤ ‖σ α‖ * house α ^ (Module.finrank ℚ K - 1) := by
  classical
  set σ' := σ.toRatAlgHom
  calc _ = ‖∏ τ : K →ₐ[ℚ] ℂ, τ α‖ := ?_
       _ = ‖(σ' α) * ∏ τ ∈ univ.erase σ', τ α‖ := by rw [mul_prod_erase univ (· α) (mem_univ σ')]
       _ ≤ ‖σ' α‖ * ∏ τ ∈ univ.erase σ', ‖τ α‖ := ?_
       _ ≤ ‖σ' α‖ * ∏ τ ∈ univ.erase σ', house α := by gcongr; apply norm_embedding_le_house
       _ = ‖σ' α‖ * house α ^ (Module.finrank ℚ K - 1) := by simp
  · rw [← Algebra.norm_eq_prod_embeddings, ← Rat.norm_cast_real,
      Real.norm_eq_abs, eq_ratCast, Complex.norm_ratCast]
  · rw [Complex.norm_mul]
    gcongr
    exact norm_prod_le (univ.erase σ') (· α)

end

end NumberField

namespace NumberField.house

theorem prod_eq_abs_norm (x : K) :
    ∏ w : InfinitePlace K, w x ^ mult w = abs (Algebra.norm ℚ x) := by
  classical
  convert! (congr_arg (‖·‖) (Algebra.norm_eq_prod_embeddings ℚ ℂ x)).symm
  · rw [norm_prod, ← Fintype.prod_equiv (RingHom.equivRatAlgHom K ℂ) (fun f => ‖f x‖)
      (fun φ => ‖φ x‖) fun _ => by simp [RingHom.equivRatAlgHom_apply]]
    rw [← Finset.prod_fiberwise Finset.univ mk (fun φ => ‖φ x‖)]
    have (w : InfinitePlace K) (φ) (hφ : φ ∈ ({φ | mk φ = w} : Finset _)) :
        ‖φ x‖ = w x := by rw [← (Finset.mem_filter.mp hφ).2, apply]
    simp_rw [Finset.prod_congr rfl (this _), Finset.prod_const, card_filter_mk_eq]
  · r

$ /bin/bash -lc "rg -n \"trace_eq_sum.*embedding|trace_eq_sum_embeddings|sum_embeddings.*trace|trace.*prod_embeddings|Algebra.trace_eq_sum\" Mathlib | sed -n '1,260p'"
Mathlib/RingTheory/Trace/Basic.lean:36:* `trace_eq_sum_embeddings`: the trace of `x : K(x)` is the sum of all embeddings of `x` into an
Mathlib/RingTheory/Trace/Basic.lean:218:theorem trace_eq_sum_embeddings_gen (pb : PowerBasis K L)
Mathlib/RingTheory/Trace/Basic.lean:252:theorem trace_eq_sum_embeddings [FiniteDimensional K L] [Algebra.IsSeparable K L] {x : L} :
Mathlib/RingTheory/Trace/Basic.lean:257:    trace_eq_sum_embeddings_gen E pb (IsAlgClosed.splits _), ← Algebra.smul_def,
Mathlib/RingTheory/Trace/Basic.lean:268:  · rw [← trace_eq_sum_embeddings (AlgebraicClosure L) (x := x)]
Mathlib/RingTheory/Trace/Basic.lean:446:  ext (i j); simp [trace_eq_sum_embeddings, embeddingsMatrix, Matrix.mul_apply]
Mathlib/RingTheory/Trace/Basic.lean:623:  rw [← this, trace_eq_sum_embeddings (E := AlgebraicClosure K)]

$ /bin/bash -lc "sed -n '240,265p' Mathlib/RingTheory/Trace/Basic.lean; sed -n '45,65p' Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.lean; sed -n '210,250p' Mathlib/NumberTheory/NumberField/InfinitePlace/Embeddings.lean"
have : FiniteDimensional L F := FiniteDimensional.right K L F
  have : Algebra.IsSeparable L F := Algebra.isSeparable_tower_top_of_isSeparable K L F
  let : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
  rw [Fintype.sum_equiv algHomEquivSigma (fun σ : F →ₐ[K] E => _) fun σ => σ.1 pb.gen,
    ← Finset.univ_sigma_univ, Finset.sum_sigma, ← Finset.sum_nsmul]
  · refine Finset.sum_congr rfl fun σ _ => ?_
    let : Algebra L E := σ.toRingHom.toAlgebra
    simp_rw [Finset.sum_const, Finset.card_univ, ← AlgHom.card L F E]
  · intro σ
    simp only [algHomEquivSigma, Equiv.coe_fn_mk, AlgHom.domRestrict, AlgHom.comp_apply,
      IsScalarTower.coe_toAlgHom']

theorem trace_eq_sum_embeddings [FiniteDimensional K L] [Algebra.IsSeparable K L] {x : L} :
    algebraMap K E (Algebra.trace K L x) = ∑ σ : L →ₐ[K] E, σ x := by
  have hx := Algebra.IsSeparable.isIntegral K x
  let pb := adjoin.powerBasis hx
  rw [trace_eq_trace_adjoin K x, Algebra.smul_def, map_mul, ← adjoin.powerBasis_gen hx,
    trace_eq_sum_embeddings_gen E pb (IsAlgClosed.splits _), ← Algebra.smul_def,
    algebraMap_smul]
  · exact (sum_embeddings_eq_finrank_mul L E pb).symm
  · have := Algebra.isSeparable_tower_bot_of_isSeparable K K⟮x⟯ L
    exact Algebra.IsSeparable.isSeparable K _

theorem trace_eq_sum_automorphisms (x : L) [FiniteDimensional K L] [IsGalois K L] :
    algebraMap K L (Algebra.trace K L x) = ∑ σ : Gal(L/K), σ x := by
  apply FaithfulSMul.algebraMap_injective L (AlgebraicClosure L)
/-- A field `K` is totally real if all of its infinite places are real. In other words,
the image of every ring homomorphism `K → ℂ` is a subset of `ℝ`. -/
@[mk_iff] class IsTotallyReal (K : Type*) [Field K] where
  isReal : ∀ v : InfinitePlace K, v.IsReal

variable {F : Type*} [Field F] {K : Type*} [Field K]

theorem nrComplexPlaces_eq_zero_iff [NumberField K] :
    nrComplexPlaces K = 0 ↔ IsTotallyReal K := by
  simp [Fintype.card_eq_zero_iff, isEmpty_subtype, isTotallyReal_iff]

theorem IsTotallyReal.complexEm

$ /bin/bash -lc "rg -n \"complexEmbedding_isReal|IsTotallyReal.isReal|isReal_iff.*re|IsReal.*coe|IsReal.*im|map.*re\" Mathlib/NumberTheory/NumberField/InfinitePlace | sed -n '1,260p'"
Mathlib/NumberTheory/NumberField/InfinitePlace/Embeddings.lean:221:  map_one' := by simp only [map_one, one_re]
Mathlib/NumberTheory/NumberField/InfinitePlace/Embeddings.lean:223:    simp only [Complex.conj_eq_iff_im.mp (RingHom.congr_fun hφ _), map_mul, mul_re,
Mathlib/NumberTheory/NumberField/InfinitePlace/Embeddings.lean:225:  map_zero' := by simp only [map_zero, zero_re]
Mathlib/NumberTheory/NumberField/InfinitePlace/Embeddings.lean:226:  map_add' := by simp only [map_add, add_re, forall_const]
Mathlib/NumberTheory/NumberField/InfinitePlace/Embeddings.lean:229:theorem IsReal.coe_embedding_apply {φ : K →+* ℂ} (hφ : IsReal φ) (x : K) :
Mathlib/NumberTheory/NumberField/InfinitePlace/Embeddings.lean:237:    IsReal (φ.comp f) := by ext1 x; simpa using RingHom.congr_fun hφ (f x)
Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.lean:56:theorem IsTotallyReal.complexEmbedding_isReal [IsTotallyReal K] (φ : K →+* ℂ) :
Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.lean:65:  isReal _ := (isReal_comap_iff f).mp <| IsTotallyReal.isReal _
Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.lean:72:    exact IsReal.comap _ (IsTotallyReal.isReal W)
Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean:246:  ComplexEmbedding.IsReal.coe_embedding_apply (isReal_iff.mp hw) x
Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean:344:/-- The map from real embeddings to real infinite places as an equiv -/
Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean:353:/-- The map from nonreal embeddings to complex infinite places -/
Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean:519:  rw [NumberField.InfinitePlace.isReal_iff] at hwreal
Mathlib/NumberTheory/NumberField/InfinitePlace/Ramification.lean:19:* `NumberField.InfinitePlace.comap`: the restriction of an infinite place along an embedding.
Mathlib/NumberTheory/NumberField/InfinitePlace/Ramification.lean:294:  exact isReal_iff.1 <| (isUnramified_iff.1 h).resolve_rig

$ /bin/bash -lc "rg -n \"traceForm.*pos|pos.*traceForm|trace.*sq.*pos|trace.*mul_self|trace.*pow_two|IsTotallyReal.*trace|TotallyReal.*trace|positive.*trace\" Mathlib | sed -n '1,300p'"
Mathlib/Analysis/Matrix/Order.lean:29:  positive semi-definite matrix `M`: `⟪x, y⟫ = (y * M * xᴴ).trace`.
Mathlib/Analysis/Matrix/Order.lean:328:        conjTranspose_conjTranspose, mul_assoc, trace_conjTranspose_mul_self_eq_zero_iff] at hx
Mathlib/Analysis/Matrix/PosDef.lean:54:  conv_lhs => rw [hA.1.spectral_theorem, conjStarAlgAut_apply, trace_mul_cycle, coe_star_mul_self,
Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean:201:  rw [Algebra.smul_def (trace M), ← C_eq_algebraMap, pow_two, ← mul_assoc, add_assoc,
Mathlib/LinearAlgebra/Matrix/PosDef.lean:375:theorem trace_conjTranspose_mul_self_eq_zero_iff {A : Matrix m n R} :
Mathlib/LinearAlgebra/Matrix/PosDef.lean:381:  simpa using trace_conjTranspose_mul_self_eq_zero_iff (A := Aᴴ)
Mathlib/RingTheory/Discriminant.lean:148:    traceMatrix_eq_embeddingsMatrixReindex_mul_trans, det_mul, det_transpose, pow_two]

$ /bin/bash -lc "rg -n \"rank_ker.*LinearMap|finrank_ker_add|finrank_ker|finrank.*ker|rank_nullity|finrank_quotient_add_finrank\" Mathlib/LinearAlgebra | sed -n '1,260p'"
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:48:  rw [← s.finrank_quotient_add_finrank, add_comm]
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:96:        ← finrank_quotient_add_finrank W] at aux
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:104:    · rw [← W.finrank_quotient_add_finrank, add_comm, add_le_add_iff_left, hW]
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:146:      rw [← @add_right_cancel_iff _ _ _ (finrank K p), Submodule.finrank_quotient_add_finrank,
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:147:        LinearEquiv.finrank_eq f₁, Submodule.finrank_quotient_add_finrank,
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:155:    rw [← add_right_cancel_iff, Submodule.finrank_quotient_add_finrank, ← LinearEquiv.finrank_eq f,
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:156:      add_comm, Submodule.finrank_quotient_add_finrank]
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:173:theorem finrank_range_add_finrank_ker [FiniteDimensional K V] (f : V →ₗ[K] V₂) :
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:174:    finrank K (LinearMap.range f) + finrank K (LinearMap.ker f) = finrank K V := by
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:176:  exact Submodule.finrank_quotient_add_finrank _
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:181:  have h₁ := f.finrank_range_add_finrank_ker
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:183:  suffices 0 < finrank K (LinearMap.ker f) from Submodule.one_le_finrank_iff.mp this
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:200:  have := finrank_range_add_finrank_ker f
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:400:    ∃ k : ℕ, k ≤ finrank K V ∧ LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ) := by
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:404:      n ≤ finrank K (LinearMap.ker (f ^ n)) := by
Mathlib/LinearAlgebra/FiniteDimensional/Lemmas.lean:414:          finrank K (LinearMap.ker (f ^ n)) < finrank K (LinearMap.ker (f ^ n.

$ /bin/bash -lc "rg -n \"exists.*IsIntegral.*nsmul|exists.*isIntegral.*mul|exists_nat.*IsIntegral|exists_int.*IsIntegral|IsIntegral.*exists.*nat|denom.*integral|multiple.*integral\" Mathlib/FieldTheory Mathlib/NumberTheory/RingOfIntegers Mathlib/NumberTheory/NumberField | sed -n '1,300p'"
rg: Mathlib/NumberTheory/RingOfIntegers: No such file or directory (os error 2)

$ /bin/bash -lc "rg -n \"Basis.*ker|basis.*Submodule|Submodule.*basis|Module.Free.chooseBasis.*ker|basisOf.*ker\" Mathlib/LinearAlgebra | sed -n '1,220p'"
Mathlib/LinearAlgebra/Dual/Basis.lean:179:theorem eval_ker {ι : Type*} (b : Basis ι R M) : LinearMap.ker (Dual.eval R M) = ⊥ :=
Mathlib/LinearAlgebra/FreeModule/PID.lean:34:- `Submodule.exists_basis_of_pid`: if `R` is a PID, then `N : Submodule R M` is
Mathlib/LinearAlgebra/FreeModule/PID.lean:38:- `Submodule.smithNormalForm`: if `R` is a PID, then `M` has a basis
Mathlib/LinearAlgebra/FreeModule/PID.lean:142:/-- The induction hypothesis of `Submodule.basisOfPid` and `Submodule.smithNormalForm`.
Mathlib/LinearAlgebra/FreeModule/PID.lean:153:theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
Mathlib/LinearAlgebra/FreeModule/PID.lean:276:see `Submodule.basisOfPid`.
Mathlib/LinearAlgebra/FreeModule/PID.lean:280:theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
Mathlib/LinearAlgebra/FreeModule/PID.lean:290:    Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top
Mathlib/LinearAlgebra/FreeModule/PID.lean:300:noncomputable def Submodule.basisOfPid {ι : Type*} [Finite ι] (b : Basis ι R M)
Mathlib/LinearAlgebra/FreeModule/PID.lean:304:theorem Submodule.basisOfPid_bot {ι : Type*} [Finite ι] (b : Basis ι R M) :
Mathlib/LinearAlgebra/FreeModule/PID.lean:305:    Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ := by
Mathlib/LinearAlgebra/FreeModule/PID.lean:306:  obtain ⟨n, b'⟩ := Submodule.basisOfPid b ⊥
Mathlib/LinearAlgebra/FreeModule/PID.lean:316:noncomputable def Submodule.basisOfPidOfLE {ι : Type*} [Finite ι] {N O : Submodule R M}
Mathlib/LinearAlgebra/FreeModule/PID.lean:318:  let ⟨n, bN'⟩ := Submodule.basisOfPid b (N.comap O.subtype)
Mathlib/LinearAlgebra/FreeModule/PID.lean:323:noncomputable def Submodule.basisOfPidOfLESpan {ι : Type*} [Finite ι] {b : ι → M}
Mathlib/LinearAlgebra/FreeModule/PID.lean:326:  Submodule.basisOfPidOfLE le (Basis.span hb)
Mathlib/LinearAlgebra/FreeModule/PID.lean:372:    obtain ⟨n, b : Basis (Fin n) R (LinearMap.range φ)⟩ := Submodule.basisOfPidOfLE this sI_basis
Mathlib/LinearAlgebra

$ /bin/bash -lc "rg -n \"exists.*(nsmul|mul).*IsIntegral|exists.*IsIntegral.*(nsmul|mul)|IsIntegral.*(nsmul|mul)|multiple.*integral|integer.*multiple|denominator.*integral\" Mathlib/FieldTheory Mathlib/NumberTheory | sed -n '1,320p'"
Mathlib/NumberTheory/MahlerMeasure.lean:164:  grind [IsIntegral, RingHom.IsIntegralElem, mem_roots', IsRoot.def, eval₂_mul, eval_map]
Mathlib/NumberTheory/MahlerMeasure.lean:218:/-- If an integer non-constant polynomial has Mahler measure equal to 1 and is not a multiple of
Mathlib/NumberTheory/Height/NumberField.lean:386:    {x : K | IsIntegral ℤ (n * x) ∧ mulHeight₁ x ≤ B}.Finite := by
Mathlib/NumberTheory/Height/NumberField.lean:389:      {x | IsIntegral ℤ (n * x) ∧ mulHeight₁ x ≤ B} from
Mathlib/NumberTheory/Height/NumberField.lean:406:      ⋃ n : Fin ⌊B⌋₊, {x : K | IsIntegral ℤ ((n + 1) * x) ∧ mulHeight₁ x ≤ B} := by
Mathlib/NumberTheory/FrobeniusNumber.lean:21:largest positive integer that cannot be expressed as a sum of nonnegative multiples of these
Mathlib/NumberTheory/RamificationInertia/Basic.lean:200:  obtain ⟨a, hag, j, hjs, hgI⟩ := Ideal.exist_integer_multiples_notMem hRS s g hj's hj'g
Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean:175:lemma _root_.IsIntegrallyClosed.minpoly_smul {r : R} (hr : r ≠ 0) {s : S} (hs : IsIntegral R s) :
Mathlib/FieldTheory/RatFunc/Luroth.lean:126:-- we first define `Φ'` as any integer multiple of `φ`, and then set `Φ` to be its

$ /bin/bash -lc "rg -n \"def trace\" Mathlib/RingTheory/Trace Mathlib/LinearAlgebra/Trace.lean | sed -n '1,160p'; sed -n '120,190p' Mathlib/RingTheory/Trace/Defs.lean 2>/dev/null || true"
Mathlib/LinearAlgebra/Trace.lean:48:def traceAux : (M →ₗ[R] M) →ₗ[R] R :=
Mathlib/LinearAlgebra/Trace.lean:75:def trace : (M →ₗ[R] M) →ₗ[R] R :=
Mathlib/RingTheory/Trace/Defs.lean:71:noncomputable def trace : S →ₗ[R] R :=
Mathlib/RingTheory/Trace/Defs.lean:171:noncomputable def traceForm : BilinForm R S :=
Mathlib/RingTheory/Trace/Defs.lean:176:-- This is a nicer lemma than the one produced by `@[simps] def traceForm`.
Mathlib/RingTheory/Trace/Basic.lean:341:noncomputable def traceMatrix (b : κ → B) : Matrix κ κ A :=
    trace R S (trace S T x) = trace R T x := by
  have := Classical.decEq ι
  have := Classical.decEq κ
  cases nonempty_fintype ι
  cases nonempty_fintype κ
  rw [trace_eq_matrix_trace (b.smulTower c), trace_eq_matrix_trace b, trace_eq_matrix_trace c,
    Matrix.trace, Matrix.trace, Matrix.trace, ← Finset.univ_product_univ, Finset.sum_product]
  refine Finset.sum_congr rfl fun i _ ↦ ?_
  simp only [map_sum, smulTower_leftMulMatrix, Finset.sum_apply, Matrix.diag,
    Finset.sum_apply i (Finset.univ : Finset κ) fun y => leftMulMatrix b (leftMulMatrix c x y y)]

theorem trace_comp_trace_of_basis [Algebra S T] [IsScalarTower R S T] {ι κ : Type*} [Finite ι]
    [Finite κ] (b : Basis ι R S) (c : Basis κ S T) :
    (trace R S).comp ((trace S T).restrictScalars R) = trace R T := by
  ext
  rw [LinearMap.comp_apply, LinearMap.restrictScalars_apply, trace_trace_of_basis b c]

@[simp]
theorem trace_trace [Algebra S T] [IsScalarTower R S T]
    [Module.Free R S] [Module.Finite R S] [Module.Free S T] [Module.Finite S T] (x : T) :
    trace R S (trace S T x) = trace R T x :=
  trace_trace_of_basis (Module.Free.chooseBasis R S) (Module.Free.chooseBasis S T) x

/-- Let `T / S / R` be a tower of finite extensions of fields. Then
$\text{Trace}_{T/R} = \text{Trace}_{S/R} \circ \text{Trace}_{T/S}$. -/
@[simp, stacks 0BIJ "Trace"]
theorem trace_comp_trace [Algebra S T] [IsScalarTower R S T]
    [Module.Free R S] [Module.Finite R S] [Module.Free S T] [Module.Finite S T] :
 

$ /bin/bash -lc "rg -n \"RingOfIntegers.rank|theorem rank\" Mathlib/NumberTheory/NumberField/Basic.lean | sed -n '1,140p'; sed -n '330,410p' Mathlib/NumberTheory/NumberField/Basic.lean"
413:theorem RingOfIntegers.rank : Module.finrank ℤ (𝓞 K) = Module.finrank ℚ K :=
`M → 𝓞 K`. -/
def restrict (f : M → K) (h : ∀ x, IsIntegral ℤ (f x)) (x : M) : 𝓞 K :=
  ⟨f x, h x⟩

/-- Given `f : M →+ K` such that `∀ x, IsIntegral ℤ (f x)`, the corresponding function
`M →+ 𝓞 K`. -/
def restrict_addMonoidHom [AddZeroClass M] (f : M →+ K) (h : ∀ x, IsIntegral ℤ (f x)) :
    M →+ 𝓞 K where
  toFun := restrict f h
  map_zero' := by simp only [restrict, map_zero]; rfl
  map_add' x y := by simp only [restrict, map_add]; rfl

/-- Given `f : M →* K` such that `∀ x, IsIntegral ℤ (f x)`, the corresponding function
`M →* 𝓞 K`. -/
def restrict_monoidHom [MulOneClass M] (f : M →* K) (h : ∀ x, IsIntegral ℤ (f x)) : M →* 𝓞 K where
  toFun := restrict f h
  map_one' := by simp only [restrict, map_one]; rfl
  map_mul' x y := by simp only [restrict, map_mul]; rfl

section extension

variable (K L : Type*) [Field K] [Field L] [Algebra K L]

instance : IsScalarTower (𝓞 K) (𝓞 L) L :=
  IsScalarTower.of_algebraMap_eq' rfl

instance : IsIntegralClosure (𝓞 L) (𝓞 K) L :=
  IsIntegralClosure.tower_top (R := ℤ)

/-- The ring of integers of `L` is isomorphic to any integral closure of `𝓞 K` in `L` -/
protected noncomputable def algEquiv (R : Type*) [CommRing R] [Algebra (𝓞 K) R] [Algebra R L]
    [IsScalarTower (𝓞 K) R L] [IsIntegralClosure R (𝓞 K) L] : 𝓞 L ≃ₐ[𝓞 K] R :=
  (IsIntegralClosure.equiv (𝓞 K) R L _).symm

/-- Any extension between ring of integers is integral. -/
instance extension_algebra_isIntegral : Algebra.IsIntegral (𝓞 K) (𝓞 L) :=
  IsIntegralClosure.isIntegral_algebra (𝓞 K) L

/-- Any extension between ring of integers of number fields is Noetherian. -/
instance extension_isNoetherian [NumberField K] [NumberField L] : IsNoetherian (𝓞 K) (𝓞 L) :=
  IsIntegralClosure.isNoetherian (𝓞 K) K L (𝓞 L)

/-- The kernel of the algebraMap between ring of integers is `⊥`. -/
theorem ker_algebraMap_eq_bot : RingHom.ker (algebraMap (𝓞 K) (𝓞 L)) = ⊥ :=
  (RingHom.ker_eq_bot_iff_eq_zero (algebr

$ /bin/bash -lc "rg -n \"theorem trace_one|lemma trace_one|\\[simp\\].*trace.*1|trace.*one\" Mathlib/RingTheory/Trace Mathlib | rg \"Algebra|trace_one\" | sed -n '1,220p'"
Mathlib/RingTheory/DedekindDomain/Different.lean:792:      simp only [Algebra.traceForm_apply, mul_one] at hz
Mathlib/LinearAlgebra/Trace.lean:190:theorem trace_one : trace R M 1 = (finrank R M : R) := by
Mathlib/LinearAlgebra/Trace.lean:194:  rw [trace_eq_matrix_trace R b, toMatrix_one, finrank_eq_card_chooseBasisIndex]
Mathlib/LinearAlgebra/Trace.lean:199:theorem trace_id : trace R M id = (finrank R M : R) := by rw [← Module.End.one_eq_id, trace_one]
Mathlib/LinearAlgebra/Matrix/Kronecker.lean:227:/-- `trace` distributes over `Matrix.kroneckerMapBilinear`.
Mathlib/LinearAlgebra/Matrix/Kronecker.lean:229:This is primarily used with `R = ℕ` to prove `Matrix.trace_kronecker`. -/
Mathlib/LinearAlgebra/Matrix/Kronecker.lean:230:theorem trace_kroneckerMapBilinear [Semiring S] [Semiring R] [Fintype m] [Fintype n]
Mathlib/LinearAlgebra/Matrix/Kronecker.lean:235:    trace (kroneckerMapBilinear f A B) = f (trace A) (trace B) := by
Mathlib/LinearAlgebra/Matrix/Kronecker.lean:236:  simp_rw [Matrix.trace, Matrix.diag, kroneckerMapBilinear_apply_apply, LinearMap.map_sum₂,
Mathlib/LinearAlgebra/Matrix/Kronecker.lean:398:theorem trace_kronecker [Fintype m] [Fintype n] [Semiring α] (A : Matrix m m α) (B : Matrix n n α) :
Mathlib/LinearAlgebra/Matrix/Kronecker.lean:400:  trace_kroneckerMapBilinear (Algebra.lmul ℕ α).toLinearMap _ _
Mathlib/LinearAlgebra/Matrix/Kronecker.lean:527:theorem trace_kroneckerTMul [Fintype m] [Fintype n] (A : Matrix m m α) (B : Matrix n n β) :
Mathlib/LinearAlgebra/Matrix/Kronecker.lean:529:  trace_kroneckerMapBilinear (TensorProduct.mk R α β) _ _
Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean:139:theorem trace_eq_neg_charpoly_coeff [Nonempty n] (M : Matrix n n R) :
Mathlib/LinearAlgebra/Matrix/PosDef.lean:500:lemma trace_pos [Nontrivial R] [IsOrderedCancelAddMonoid R] [Nonempty n] {A : Matrix n n R}
Mathlib/LinearAlgebra/Matrix/Trace.lean:146:theorem trace_one : trace (1 : Matrix n n R) = Fintype.card n := by
Mathlib/LinearAlgebra/Matrix/Trace.lean:147:

$ /bin/bash -lc "sed -n '1,90p' Mathlib/NumberTheory/NumberField/House.lean; rg -n \"house_(add|mul|neg|nsmul|smul|pow)|house.*add|house.*mul|house_nonneg|norm_house\" Mathlib/NumberTheory/NumberField/House.lean | sed -n '1,240p'"
/-
Copyright (c) 2024 Michail Karatarakis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michail Karatarakis
-/
module

public import Mathlib.NumberTheory.SiegelsLemma
public import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
public import Mathlib.NumberTheory.NumberField.EquivReindex

/-!
# House of an algebraic number

This file defines the house of an algebraic number `α`, which is
the largest of the modulus of its conjugates.

## References
* [D. Marcus, *Number Fields*][marcus1977number]
* [Hua, L.-K., *Introduction to number theory*][hua1982house]

## Tags
number field, algebraic number, house
-/

@[expose] public section

variable {K : Type*} [Field K] [NumberField K]

namespace NumberField

noncomputable section

open Module.Free Module canonicalEmbedding Matrix Finset

attribute [local instance] Matrix.seminormedAddCommGroup

/-- The house of an algebraic number as the norm of its image by the canonical embedding. -/
def house (α : K) : ℝ := ‖canonicalEmbedding K α‖

/-- The house is the largest of the modulus of the conjugates of an algebraic number. -/
theorem house_eq_sup' (α : K) :
    house α = univ.sup' univ_nonempty (fun φ : K →+* ℂ ↦ ‖φ α‖₊) := by
  rw [house, ← coe_nnnorm, nnnorm_eq, ← sup'_eq_sup univ_nonempty]

theorem house_sum_le_sum_house {ι : Type*} (s : Finset ι) (α : ι → K) :
    house (∑ i ∈ s, α i) ≤ ∑ i ∈ s, house (α i) := by
  simp only [house, map_sum]; apply norm_sum_le_of_le; intros; rfl

theorem house_nonneg (α : K) : 0 ≤ house α := norm_nonneg _

theorem house_mul_le (α β : K) : house (α * β) ≤ house α * house β := by
  simp only [house, map_mul]; apply norm_mul_le

lemma house_prod_le (s : Finset K) : house (∏ x ∈ s, x) ≤ ∏ x ∈ s, house x := by
  simpa [house, map_prod] using Finset.norm_prod_le _ _

theorem house_add_le (α β : K) : house (α + β) ≤ house α + house β := by
  simp only [house, map_add]; apply norm_add_le

theorem house_pow_le (α : K) (i : ℕ) : house (

$ /bin/bash -lc "rg -n \"norm_norm_le_norm_mul_house_pow|one_le_house_of_isIntegral\" Mathlib/NumberTheory/NumberField/House.lean; rg -n \"norm_eq.*ofReal|ofReal.*norm|Complex.ofRealHom\" Mathlib/Data/Complex Mathlib/Analysis/Complex | sed -n '1,180p'"
88:lemma one_le_house_of_isIntegral {α : K} (hα : IsIntegral ℤ α) (hα0 : α ≠ 0) :
95:lemma norm_norm_le_norm_mul_house_pow (α : K) (σ : K →+* ℂ) :
Mathlib/Analysis/Complex/JensenFormula.lean:74:    rw [← ofReal_inj, ← normSq_eq_norm_sq, normSq_sub ]
Mathlib/Analysis/Complex/Poisson.lean:115:  simp only [Complex.norm_mul, norm_real, norm_eq_abs, norm_exp_ofReal_mul_I, mul_one, sq_abs,
Mathlib/Analysis/Complex/Poisson.lean:183:    I_re, I_im, add_im, sub_im, mul_im, div_im, ofReal_im, normSq_apply]
Mathlib/Analysis/Complex/Trigonometric.lean:874:  simpa [← ofReal_cos, ← norm_eq_abs, ← norm_real] using Complex.cos_bound (x := x) (by simpa)
Mathlib/Analysis/Complex/Trigonometric.lean:877:  simpa [← ofReal_sin, ← norm_eq_abs, ← norm_real] using Complex.sin_bound (x := x) (by simpa)
Mathlib/Analysis/Complex/Trigonometric.lean:978:  rw [add_neg_cancel, ofReal_zero, zero_mul, exp_zero, add_halves, ← neg_mul, Complex.norm_mul,
Mathlib/Analysis/Complex/Trigonometric.lean:980:    show -(ofReal (x / 2)) = ofReal (-x / 2) by norm_cast; exact neg_div' 2 x,
Mathlib/Analysis/Complex/Trigonometric.lean:981:    norm_exp_ofReal_mul_I, one_mul, ← norm_neg, neg_sub, mul_comm]
Mathlib/Analysis/Complex/Trigonometric.lean:984:  rw [exp_eq_exp_re_mul_sin_add_cos, Complex.norm_mul, norm_exp_ofReal, norm_cos_add_sin_mul_I,
Mathlib/Analysis/Complex/PhragmenLindelof.lean:147:    apply ((continuous_ofReal.mul continuous_const).cexp.smul continuous_const).norm.tendsto'
Mathlib/Analysis/Complex/Norm.lean:130:  rw [← ofReal_intCast, norm_real, Real.norm_eq_abs]
Mathlib/Analysis/Complex/Norm.lean:281:    simpa only [← ofReal_sub, norm_real, sub_re, sub_im] using (abs_im_le_norm _).trans_lt <| H _ ij
Mathlib/Analysis/Complex/AbelLimit.lean:70:  simp only [Set.mem_ofPred_eq, stolzSet, ← ofReal_one, ← ofReal_sub, norm_real,
Mathlib/Analysis/Complex/Hadamard.lean:403:  simp only [norm_mul, ← ofReal_add, norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε _) _,
Mathlib/Analysis/Complex/OperatorNorm.lean

$ /bin/bash -lc "rg -n \"card_fun|Fintype.card.*→|card_pi|card_congr.*fun|card_fin\" Mathlib/Data/Fintype Mathlib | sed -n '1,220p'"
Mathlib/Data/Fintype/Card.lean:386:theorem Fintype.card_subtype_le [Fintype α] (p : α → Prop) [Fintype {a // p a}] :
Mathlib/Data/Fintype/Card.lean:390:lemma Fintype.card_subtype_lt [Fintype α] {p : α → Prop} [Fintype {a // p a}] {x : α} (hx : ¬p x) :
Mathlib/Data/Fintype/Card.lean:395:theorem Fintype.card_subtype [Fintype α] (p : α → Prop) [Fintype {a // p a}] [DecidablePred p] :
Mathlib/Data/Fintype/Card.lean:401:theorem Fintype.card_subtype_compl [Fintype α] (p : α → Prop) [Fintype { x // p x }]
Mathlib/Data/Fintype/Card.lean:410:theorem Fintype.card_subtype_mono (p q : α → Prop) (h : p ≤ q) [Fintype { x // p x }]
Mathlib/Data/Fintype/Card.lean:415:theorem Fintype.card_compl_eq_card_compl [Finite α] (p q : α → Prop) [Fintype { x // p x }]
Mathlib/Data/Fintype/Card.lean:483:      (∀ (β) [Fintype β], Fintype.card β < Fintype.card α → P β) → P α) :
Mathlib/Data/Fintype/Card.lean:497:theorem Fintype.card_fin (n : ℕ) : Fintype.card (Fin n) = n :=
Mathlib/Data/Fintype/Card.lean:500:theorem Fintype.card_fin_lt_of_le {m n : ℕ} (h : m ≤ n) :
Mathlib/Data/Fintype/Card.lean:502:  conv_rhs => rw [← Fintype.card_fin m]
Mathlib/Data/Fintype/Card.lean:507:theorem Finset.card_fin (n : ℕ) : #(univ : Finset (Fin n)) = n := by simp
Mathlib/Data/Fintype/Card.lean:512:  (Fintype.card_fin m).symm.trans <| (Fintype.card_congr <| Equiv.cast h).trans (Fintype.card_fin n)
Mathlib/Data/Fintype/Card.lean:524:theorem card_finset_fin_le {n : ℕ} (s : Finset (Fin n)) : #s ≤ n := by
Mathlib/Data/Fintype/Card.lean:525:  simpa only [Fintype.card_fin] using s.card_le_univ
Mathlib/Data/Fintype/BigOperators.lean:116:@[simp] lemma Finset.card_pi (s : Finset ι) (t : ∀ i, Finset (α i)) :
Mathlib/Data/Fintype/BigOperators.lean:117:    #(s.pi t) = ∏ i ∈ s, #(t i) := Multiset.card_pi _ _
Mathlib/Data/Fintype/BigOperators.lean:123:@[simp] lemma card_piFinset (s : ∀ i, Finset (α i)) :
Mathlib/Data/Fintype/BigOperators.lean:128:`Fintype.card_piFinset`. -/
Mathlib/Data/Fintype/BigOperators.lean:129:lemma card_

$ /bin/bash -lc "rg -n \"theorem infDist_eq_iInf|lemma infDist_eq_iInf|infDist_le_dist_of_mem|le_infDist|infDist.*lower\" Mathlib/Topology/MetricSpace | sed -n '1,240p'"
Mathlib/Topology/MetricSpace/Closeds.lean:362:    exact infDist_le_infDist_add_hausdorffDist (edist_ne_top t s)
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:576:theorem infDist_eq_iInf : infDist x s = ⨅ y : s, dist x y := by
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:617:theorem infDist_le_dist_of_mem (h : y ∈ s) : infDist x s ≤ dist x y := by
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:622:theorem infDist_le_infDist_of_subset (h : s ⊆ t) (hs : s.Nonempty) : infDist x t ≤ infDist x s :=
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:625:lemma le_infDist {r : ℝ} (hs : s.Nonempty) : r ≤ infDist x s ↔ ∀ ⦃y⦄, y ∈ s → r ≤ dist x y := by
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:631:  simp [← not_le, le_infDist hs]
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:635:theorem infDist_le_infDist_add_dist : infDist x s ≤ infDist y s + dist x y := by
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:641:  h.not_ge <| infDist_le_dist_of_mem hy
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:656:theorem dist_le_infDist_add_diam (hs : IsBounded s) (hy : y ∈ s) :
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:665:  LipschitzWith.of_le_add fun _ _ => infDist_le_infDist_add_dist
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:719:  refine le_antisymm ?_ (infDist_le_infDist_of_subset inter_subset_left ⟨y, h⟩)
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:723:  · exact hz.not_ge (infDist_le_dist_of_mem ⟨hzs, hle⟩)
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:725:    exact (hlt.trans hz).not_ge (infDist_le_dist_of_mem h)
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:758:  LipschitzWith.of_le_add fun _ _ => infDist_le_infDist_add_dist
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:847:    exact le_trans (infDist_le_dist_of_mem yt) hy
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:850:    exact le_trans (infDist_le_dist_of_mem ys) hy
Mathlib/Topology/MetricSpace/HausdorffDistance.lean:890:theorem infDi

$ /bin/bash -lc "rg -n \"abs_inner_le_norm|abs.*inner.*norm|real_inner_comm|inner_smul|norm_smul\" Mathlib/Analysis/InnerProductSpace/Basic.lean | sed -n '1,220p'"
59:theorem real_inner_comm (x y : F) : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ :=
87:/-- See `inner_smul_left` for the common special when `𝕜 = 𝕝`. -/
88:lemma inner_smul_left_eq_star_smul (x y : E) (r : 𝕝) : ⟪r • x, y⟫ = r† • ⟪x, y⟫ := by
92:/-- Special case of `inner_smul_left_eq_star_smul` when the acting ring has a trivial star
94:lemma inner_smul_left_eq_smul [TrivialStar 𝕝] (x y : E) (r : 𝕝) : ⟪r • x, y⟫ = r • ⟪x, y⟫ := by
95:  rw [inner_smul_left_eq_star_smul, starRingEnd_apply, star_trivial]
97:/-- See `inner_smul_right` for the common special when `𝕜 = 𝕝`. -/
98:lemma inner_smul_right_eq_smul (x y : E) (r : 𝕝) : ⟪x, r • y⟫ = r • ⟪x, y⟫ := by
99:  rw [← inner_conj_symm, inner_smul_left_eq_star_smul, starRingEnd_apply, starRingEnd_apply,
104:/-- See `inner_smul_left_eq_star_smul` for the case of a general algebra action. -/
105:theorem inner_smul_left (x y : E) (r : 𝕜) : ⟪r • x, y⟫ = r† * ⟪x, y⟫ :=
106:  inner_smul_left_eq_star_smul ..
108:theorem real_inner_smul_left (x y : F) (r : ℝ) : ⟪r • x, y⟫_ℝ = r * ⟪x, y⟫_ℝ :=
109:  inner_smul_left _ _ _
111:theorem inner_smul_real_left (x y : E) (r : ℝ) : ⟪(r : 𝕜) • x, y⟫ = r • ⟪x, y⟫ := by
112:  rw [inner_smul_left, conj_ofReal, Algebra.smul_def]
114:/-- See `inner_smul_right_eq_smul` for the case of a general algebra action. -/
115:theorem inner_smul_right (x y : E) (r : 𝕜) : ⟪x, r • y⟫ = r * ⟪x, y⟫ :=
116:  inner_smul_right_eq_smul ..
118:theorem real_inner_smul_right (x y : F) (r : ℝ) : ⟪x, r • y⟫_ℝ = r * ⟪x, y⟫_ℝ :=
119:  inner_smul_right _ _ _
121:theorem inner_smul_real_right (x y : E) (r : ℝ) : ⟪x, (r : 𝕜) • y⟫ = r • ⟪x, y⟫ := by
122:  rw [inner_smul_right, Algebra.smul_def]
131:  LinearMap.mk₂'ₛₗ _ _ (fun v w => ⟪v, w⟫) inner_add_left (fun _ _ _ => inner_smul_left _ _ _)
132:    inner_add_right fun _ _ _ => inner_smul_right _ _ _
148:  exact real_inner_comm v w
188:  rw [← zero_smul 𝕜 (0 : E), inner_smul_left, map_zero, zero_mul]
229:  rw [← neg_one_smul 𝕜 x, inner_smul_left]
287:      rw [real_inner_comm y, ← norm_mul]
358:   

$ /bin/bash -lc "rg -n \"EuclideanSpace.norm_eq|norm_eq_sqrt_sum|dist_eq.*EuclideanSpace|EuclideanSpace.*dist|real_inner.*Fin|inner_apply\" Mathlib/Analysis/InnerProductSpace Mathlib/Geometry/Euclidean | sed -n '1,260p'"
Mathlib/Analysis/InnerProductSpace/ProdL2.lean:47:theorem prod_inner_apply (x y : WithLp 2 (E × F)) :
Mathlib/Analysis/InnerProductSpace/ProdL2.lean:67:        WithLp.prod_inner_apply, Sum.forall, Sum.elim_inl, Function.comp_apply, inner_zero_right,
Mathlib/Analysis/InnerProductSpace/Calculus.lean:128:theorem fderiv_inner_apply (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (y : G) :
Mathlib/Analysis/InnerProductSpace/Calculus.lean:132:theorem deriv_inner_apply {f g : ℝ → E} {x : ℝ} (hf : DifferentiableAt ℝ f x)
Mathlib/Analysis/InnerProductSpace/Symmetric.lean:147:theorem IsSymmetric.im_inner_apply_self {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (x : E) :
Mathlib/Analysis/InnerProductSpace/Symmetric.lean:157:theorem IsSymmetric.coe_re_inner_apply_self {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (x : E) :
Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean:22:+ `ContinuousLinearMapWOT.tendsto_iff_forall_inner_apply_tendsto`: a function `f : α → E →WOT[𝕜] F`
Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean:48:lemma tendsto_iff_forall_inner_apply_tendsto {α : Type*} {l : Filter α}
Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean:55:lemma le_nhds_iff_forall_inner_apply_le_nhds {l : Filter (E →WOT[𝕜] F)}
Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean:57:  tendsto_iff_forall_inner_apply_tendsto (f := id)
Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean:62:  tendsto_iff_forall_inner_apply_tendsto
Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean:66:  tendsto_iff_forall_inner_apply_tendsto
Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean:77:@[fun_prop] alias ⟨continuousWithinAt_inner_apply, continuousWithinAt⟩ := continuousWithinAt_iff
Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean:78:@[fun_prop] alias ⟨continuousOn_inner_apply, continuousOn⟩ := continuousOn_iff
Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean:79:@[fun_prop] alias ⟨continuousAt_inner_apply, continuou

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:21:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.coe_realEmbedding`:
  [NumberField K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [NumberField K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:29:12: error(lean.unknownIdentifier): Unknown constant `Algebra.trace_eq_sum_embeddings`
SharpDistanceScratch.lean:28:48: error: unsolved goals
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyReal K
z : K
⊢ ↑((Algebra.trace ℚ K) (z ^ 2)) = ∑ φ, (realEmbedding φ) z ^ 2
SharpDistanceScratch.lean:43:6: error: Type mismatch
  (Finset.sum_eq_zero_iff_of_nonneg fun x x_1 => sq_nonneg ((realEmbedding x) z)).mp hs
has type
  ∀ i ∈ Finset.univ, (realEmbedding i) z ^ 2 = 0
but is expected to have type
  ∀ (φ : K →+* ℂ), (realEmbedding φ) z ^ 2 = 0
SharpDistanceScratch.lean:46:38: error: Application type mismatch: The argument
  this
has type
  (realEmbedding φ) z = 0
but is expected to have type
  (realEmbedding φ) z = (realEmbedding φ) 0
in the application
  RingHom.injective (realEmbedding φ) this

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:21:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.coe_realEmbedding`:
  [NumberField K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [NumberField K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:29:12: error: typeclass instance problem is stuck
  Algebra.IsSeparable ?m.52 K

Note: Lean will not try to resolve this typeclass instance problem because the first, third, and fifth type arguments to `Algebra.IsSeparable` contain metavariables. These arguments must be fully determined before Lean will try to resolve the typeclass.

Hint: Adding type annotations and supplying implicit arguments to functions can give Lean more information for typeclass resolution. For example, if you have a variable `x` that you intend to be a `Nat`, but Lean reports it as having an unresolved type like `?m`, replacing `x` with `(x : Nat)` can get typeclass resolution un-stuck.

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:21:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.coe_realEmbedding`:
  [NumberField K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [NumberField K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:31:36: error(lean.unknownIdentifier): Unknown constant `Complex.sum_re`
SharpDistanceScratch.lean:31:2: error: Type mismatch: After simplification, term
  hre
 has type
  (realEmbedding (algebraMap ℚ ℂ)) ((Algebra.trace ℚ K) (z * z)) = (∑ x, x (z * z)).re
but is expected to have type
  ↑((Algebra.trace ℚ K) (z * z)) = ∑ x, (realEmbedding x) z * (realEmbedding x) z

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:21:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.coe_realEmbedding`:
  [NumberField K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [NumberField K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:35:38: error: Type mismatch
  h
has type
  (algebraMap ℚ ℂ) ((Algebra.trace ℚ K) (z ^ 2)) =
    ∑ σ ∈ @Finset.univ (K →ₐ[ℚ] ℂ) (minpoly.AlgHom.fintype ℚ K ℂ), σ (z ^ 2)
but is expected to have type
  (algebraMap ℚ ℂ) ((Algebra.trace ℚ K) (z ^ 2)) =
    ∑ φ ∈ @Finset.univ (K →+* ℂ) (Embeddings.instFintypeRingHom K ℂ), φ (z ^ 2)

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:54:43: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  Nonempty (K →ₐ[ℚ] ℂ)

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:20:31: error: Type mismatch
  ![x, y]
has type
  Fin (Nat.succ 0).succ → ℝ
but is expected to have type
  Point
SharpDistanceScratch.lean:22:0: error: Not a definitional equality: the left-hand side
  (point x y).ofLp 0
is not definitionally equal to the right-hand side
  x
SharpDistanceScratch.lean:22:56: error: Type mismatch
  rfl
has type
  ?m.6 = ?m.6
but is expected to have type
  (point x y).ofLp 0 = x
SharpDistanceScratch.lean:23:0: error: Not a definitional equality: the left-hand side
  (point x y).ofLp 1
is not definitionally equal to the right-hand side
  y
SharpDistanceScratch.lean:23:55: error: Type mismatch
  rfl
has type
  ?m.6 = ?m.6
but is expected to have type
  (point x y).ofLp 1 = y
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:43:36: error: expected token

$ /bin/bash -lc "sed -n '120,170p' Mathlib/Analysis/InnerProductSpace/PiL2.lean; nl -ba SharpDistanceScratch.lean | sed -n '14,52p'"
/-- Notation for vectors in Lp space. `!₂[x, y, ...]` is a shorthand for
`WithLp.toLp 2 ![x, y, ...]`, of type `EuclideanSpace _ (Fin _)`.

This also works for other subscripts. -/
syntax (name := PiLp.vecNotation) "!" noWs subscriptTerm noWs "[" term,* "]" : term
macro_rules | `(!$p:subscript[$e:term,*]) => do
  -- override the `Fin n.succ` to a literal
  let n := e.getElems.size
  `(WithLp.toLp $p (V := ∀ _ : Fin $(quote n), _) ![$e,*])

/-- Unexpander for the `!₂[x, y, ...]` notation. -/
@[app_delab WithLp.toLp]
meta def EuclideanSpace.delabVecNotation : Delab :=
  whenNotPPOption getPPExplicit <| whenPPOption getPPNotation <| withOverApp 3 do
    -- check that the `WithLp.toLp _` is present
    let p : Term ← withNaryArg 0 <| delab
    -- to be conservative, only allow subscripts which are numerals
    guard <| p matches `($_:num)
    let `(![$elems,*]) ← withNaryArg 2 delab | failure
    `(!$p[$elems,*])

end Notation

theorem EuclideanSpace.nnnorm_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x : EuclideanSpace 𝕜 n) : ‖x‖₊ = NNReal.sqrt (∑ i, ‖x i‖₊ ^ 2) :=
  PiLp.nnnorm_eq_of_L2 x

theorem EuclideanSpace.norm_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x : EuclideanSpace 𝕜 n) : ‖x‖ = √(∑ i, ‖x i‖ ^ 2) := by
  simpa only [Real.coe_sqrt, NNReal.coe_sum] using! congr_arg ((↑) : ℝ≥0 → ℝ) x.nnnorm_eq

theorem EuclideanSpace.norm_sq_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x : EuclideanSpace 𝕜 n) : ‖x‖ ^ 2 = ∑ i, ‖x i‖ ^ 2 :=
  PiLp.norm_sq_eq_of_L2 _ x

theorem EuclideanSpace.real_norm_sq_eq {n : Type*} [Fintype n] (x : EuclideanSpace ℝ n) :
    ‖x‖ ^ 2 = ∑ i, (x i) ^ 2 := by
  simp [EuclideanSpace.norm_sq_eq]

@[wikidata Q847073]
theorem EuclideanSpace.dist_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x y : EuclideanSpace 𝕜 n) : dist x y = √(∑ i, dist (x i) (y i) ^ 2) :=
  PiLp.dist_eq_of_L2 x y

theorem EuclideanSpace.dist_sq_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x y : EuclideanSpace 𝕜 n) : dist x y ^ 

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:43:36: error: expected token

$ /bin/bash -lc "sed -n '42,44p' SharpDistanceScratch.lean | od -An -tx1c"
6c  65  6d  6d  61  20  70  6f  69  6e  74  4c  69  6e  65  44
   l   e   m   m   a       p   o   i   n   t   L   i   n   e   D
  69  73  74  5f  67  65  5f  69  6e  6e  65  72  5f  64  69  76
   i   s   t   _   g   e   _   i   n   n   e   r   _   d   i   v
  5f  6e  6f  72  6d  20  28  70  20  3a  20  50  6f  69  6e  74
   _   n   o   r   m       (   p       :       P   o   i   n   t
  29  20  28  e2  84  93  20  3a  20  52  65  61  6c  4c  69  6e
   )       ( 342 204 223       :       R   e   a   l   L   i   n
  65  29  20  28  6e  6f  72  6d  61  6c  20  3a  20  50  6f  69
   e   )       (   n   o   r   m   a   l       :       P   o   i
  6e  74  29  0a  20  20  20  20  28  68  6e  6f  72  6d  61  6c
   n   t   )  \n                   (   h   n   o   r   m   a   l
  20  3a  20  6e  6f  72  6d  61  6c  20  e2  89  a0  20  30  29
       :       n   o   r   m   a   l     342 211 240       0   )
  20  28  68  6f  72  74  68  20  3a  20  e2  9f  aa  28  e2  84
       (   h   o   r   t   h       :     342 237 252   ( 342 204
  93  2e  64  69  72  65  63  74  69  6f  6e  29  2c  20  6e  6f
 223   .   d   i   r   e   c   t   i   o   n   )   ,       n   o
  72  6d  61  6c  e2  9f  ab  5f  e2  84  9d  20  3d  20  30  29
   r   m   a   l 342 237 253   _ 342 204 235       =       0   )
  20  3a  0a  20  20  20  20  7c  e2  9f  aa  70  20  2d  20  e2
       :  \n                   | 342 237 252   p       -     342
  84  93  2e  62  61  73  65  2c  20  6e  6f  72  6d  61  6c  e2
 204 223   .   b   a   s   e   ,       n   o   r   m   a   l 342
  9f  ab  5f  e2  84  9d  7c  20  2f  20  e2  80  96  6e  6f  72
 237 253   _ 342 204 235   |       /     342 200 226   n   o   r
  6d  61  6c  e2  80  96  20  e2  89  a4  20  70  6f  69  6e  74
   m   a   l 342 200 226     342 211 244       p   o   i   n   t
  4c  69  6e  65  44  69  73  74  20  70  20  e2  84  93  20  3a
   L   i   n   e   D   i   s   t       p     342 204 223       :
  3d  20  62  79  0a
   =       b   y  \n

$ /bin/bash -lc "python3 - <<'PY'
s=open('SharpDistanceScratch.lean').read().splitlines()[42]
for i,c in enumerate(s,1): print(i,repr(c))
PY"
1 ' '
2 ' '
3 ' '
4 ' '
5 '('
6 'h'
7 'n'
8 'o'
9 'r'
10 'm'
11 'a'
12 'l'
13 ' '
14 ':'
15 ' '
16 'n'
17 'o'
18 'r'
19 'm'
20 'a'
21 'l'
22 ' '
23 '≠'
24 ' '
25 '0'
26 ')'
27 ' '
28 '('
29 'h'
30 'o'
31 'r'
32 't'
33 'h'
34 ' '
35 ':'
36 ' '
37 '⟪'
38 '('
39 'ℓ'
40 '.'
41 'd'
42 'i'
43 'r'
44 'e'
45 'c'
46 't'
47 'i'
48 'o'
49 'n'
50 ')'
51 ','
52 ' '
53 'n'
54 'o'
55 'r'
56 'm'
57 'a'
58 'l'
59 '⟫'
60 '_'
61 'ℝ'
62 ' '
63 '='
64 ' '
65 '0'
66 ')'
67 ' '
68 ':'

$ /bin/bash -lc "rg -n 'notation.*⟪|scoped.*⟪|⟫_ℝ' Mathlib/Analysis/InnerProductSpace/Basic.lean Mathlib | sed -n '1,100p'"
Mathlib/Analysis/InnerProductSpace/Basic.lean:49:local notation "⟪" x ", " y "⟫" => inner 𝕜 x y
Mathlib/Analysis/InnerProductSpace/Basic.lean:59:theorem real_inner_comm (x y : F) : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ :=
Mathlib/Analysis/InnerProductSpace/Basic.lean:108:theorem real_inner_smul_left (x y : F) (r : ℝ) : ⟪r • x, y⟫_ℝ = r * ⟪x, y⟫_ℝ :=
Mathlib/Analysis/InnerProductSpace/Basic.lean:118:theorem real_inner_smul_right (x y : F) (r : ℝ) : ⟪x, r • y⟫_ℝ = r * ⟪x, y⟫_ℝ :=
Mathlib/Analysis/InnerProductSpace/Basic.lean:152:@[simp] lemma innerₗ_apply_apply (v w : F) : innerₗ F v w = ⟪v, w⟫_ℝ := rfl
Mathlib/Analysis/InnerProductSpace/Basic.lean:203:theorem real_inner_self_nonneg {x : F} : 0 ≤ ⟪x, x⟫_ℝ :=
Mathlib/Analysis/InnerProductSpace/Basic.lean:222:theorem real_inner_self_abs (x : F) : |⟪x, x⟫_ℝ| = ⟪x, x⟫_ℝ :=
Mathlib/Analysis/InnerProductSpace/Basic.lean:254:/-- Expand `⟪x + y, x + y⟫_ℝ` -/
Mathlib/Analysis/InnerProductSpace/Basic.lean:256:    ⟪x + y, x + y⟫_ℝ = ⟪x, x⟫_ℝ + 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ := by
Mathlib/Analysis/InnerProductSpace/Basic.lean:257:  have : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ := by rw [← inner_conj_symm]; rfl
Mathlib/Analysis/InnerProductSpace/Basic.lean:265:/-- Expand `⟪x - y, x - y⟫_ℝ` -/
Mathlib/Analysis/InnerProductSpace/Basic.lean:267:    ⟪x - y, x - y⟫_ℝ = ⟪x, x⟫_ℝ - 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ := by
Mathlib/Analysis/InnerProductSpace/Basic.lean:268:  have : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ := by rw [← inner_conj_symm]; rfl
Mathlib/Analysis/InnerProductSpace/Basic.lean:284:theorem real_inner_mul_inner_self_le (x y : F) : ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ ≤ ⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ :=
Mathlib/Analysis/InnerProductSpace/Basic.lean:286:    ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ ≤ ‖⟪x, y⟫_ℝ‖ * ‖⟪y, x⟫_ℝ‖ := by
Mathlib/Analysis/InnerProductSpace/Basic.lean:289:    _ ≤ ⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ := @inner_mul_inner_self_le ℝ _ _ _ _ x y
Mathlib/Analysis/InnerProductSpace/Basic.lean:294:theorem inner_eq_norm_sq_left_iff {v w : F} : ⟪v, w⟫_ℝ = ‖v‖ ^ 2 ↔ ⟪v, v - w⟫_ℝ = 0 :=
Mathlib/Analysis/InnerProductSpace/Basic.lean:300:theorem inne

$ /bin/bash -lc "rg -n 'scoped.*notation.*⟪|notation.*⟪.*_ℝ|syntax.*⟪' Mathlib/Analysis/InnerProductSpace Mathlib | head -40"
Mathlib/Analysis/InnerProductSpace/Defs.lean:86:scoped[InnerProductSpace] notation:max "⟪" x ", " y "⟫_" 𝕜:max => inner 𝕜 x y
Mathlib/Analysis/InnerProductSpace/Defs.lean:91:scoped[RealInnerProductSpace] notation "⟪" x ", " y "⟫" => inner ℝ x y
Mathlib/Analysis/InnerProductSpace/Defs.lean:94:scoped[ComplexInnerProductSpace] notation "⟪" x ", " y "⟫" => inner ℂ x y
Mathlib/Topology/Sheaves/Presheaf.lean:117:scoped[AlgebraicGeometry] notation:80 x " |_ₗ " U " ⟪" e "⟫ " =>
Mathlib/Analysis/InnerProductSpace/Defs.lean:86:scoped[InnerProductSpace] notation:max "⟪" x ", " y "⟫_" 𝕜:max => inner 𝕜 x y
Mathlib/Analysis/InnerProductSpace/Defs.lean:91:scoped[RealInnerProductSpace] notation "⟪" x ", " y "⟫" => inner ℝ x y
Mathlib/Analysis/InnerProductSpace/Defs.lean:94:scoped[ComplexInnerProductSpace] notation "⟪" x ", " y "⟫" => inner ℂ x y

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:122:38: error: unsolved goals
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyReal K
hd : 0 < Module.finrank ℚ K
h : ∀ (j : IntegerIndex K), (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) j) = 0
ht : Algebra.trace ℤ (𝓞 K) = 0
h1 : (Algebra.trace ℤ (𝓞 K)) 1 = 0 1
hrank : 0 < Module.finrank ℤ (𝓞 K)
⊢ (LinearMap.trace ℤ (𝓞 K)) ((LinearMap.mul ℤ (𝓞 K)) 1) = ↑(Module.finrank ℤ (𝓞 K))
SharpDistanceScratch.lean:125:2: error: Type mismatch: After simplification, term
  h1
 has type
  Module.finrank ℤ (𝓞 K) = 0
but is expected to have type
  False

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:127:8: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (Algebra.lmul ℤ (𝓞 K)) 1
in the target expression
  (LinearMap.trace ℤ (𝓞 K) ∘ₗ (Algebra.lmul ℤ (𝓞 K)).toLinearMap) 1 = ↑(Module.finrank ℤ (𝓞 K))

K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyReal K
hd : 0 < Module.finrank ℚ K
h : ∀ (j : IntegerIndex K), (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) j) = 0
ht : Algebra.trace ℤ (𝓞 K) = 0
h1 : (Algebra.trace ℤ (𝓞 K)) 1 = 0 1
hrank : 0 < Module.finrank ℤ (𝓞 K)
hlmul : (Algebra.lmul ℤ (𝓞 K)) 1 = 1
⊢ (LinearMap.trace ℤ (𝓞 K) ∘ₗ (Algebra.lmul ℤ (𝓞 K)).toLinearMap) 1 = ↑(Module.finrank ℤ (𝓞 K))
SharpDistanceScratch.lean:129:2: error: omega could not prove the goal:
a possible counterexample may satisfy the constraints
  b ≥ 1
  a ≥ 1
where
 a := ↑(Module.finrank ℚ K)
 b := (Algebra.trace ℤ (𝓞 K)) 1

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:131:2: error: omega could not prove the goal:
a possible counterexample may satisfy the constraints
  b ≥ 1
  a ≥ 1
where
 a := ↑(Module.finrank ℚ K)
 b := (Algebra.trace ℤ (𝓞 K)) 1

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:152:54: error: unsolved goals
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyReal K
B : TraceBasisData K
i : TraceIndex B
⊢ (Algebra.trace ℤ (𝓞 K))
        (↑((Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) B.pivot)) * (RingOfIntegers.basis K) ↑i) -
      (Algebra.trace ℤ (𝓞 K))
        (↑((Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) ↑i)) * (RingOfIntegers.basis K) B.pivot) =
    0
SharpDistanceScratch.lean:165:2: error: Type mismatch: After simplification, term
  hi
 has type
  ∑ c_1, ((RingOfIntegers.basis K).repr (↑(c c_1) * (RingOfIntegers.basis K) c_1)) i =
    ∑ c, ((RingOfIntegers.basis K).repr (↑(c' c) * (RingOfIntegers.basis K) c)) i
but is expected to have type
  c i = c' i
SharpDistanceScratch.lean:174:35: error(lean.unknownIdentifier): Unknown constant `LinearEquiv.map_nsmul`
SharpDistanceScratch.lean:175:4: error(lean.unknownIdentifier): Unknown constant `LinearEquiv.map_sub`
SharpDistanceScratch.lean:175:25: error(lean.unknownIdenti

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:2: error: ring failed, ring expressions not equal
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyReal K
B : TraceBasisData K
i : TraceIndex B
⊢ (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) B.pivot) = 0
SharpDistanceScratch.lean:168:2: error: Type mismatch: After simplification, term
  hi
 has type
  ∑ c_1, (Finsupp.single c_1 ↑(c c_1)) i = ∑ c, (Finsupp.single c ↑(c' c)) i
but is expected to have type
  c i = c' i
SharpDistanceScratch.lean:167:52: warning: This simp argument is unused:
  Finsupp.single_apply

Hint: Omit it from the simp argument list.
  [apply] simp only [linearCombo, map_sum, map_nsmul, (NumberField.RingOfIntegers.basis K).repr_self] at hi

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:180:2: error: `simp` made no progress
SharpDistanceScratch.lean:179:4: warning: This simp argument is unused:
  Finsupp.single_apply

Hint: Omit it from the simp argument list.
  [apply

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:2: error: ring failed, ring expressions not equal
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyReal K
B : TraceBasisData K
i : TraceIndex B
⊢ (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) B.pivot) = 0
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:184:11: error(lean.unknownIdentifier): Unknown identifier `smul_left_cancel₀`
SharpDistanceScratch.lean:181:4: warning: This simp argument is unused:
  Finsupp.single_apply

Hint: Omit it from the simp argument list.
  [apply] simp only [linearCombo, map_sum, map_nsmul, traceGenerator, map_sub, map_smul,
    (Numb

$ /bin/bash -lc "lake env lean SharpDistanceScratch.lean 2>&1 | sed -n '1,220p'"
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsTotallyReal K
B : TraceBasisData K
i : TraceIndex B
⊢ (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) B.pivot) • (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) ↑i) -
      (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) ↑i) •
        (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) B.pivot) =
    0
SharpDistanceScratch.lean:151:6: warning: declaration uses `sorry`
SharpDistanceScratch.lean:160:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:185:11: error(lean.unknownIdentifier): Unknown identifier `smul_left_cancel₀`
SharpDistanceScratch.lean:182:4: warning: This simp argumen

$ /bin/bash -lc "lake env lean SharpDistanceScratch.lean 2>&1 | sed -n '1,260p'"
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "nl -ba SharpDistanceScratch.lean | sed -n '174,188p'"
174	  intro c c' h
   175	  funext i
   176	  have hi := congrArg (fun z : 𝓞 K ↦
   177	    (NumberField.RingOfIntegers.basis K).repr z i.1) h
   178	  have ht := B.pivot_trace_ne_zero
   179	  simp only [linearCombo, map_sum, map_nsmul, traceGenerator,
   180	    map_sub, map_smul, (NumberField.RingOfIntegers.basis K).repr_self,
   181	    Finsupp.single_apply] at hi
   182	  simp only [Subtype.coe_inj] at hi
   183	  simp [Finsupp.single_apply, i.2] at hi
   184	  have hcast : (c i : ℤ) = c' i := by
   185	    exact mul_right_cancel₀ ht hi
   186	  exact_mod_cast hcast
   187	
   188	end

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "nl -ba SharpDistanceScratch.lean | sed -n '178,188p'"
178	  have ht := B.pivot_trace_ne_zero
   179	  simp only [linearCombo, map_sum, map_nsmul, traceGenerator,
   180	    map_sub, map_smul, (NumberField.RingOfIntegers.basis K).repr_self,
   181	    Finsupp.single_apply] at hi
   182	  have hcoe (x : TraceIndex B) : (x.1 = i.1) = (x = i) :=
   183	    propext Subtype.ext_iff.symm
   184	  simp_rw [hcoe] at hi
   185	  simp [Finsupp.single_apply, i.2] at hi
   186	  have hcast : (c i : ℤ) = c' i := by
   187	    exact mul_right_cancel₀ ht hi
   188	  exact_mod_cast hcast

$ /bin/bash -lc "lake env lean SharpDistanceScratch.lean 2>&1 | sed -n '45,180p'"
B : TraceBasisData K
c c' : TraceIndex B → ℕ
h : linearCombo (traceGenerator B) c = linearCombo (traceGenerator B) c'
i : TraceIndex B
ht : (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) B.pivot) ≠ 0
hi :
  (∑ x,
        c x •
          ((Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) B.pivot) • Finsupp.single (↑x) 1 -
            (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) ↑x) • Finsupp.single B.pivot 1))
      ↑i =
    (∑ x,
        c' x •
          ((Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) B.pivot) • Finsupp.single (↑x) 1 -
            (Algebra.trace ℤ (𝓞 K)) ((RingOfIntegers.basis K) ↑x) • Finsupp.single B.pivot 1))
      ↑i
hcoe : ∀ (x : TraceIndex B), (↑x = ↑i) = (x = i)
⊢ c i = c' i
SharpDistanceScratch.lean:181:4: warning: This simp argument is unused:
  Finsupp.single_apply

Hint: Omit it from the simp argument list.
  [apply] simp only [linearCombo, map_sum, map_nsmul, traceGenerator, map_sub, map_smul,
    (NumberField.RingOfIntegers.basis K).repr_self] at hi

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "nl -ba SharpDistanceScratch.lean | sed -n '180,192p'"
180	    map_sub, map_smul, (NumberField.RingOfIntegers.basis K).repr_self,
   181	    Finsupp.single_apply] at hi
   182	  have hcoe (x : TraceIndex B) : (x.1 = i.1) = (x = i) :=
   183	    propext Subtype.ext_iff.symm
   184	  simp [Finset.sum_apply, Finsupp.single_apply, i.2, hcoe] at hi
   185	  have hcast : (c i : ℤ) = c' i := by
   186	    exact mul_right_cancel₀ ht hi
   187	  exact_mod_cast hcast
   188	  have hcast : (c i : ℤ) = c' i := by
   189	    exact mul_right_cancel₀ ht hi
   190	  exact_mod_cast hcast
   191	
   192	end

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "nl -ba SharpDistanceScratch.lean | sed -n '356,455p'"
356	          congr 1
   357	          simp [linearCombo]
   358	    _ ≤ ∑ i, NumberField.house ((c i) * (g i : K)) := by
   359	      simpa using NumberField.house_sum_le_sum_house Finset.univ
   360	        (fun i ↦ (c i) * (g i : K))
   361	    _ = ∑ i, (c i : ℝ) * NumberField.house (g i : K) := by
   362	      apply Finset.sum_congr rfl
   363	      intro i _
   364	      simpa using NumberField.house_nat_mul (g i : K) (c i)
   365	    _ ≤ ∑ i, (q : ℝ) * NumberField.house (g i : K) := by
   366	      gcongr with i hi
   367	      · exact_mod_cast hc i
   368	      · exact NumberField.house_nonneg _
   369	    _ = q * ∑ i, NumberField.house (g i : K) := by
   370	      rw [Finset.mul_sum]
   371	
   372	lemma house_aZero_le (B : TraceBasisData K) {m : ℕ} (c : ACoeff B m) :
   373	    NumberField.house (aZero B c : K) ≤ (m : ℝ) ^ 2 * aConstant B := by
   374	  calc
   375	    NumberField.house (aZero B c : K) ≤
   376	        (m ^ 2 : ℕ) * ∑ i, NumberField.house (traceGenerator B i : K) := by
   377	      apply house_linearCombo_le
   378	      intro i
   379	      omega
   380	    _ ≤ (m : ℝ) ^ 2 * aConstant B := by
   381	      gcongr
   382	      exact le_max_right _ _
   383	
   384	lemma house_yHalf_le {m : ℕ} (c : YCoeff K m) :
   385	    NumberField.house (yHalf c : K) ≤ (m : ℝ) * yConstant K := by
   386	  calc
   387	    NumberField.house (yHalf c : K) ≤
   388	        m * ∑ i, NumberField.house (NumberField.RingOfIntegers.basis K i : K) := by
   389	      apply house_linearCombo_le
   390	      intro i
   391	      omega
   392	    _ ≤ (m : ℝ) * yConstant K := by
   393	      gcongr
   394	      exact le_max_right _ _
   395	
   396	lemma house_sourceY_le (B : TraceBasisData K) {m : ℕ} (s : ArithmeticSource B m) :
   397	    NumberField.house (sourceY s : K) ≤
   398	      2 * yConstant K * m := by
   399	  calc
   400	    NumberField.house (sourceY s : K) =
   401	        NumberField.house (2 * (yHalf s.2 : K)) := by
   402	      simp [sourceY]
   403	 

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=500000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=800000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "nl -ba SharpDistanceScratch.lean | sed -n '446,475p'"
446	  let CX : ℝ := aConstant B + 2 * yConstant K ^ 2
   447	  let CY : ℝ := 2 * yConstant K
   448	  have hXs := house_sourceX_le B s
   449	  have hXt := house_sourceX_le B t
   450	  have hYs := house_sourceY_le B s
   451	  have hYt := house_sourceY_le B t
   452	  have hCY : 0 ≤ CY := by
   453	    dsimp [CY]
   454	    positivity
   455	  have hCX : 0 ≤ CX := by
   456	    dsimp [CX]
   457	    positivity
   458	  have hYdiff :
   459	      NumberField.house ((sourceY t : K) - (sourceY s : K)) ≤ 2 * CY * m := by
   460	    calc
   461	      _ ≤ NumberField.house (sourceY t : K) +
   462	          NumberField.house (sourceY s : K) := house_sub_le _ _
   463	      _ ≤ CY * m + CY * m := by
   464	        dsimp [CY]
   465	        constructor
   466	        · exact hYt
   467	        · exact hYs
   468	      _ = 2 * CY * m := by ring
   469	  change NumberField.house
   470	    ((sourceX B s : K) - (sourceX B t : K) +
   471	      (sourceY s : K) * ((sourceY t : K) - (sourceY s : K))) ≤ _
   472	  calc
   473	    _ ≤ NumberField.house ((sourceX B s : K) - (sourceX B t : K)) +
   474	        NumberField.house ((sourceY s : K) *
   475	          ((sourceY t : K) - (sourceY s : K))) :=

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=800000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "rg -n \"def ofReal.*RingHom|ofRealHom|ofReal.*comp.*RingHom|ofRealAm\" Mathlib/Data/Complex Mathlib/Analysis/Complex | sed -n '1,160p'"
Mathlib/Data/Complex/BigOperators.lean:26:  map_prod ofRealHom _ _
Mathlib/Data/Complex/BigOperators.lean:30:  map_sum ofRealHom _ _
Mathlib/Data/Complex/BigOperators.lean:34:  map_expect ofRealHom ..
Mathlib/Data/Complex/Basic.lean:593:def ofRealHom : ℝ →+* ℂ where
Mathlib/Data/Complex/Basic.lean:600:@[simp] lemma ofRealHom_eq_coe (r : ℝ) : ofRealHom r = r := rfl
Mathlib/Data/Complex/Basic.lean:605:  map_comp_add ofRealHom ..
Mathlib/Data/Complex/Basic.lean:608:  map_comp_sub ofRealHom ..
Mathlib/Data/Complex/Basic.lean:611:  map_comp_neg ofRealHom _
Mathlib/Data/Complex/Basic.lean:614:  map_comp_nsmul ofRealHom ..
Mathlib/Data/Complex/Basic.lean:617:  map_comp_zsmul ofRealHom ..
Mathlib/Data/Complex/Basic.lean:620:  map_comp_mul ofRealHom ..
Mathlib/Data/Complex/Basic.lean:623:  map_comp_pow ofRealHom ..
Mathlib/Data/Complex/Basic.lean:719:theorem ofReal_div (r s : ℝ) : ((r / s : ℝ) : ℂ) = r / s := map_div₀ ofRealHom r s
Mathlib/Data/Complex/Basic.lean:722:theorem ofReal_zpow (r : ℝ) (n : ℤ) : ((r ^ n : ℝ) : ℂ) = (r : ℂ) ^ n := map_zpow₀ ofRealHom r n
Mathlib/Analysis/Complex/Basic.lean:282:  ⟨ofRealAm.toLinearMap, norm_real⟩
Mathlib/Analysis/Complex/Basic.lean:315:theorem ringHom_eq_ofReal_of_continuous {f : ℝ →+* ℂ} (h : Continuous f) : f = ofRealHom := by
Mathlib/Analysis/Complex/Basic.lean:325:theorem ofRealCLM_coe : (ofRealCLM : ℝ →ₗ[ℝ] ℂ) = ofRealAm.toLinearMap :=

$ /bin/bash -lc "rg -n \"norm_embedding_le_house.*ofReal|abs.*le_house|embedding_le_house\" Mathlib/NumberTheory/NumberField/House.lean Mathlib | sed -n '1,160p'"
Mathlib/NumberTheory/NumberField/House.lean:83:lemma norm_embedding_le_house (α : K) (σ : K →+* ℂ) : ‖σ α‖ ≤ house α := by
Mathlib/NumberTheory/NumberField/House.lean:93:  apply hσ.trans (norm_embedding_le_house α σ)
Mathlib/NumberTheory/NumberField/House.lean:102:       _ ≤ ‖σ' α‖ * ∏ τ ∈ univ.erase σ', house α := by gcongr; apply norm_embedding_le_house
Mathlib/NumberTheory/NumberField/House.lean:83:lemma norm_embedding_le_house (α : K) (σ : K →+* ℂ) : ‖σ α‖ ≤ house α := by
Mathlib/NumberTheory/NumberField/House.lean:93:  apply hσ.trans (norm_embedding_le_house α σ)
Mathlib/NumberTheory/NumberField/House.lean:102:       _ ≤ ‖σ' α‖ * ∏ τ ∈ univ.erase σ', house α := by gcongr; apply norm_embedding_le_house

$ /bin/bash -lc 'rg -n "norm_point|norm.*!₂|EuclideanSpace.*vecNotation|PiLp.inner_apply.*Fin 2" Mathlib | sed -n '"'1,100p'"

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=800000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "rg -n \"Rat.*norm|norm.*Rat|norm_eq_abs\" Mathlib/Data/Rat Mathlib/Analysis/Normed | sed -n '1,180p'"
Mathlib/Data/Rat/Init.lean:75:@[norm_cast] lemma cast_id (n : ℚ≥0) : NNRat.cast n = n := rfl
Mathlib/Data/Rat/Init.lean:82:@[norm_cast] lemma cast_id (n : ℚ) : Rat.cast n = n := rfl
Mathlib/Data/Rat/Defs.lean:109:  rw [mul_def, normalize_eq_mkRat]
Mathlib/Data/Rat/Defs.lean:133:@[simp] lemma divInt_one (n : ℤ) : n /. 1 = n := by simp [divInt, mkRat, normalize]
Mathlib/Analysis/Normed/Group/AddCircle.lean:50:  simp only [norm_eq_infDist, ← Real.norm_eq_abs, ← infDist_smul₀ ht, smul_zero]
Mathlib/Analysis/Normed/Group/AddCircle.lean:77:  simp only [le_antisymm_iff, le_norm_iff, Real.norm_eq_abs]
Mathlib/Analysis/Normed/Group/AddCircle.lean:146:  simp only [dist_eq_norm, mem_preimage, mem_closedBall, zsmul_eq_mul, mem_iUnion, Real.norm_eq_abs,
Mathlib/Analysis/Normed/Group/Real.lean:56:theorem norm_eq_abs (r : ℝ) : ‖r‖ = |r| :=
Mathlib/Analysis/Normed/Group/Real.lean:60:  ⟨fun _r _y => by rw [Real.dist_eq, ← abs_neg, neg_sub, add_comm, sub_eq_add_neg, norm_eq_abs]⟩
Mathlib/Analysis/Normed/Group/Real.lean:116:  rw [Real.norm_eq_abs r, abs_of_nonneg hr]
Mathlib/Analysis/Normed/Group/Rat.lean:30:  rw [← Rat.norm_cast_real, ← Int.norm_cast_real]; congr 1
Mathlib/Analysis/Normed/Field/Lemmas.lean:274:  norm_mul a b := by simp only [norm, Rat.cast_mul, abs_mul]
Mathlib/Analysis/Normed/Field/Lemmas.lean:279:    ⟨q, by rwa [← Rat.norm_cast_real, Real.norm_eq_abs, abs_of_pos (h₀.trans_lt h.1)]⟩
Mathlib/Analysis/Normed/Field/Basic.lean:30:  IsOfFinOrder Isometry.norm_map_of_map_one NNReal.isOpen_Ico_zero Rat.norm_cast_real
Mathlib/Analysis/Normed/Field/Basic.lean:308:    ⟨x, by rwa [Real.norm_eq_abs, abs_of_nonneg (h₀.trans h.1.le)]⟩
Mathlib/Analysis/Normed/Algebra/QuaternionExponential.lean:141:    Real.norm_eq_abs]
Mathlib/Analysis/Normed/Group/Int.lean:30:theorem norm_eq_abs (n : ℤ) : ‖n‖ = |(n : ℝ)| :=
Mathlib/Analysis/Normed/Group/Int.lean:33:theorem norm_natCast (n : ℕ) : ‖(n : ℤ)‖ = n := by simp [Int.norm_eq_abs]
Mathlib/Analysis/Normed/Group/Int.lean:40:      _ = ‖n‖ := 

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=800000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=800000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "lake env lean -DmaxHeartbeats=800000 SharpDistanceScratch.lean 2>&1 | sed -n '1,20p'"
SharpDistanceScratch.lean:12:7: error(lean.unknownIdentifier): Unknown identifier `abs_eq_norm`
SharpDistanceScratch.lean:13:7: error(lean.unknownIdentifier): Unknown identifier `norm_eq_abs`
Rat.norm_cast_real (r : ℚ) : ‖↑r‖ = ‖r‖
SharpDistanceScratch.lean:41:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:117:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:113:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:

$ /bin/bash -lc "sed -n '1,48p' Mathlib/Analysis/Normed/Group/Rat.lean; rg -n \"instance.*Norm ℚ|def norm.*ℚ|Rat.instNorm\" Mathlib/Analysis/Normed Mathlib/Data/Rat | head -50"
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
module

public import Mathlib.Analysis.Normed.Group.Int
public import Mathlib.Topology.Instances.Rat

/-! # ℚ as a normed group -/

public section

namespace Rat

instance instNormedAddCommGroup : NormedAddCommGroup ℚ where
  norm r := ‖(r : ℝ)‖
  dist_eq r₁ r₂ := by
    simp only [dist_eq, norm, cast_add, cast_neg]
    rw [← abs_neg, neg_sub]
    abel_nf

@[norm_cast, simp high] -- increase priority to prevent the left-hand side from simplifying
theorem norm_cast_real (r : ℚ) : ‖(r : ℝ)‖ = ‖r‖ :=
  rfl

@[norm_cast, simp]
theorem _root_.Int.norm_cast_rat (m : ℤ) : ‖(m : ℚ)‖ = ‖m‖ := by
  rw [← Rat.norm_cast_real, ← Int.norm_cast_real]; congr 1

end Rat
Mathlib/Analysis/Normed/Order/Lattice.lean:62:instance : HasSolidNorm ℚ := ⟨fun _ _ _ => by simpa only [norm, ← Rat.cast_abs, Rat.cast_le]⟩
Mathlib/Analysis/Normed/Field/Lemmas.lean:271:instance Rat.instNormedField : NormedField ℚ where

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=800000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=800000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1200000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "nl -ba SharpDistanceScratch.lean | sed -n '580,705p'"
580	  ext i
   581	  fin_cases i <;> simp
   582	
   583	def yScale (m : ℕ) : ℝ :=
   584	  2 * yConstant K * (m + 1)
   585	
   586	def xScale (B : TraceBasisData K) (m : ℕ) : ℝ :=
   587	  (aConstant B + 2 * yConstant K ^ 2) * (m + 1) ^ 2
   588	
   589	lemma yScale_pos (m : ℕ) : 0 < yScale (K := K) m := by
   590	  unfold yScale
   591	  positivity
   592	
   593	lemma xScale_pos (B : TraceBasisData K) (m : ℕ) : 0 < xScale B m := by
   594	  unfold xScale
   595	  apply mul_pos
   596	  · exact add_pos_of_pos_of_nonneg (aConstant_pos B)
   597	      (mul_nonneg (by norm_num) (sq_nonneg _))
   598	  · positivity
   599	
   600	def scaledPoint (σ : K →+* ℝ) (B : TraceBasisData K) {m : ℕ}
   601	    (s : ArithmeticSource B m) : Point :=
   602	  point
   603	    ((σ (sourceY s : K) + yScale (K := K) m) / (2 * yScale (K := K) m))
   604	    ((σ (sourceX B s : K) + xScale B m) / (2 * xScale B m))
   605	
   606	def scaledDirection (σ : K →+* ℝ) (B : TraceBasisData K) {m : ℕ}
   607	    (s : ArithmeticSource B m) : Point :=
   608	  point (xScale B m) (yScale (K := K) m * σ (sourceY s : K))
   609	
   610	def scaledNormal (σ : K →+* ℝ) (B : TraceBasisData K) {m : ℕ}
   611	    (s : ArithmeticSource B m) : Point :=
   612	  point (-(yScale (K := K) m * σ (sourceY s : K))) (xScale B m)
   613	
   614	def scaledLine (σ : K →+* ℝ) (B : TraceBasisData K) {m : ℕ}
   615	    (s : ArithmeticSource B m) : RealLine where
   616	  base := scaledPoint σ B s
   617	  direction := scaledDirection σ B s
   618	  direction_ne_zero := by
   619	    intro h
   620	    have h0 := congrArg (fun p : Point ↦ p 0) h
   621	    simp [scaledDirection] at h0
   622	    exact (xScale_pos B m).ne' h0
   623	
   624	lemma scaledPoint_mem_unitSquare (σ : K →+* ℝ) (B : TraceBasisData K)
   625	    {m : ℕ} (s : ArithmeticSource B m) :
   626	    InUnitSquare (scaledPoint σ B s) := by
   627	  have hy0 : |σ (sourceY s : K)| ≤ yScale (K := K) m := by
   628	    calc
   629	      _ ≤ NumberField.house (

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1200000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1200000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1200000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "rg -n \"Nat.ceil.*rpow|ceil.*rpow|Nat.ceil_le|Nat.le_ceil|ceil_lt_add_one|Nat.ceil_lt\" Mathlib/Algebra/Order Mathlib/Data/Nat Mathlib/Analysis | sed -n '1,260p'"
Mathlib/Algebra/Order/Archimedean/Real/Basic.lean:87:    replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)
Mathlib/Algebra/Order/Archimedean/Real/Basic.lean:88:    replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)
Mathlib/Algebra/Order/Archimedean/Basic.lean:341:    let ⟨q, h⟩ := H x; ⟨⌈q⌉₊, lt_of_lt_of_le h <| mod_cast Nat.le_ceil _⟩
Mathlib/Algebra/Order/Floor/Semiring.lean:357:theorem ceil_lt_add_one (ha : 0 ≤ a) : (⌈a⌉₊ : R) < a + 1 :=
Mathlib/Algebra/Order/Floor/Semiring.lean:369:  · rwa [ceil_eq_zero.2 (tsub_nonpos_of_le han), eq_comm, tsub_eq_zero_iff_le, Nat.ceil_le]
Mathlib/Algebra/Order/Floor/Semiring.lean:405:  · simpa using (ceil_lt_add_one ha).le
Mathlib/Algebra/Order/Floor/Semifield.lean:71:      ⌈a⌉₊ < a + 1 := ceil_lt_add_one <| hba.trans' <| by positivity
Mathlib/Algebra/Order/Floor/Ring.lean:681:theorem ceil_lt_add_one (a : R) : (⌈a⌉ : R) < a + 1 := by
Mathlib/Algebra/Order/Floor/Ring.lean:693:  refine (ceil_lt_add_one _).le.trans ?_
Mathlib/Algebra/Order/Floor/Ring.lean:766:  convert ceil_lt_add_one (R := k) _
Mathlib/Algebra/Order/Floor/Ring.lean:776:      ⌈a⌉ < a + 1 := ceil_lt_add_one _
Mathlib/Algebra/Order/Floor/Ring.lean:868:/-- a variant of `Nat.ceil_lt_add_one` with its condition `0 ≤ a` generalized to `-1 < a` -/
Mathlib/Algebra/Order/Floor/Ring.lean:870:lemma ceil_lt_add_one_of_gt_neg_one (ha : -1 < a) : ⌈a⌉₊ < a + 1 := by
Mathlib/Algebra/Order/Floor/Ring.lean:872:  · exact ceil_lt_add_one h
Mathlib/Algebra/Order/Floor/Extended.lean:137:  | (r : ℝ≥0) => by simpa using mod_cast Nat.ceil_le_floor_add_one r
Mathlib/Algebra/Order/Floor/Extended.lean:211:lemma ceil_lt_add_one (hr : r ≠ ∞) : (⌈r⌉ₑ : ℝ≥0∞) < r + 1 := by
Mathlib/Algebra/Order/Floor/Extended.lean:212:  lift r to ℝ≥0 using hr; simpa using mod_cast Nat.ceil_lt_add_one zero_le
Mathlib/Analysis/BoxIntegral/Integrability.lean:140:    exact hJ.2 ▸ Nat.le_ceil _
Mathlib/Analysis/BoxIntegral/UnitPartition.lean:86:    le_trans (Metric.ball_subset_ball (Nat.le_ceil R)) this

$ /bin/bash -lc "rg -n \"card_le_iff.*Embedding|Fintype.card_le_iff|exists_embedding.*card|Embedding.*card_le\" Mathlib/Data/Fintype Mathlib | sed -n '1,200p'"
Mathlib/Data/Set/Card.lean:513:theorem _root_.Function.Embedding.encard_le (e : s ↪ t) : s.encard ≤ t.encard :=
Mathlib/Data/Set/Card.lean:542:  (Function.Embedding.subtype _).encard_le
Mathlib/Data/Set/Card.lean:547:  Function.Embedding.encard_le ⟨fun ⟨⟨x, _⟩, hx⟩ ↦ ⟨x, hx⟩, fun _ _ h ↦ by
Mathlib/GroupTheory/GroupAction/Jordan.lean:325:      rwa [← not_nonempty_iff, Function.Embedding.nonempty_iff_card_le, Fintype.card_fin,
Mathlib/Combinatorics/SimpleGraph/CompleteMultipartite.lean:320:    rw [Embedding.nonempty_iff_card_le, Fintype.card_fin]
Mathlib/Combinatorics/SimpleGraph/CompleteMultipartite.lean:396:      rw [Embedding.nonempty_iff_card_le,
Mathlib/Combinatorics/SimpleGraph/CompleteMultipartite.lean:400:      rw [Embedding.nonempty_iff_card_le, Fintype.card_fin, card_coe, K.card_mem_parts p.prop]
Mathlib/Combinatorics/SimpleGraph/VertexCover.lean:191:  exact Function.Embedding.encard_le <| Function.Embedding.mk f hf |>.subtypeMap (by simp)
Mathlib/Combinatorics/SimpleGraph/Bipartite.lean:334:    exact Function.Embedding.nonempty_of_card_le card_left.symm.le
Mathlib/Combinatorics/SimpleGraph/Bipartite.lean:338:    exact Function.Embedding.nonempty_of_card_le card_right.symm.le
Mathlib/Combinatorics/SimpleGraph/Bipartite.lean:509:  grw [hG.edgeSetEmbeddingCompleteBipartiteGraph.encard_le]
Mathlib/Combinatorics/SimpleGraph/Coloring/Vertex.lean:275:  G.recolorOfEmbedding <| (Function.Embedding.nonempty_of_card_le hn).some
Mathlib/Combinatorics/SimpleGraph/Coloring/Vertex.lean:280:      (Embedding.completeGraph (Embedding.nonempty_of_card_le hαβ).some).toHom.comp := rfl
Mathlib/Combinatorics/SimpleGraph/Copy.lean:299:    fun h ↦ ⟨Copy.bot (Function.Embedding.nonempty_of_card_le h).some⟩⟩
Mathlib/SetTheory/Cardinal/Embedding.lean:17:* `Fin.Embedding.exists_embedding_disjoint_range_of_add_le_ENat_card`
Mathlib/SetTheory/Cardinal/Embedding.lean:22:* `Fin.Embedding.exists_embedding_disjoint_range_of_add_le_Nat_card`
Mathlib/SetTheory/Cardinal/Embedding.lean:44:theor

$ /bin/bash -lc "rg -n \"rpow_nat_inv_pow|rpow_inv_natCast|rpow_one_div_nat|rpow_mul|rpow_natCast|Nat.cast.*ceil\" Mathlib/Analysis/SpecialFunctions/Pow Mathlib | sed -n '1,260p'"
Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean:329:  rwa [rpow_natCast]
Mathlib/NumberTheory/SiegelsLemma.lean:169:        rw [← rpow_mul (mul_nonneg (Nat.cast_nonneg' n) (le_trans zero_le_one (le_max_left ..))),
Mathlib/NumberTheory/SiegelsLemma.lean:170:          ← Real.rpow_natCast, div_mul_cancel₀]
Mathlib/NumberTheory/SiegelsLemma.lean:177:        rw [← rpow_natCast, ← rpow_add (Nat.cast_add_one_pos B), ← rpow_natCast, sub_add_cancel]
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean:155:    rw [convexBodySum_volume, ← ENNReal.ofReal_pow (by positivity), ← Real.rpow_natCast,
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean:156:      ← Real.rpow_mul toReal_nonneg, div_mul_cancel₀, Real.rpow_one, ofReal_toReal, mul_comm,
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean:162:  rw [div_pow B, ← Real.rpow_natCast B, ← Real.rpow_mul (by positivity), div_mul_cancel₀ _
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean:235:    (by positivity) (by positivity), ← Real.rpow_mul (by positivity), inv_mul_cancel₀ h.ne',
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean:236:    Real.rpow_one, Real.mul_rpow (by positivity) (by positivity), Real.rpow_natCast,
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean:237:    Real.rpow_natCast, ← pow_mul, ← Real.rpow_mul (by positivity),
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean:357:      rw [← Real.rpow_natCast]
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean:129:      ← NNReal.rpow_natCast, ← NNReal.rpow_mul, inv_mul_cancel₀, NNReal.rpow_one, mul_assoc,
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean:600:  rw [← rpow_natCast, ← rpow_le_rpow_iff (by simp only [Rat.cast_abs, abs_nonneg])
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean:601:      (rpow_nonneg h2 _) h1, ← rpow_mul h2, mul_inv_cancel₀ (Nat.cast_ne_zero.mpr
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean:607:  · simp_rw [← 

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1200000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1200000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "nl -ba SharpDistanceScratch.lean | sed -n '804,860p'"
804	def constructionConstant (B : TraceBasisData K) : ℝ :=
   805	  8 * (aConstant B + 6 * yConstant K ^ 2) *
   806	    (2 * aConstant B + 12 * yConstant K ^ 2) ^
   807	      (Module.finrank ℚ K - 1)
   808	
   809	lemma constructionConstant_pos (B : TraceBasisData K) :
   810	    0 < constructionConstant B := by
   811	  unfold constructionConstant
   812	  positivity
   813	
   814	lemma construction_denominator_le (B : TraceBasisData K) {m : ℕ}
   815	    (hm : 0 < m) (hd : 0 < Module.finrank ℚ K) :
   816	    2 * ((aConstant B + 6 * yConstant K ^ 2) * (m + 1) ^ 2) *
   817	        (((2 * aConstant B + 12 * yConstant K ^ 2) * m ^ 2) ^
   818	          (Module.finrank ℚ K - 1)) ≤
   819	      constructionConstant B * (m : ℝ) ^ (2 * Module.finrank ℚ K) := by
   820	  let d := Module.finrank ℚ K
   821	  let G := aConstant B + 6 * yConstant K ^ 2
   822	  let H := 2 * aConstant B + 12 * yConstant K ^ 2
   823	  have hG : 0 ≤ G := by
   824	    dsimp [G]
   825	    positivity
   826	  have hH : 0 ≤ H := by
   827	    dsimp [H]
   828	    positivity
   829	  have hm1 : ((m + 1 : ℕ) : ℝ) ≤ 2 * m := by
   830	    norm_num
   831	    exact_mod_cast (show m + 1 ≤ 2 * m by omega)
   832	  calc
   833	    2 * (G * (m + 1) ^ 2) * (H * m ^ 2) ^ (d - 1) ≤
   834	        2 * (G * (2 * m) ^ 2) * (H * m ^ 2) ^ (d - 1) := by
   835	      gcongr
   836	    _ = constructionConstant B * (m : ℝ) ^ (2 * d) := by
   837	      have hexp : 2 + 2 * (d - 1) = 2 * d := by omega
   838	      simp only [mul_pow]
   839	      rw [show (m : ℝ) ^ 2 * (m ^ 2) ^ (d - 1) =
   840	          m ^ (2 * d) by
   841	        rw [← pow_mul, ← pow_add, hexp]]
   842	      simp [constructionConstant, G, H]
   843	      ring
   844	
   845	lemma selectedConfiguration_simple_separated (σ : K →+* ℝ)
   846	    (B : TraceBasisData K) {m n : ℕ} (hm : 0 < m)
   847	    (hd : 0 < Module.finrank ℚ K)
   848	    (hcard : n ≤ Fintype.card (ArithmeticSource B m)) :
   849	    (selectedConfiguration σ B m n hcard).Sep

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc "sed -n '800,900p' SharpDistanceScratch.lean && lake env lean -DmaxHeartbeats=1500000 SharpDistanceScratch.lean"
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistanceScratch.lean >/tmp/sharp.log 2>&1; code=$?; tail -20 /tmp/sharp.log; exit $code'
Note: This linter can be disabled with `set_option linter.unreachableTactic false`
SharpDistanceScratch.lean:791:13: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
SharpDistanceScratch.lean:792:11: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
Try this:
  [apply] ring_nf
  
  The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.
    
  Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead.

$ /bin/bash -lc 'rg "tendsto_rpow_atTop|ceil_lt_add_one|rpow_inv_natCast|rpow_nat_inv_pow|natCast.*rpow|pow_rpow_inv" Mathlib | head -80'
Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean:      ((tendsto_rpow_atTop (sub_pos.2 hlt)).comp tendsto_natCast_atTop_atTop).eventually
Mathlib/NumberTheory/Transcendental/Liouville/Measure.lean:    rw [Real.volume_ball, mul_one_div, ← NNReal.coe_two, ← NNReal.coe_natCast, ← NNReal.coe_rpow,
Mathlib/NumberTheory/SiegelsLemma.lean:        rw [← rpow_natCast, ← rpow_add (Nat.cast_add_one_pos B), ← rpow_natCast, sub_add_cancel]
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean:  rw [div_pow B, ← Real.rpow_natCast B, ← Real.rpow_mul (by positivity), div_mul_cancel₀ _
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean:    Real.rpow_natCast, ← pow_mul, ← Real.rpow_mul (by positivity),
Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean:      log_mul (pow_ne_zero _ q_pos'.ne') hε.ne', ← rpow_natCast, ← rpow_natCast, log_rpow q_pos',
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean:      ← NNReal.rpow_natCast, ← NNReal.rpow_mul, inv_mul_cancel₀, NNReal.rpow_one, mul_assoc,
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean:  rw [← rpow_natCast, ← rpow_le_rpow_iff (by simp only [Rat.cast_abs, abs_nonneg])
Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean:    ← Nat.cast_ofNat, rpow_natCast, Nat.cast_ofNat, ← Nat.cast_ofNat (n := 60), rpow_natCast]
Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean:    refine tendsto_const_nhds.div_atTop <| (tendsto_rpow_atTop (by linarith)).comp ?_
Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean:          rw [← ofReal_natCast n, norm_cpow_eq_rpow_re_of_nonneg (by simp) (by simp; grind)]
Mathlib/NumberTheory/Ostrowski.lean:      simp only [← rpow_natCast, p.cast_nonneg, ← rpow_mul, neg_mul, mul_one, ← rpow_neg, hpm,
Mathlib/NumberTheory/Ostrowski.lean:      rw [Nat.cast_pow, map_pow, ← rpow_natCast, rpow_rpow_inv (by positivity) (by positivity)]
Mathlib/NumberTheory/Ostrowski.lean:      · rw [← rpow_natCast, rpow_le_rpow_left_iff (one_lt_of_not_bounded notbdd hm)]

$ /bin/bash -lc "cat > Query.lean <<'EOF'
import SharpDistanceScratch
#check tendsto_rpow_atTop
#check Filter.eventually_atTop.1
#check Filter.eventually_atTop.2
#check Metric.tendsto_atTop
#check Nat.le_ceil
#check Nat.ceil_lt_add_one
#check Real.rpow_le_rpow
#check Real.rpow_le_rpow_of_exponent_nonneg
#check Real.rpow_natCast
#check Real.rpow_mul
#check Real.rpow_add
#check Real.rpow_inv_rpow
#check Real.rpow_rpow_inv
#check Real.rpow_sub
#check Real.rpow_neg
#check Real.one_le_rpow
#check Real.rpow_le_rpow_of_exponent_le
#check Real.strictMono_rpow_of_base_gt_one
#check Real.rpow_pos_of_pos
#check Nat.cast_pow
#check Nat.cast_le
#check pow_le_pow_left₀
#check Real.rpow_nat_inv_pow_nat
#check Real.pow_rpow_inv_natCast
#check pow_rpow_inv_natCast
#check Real.rpow_inv_natCast_pow
#check Real.rpow_natCast
#check Real.rpow_mul_natCast
#check Real.rpow_one
#check Real.rpow_nonneg
#check Real.rpow_pos_of_pos
#check Real.rpow_mul
#check Nat.one_le_ceil
#check Nat.ceil_pos
#check Nat.ceil_le
EOF
lake env lean Query.lean 2>/dev/null"
Query.lean:1:0: error: unknown module prefix 'SharpDistanceScratch'

No directory 'SharpDistanceScratch' or file 'SharpDistanceScratch.olean' in the search path entries:
/work/repro/.lake/packages/Cli/.lake/build/lib/lean
/work/repro/.lake/packages/batteries/.lake/build/lib/lean
/work/repro/.lake/packages/Qq/.lake/build/lib/lean
/work/repro/.lake/packages/aesop/.lake/build/lib/lean
/work/repro/.lake/packages/proofwidgets/.lake/build/lib/lean
/work/repro/.lake/packages/importGraph/.lake/build/lib/lean
/work/repro/.lake/packages/LeanSearchClient/.lake/build/lib/lean
/work/repro/.lake/packages/plausible/.lake/build/lib/lean
/work/repro/.lake/build/lib/lean
/opt/elan/toolchains/leanprover--lean4---v4.33.0-rc1/lib/lean
/opt/elan/toolchains/leanprover--lean4---v4.33.0-rc1/lib/lean

$ /bin/bash -lc 'lake env lean Query.lean 2>/dev/null'
tendsto_rpow_atTop {y : ℝ} (hy : 0 < y) : Filter.Tendsto (fun x => x ^ y) Filter.atTop Filter.atTop
Filter.eventually_atTop.mp : (∀ᶠ (x : ?m.1) in Filter.atTop, ?m.4 x) → ∃ a, ∀ (b : ?m.1), a ≤ b → ?m.4 b
Filter.eventually_atTop.mpr : (∃ a, ∀ (b : ?m.1), a ≤ b → ?m.4 b) → ∀ᶠ (x : ?m.1) in Filter.atTop, ?m.4 x
Metric.tendsto_atTop.{u, v} {α : Type u} {β : Type v} [PseudoMetricSpace α] [Nonempty β] [SemilatticeSup β] {u : β → α}
  {a : α} : Filter.Tendsto u Filter.atTop (nhds a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε
Nat.le_ceil.{u_1} {R : Type u_1} [Semiring R] [LinearOrder R] [FloorSemiring R] (a : R) : a ≤ ↑⌈a⌉₊
Nat.ceil_lt_add_one.{u_1} {R : Type u_1} [Semiring R] [LinearOrder R] [FloorSemiring R] {a : R} [IsStrictOrderedRing R]
  (ha : 0 ≤ a) : ↑⌈a⌉₊ < a + 1
Real.rpow_le_rpow {x y z : ℝ} (h : 0 ≤ x) (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x ^ z ≤ y ^ z
Query.lean:14:7: error(lean.unknownIdentifier): Unknown constant `Real.rpow_le_rpow_of_exponent_nonneg`
Real.rpow_natCast (x : ℝ) (n : ℕ) : x ^ ↑n = x ^ n
Real.rpow_mul {x : ℝ} (hx : 0 ≤ x) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z
Real.rpow_add {x : ℝ} (hx : 0 < x) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z
Real.rpow_inv_rpow {x y : ℝ} (hx : 0 ≤ x) (hy : y ≠ 0) : (x ^ y⁻¹) ^ y = x
Real.rpow_rpow_inv {x y : ℝ} (hx : 0 ≤ x) (hy : y ≠ 0) : (x ^ y) ^ y⁻¹ = x
Real.rpow_sub {x : ℝ} (hx : 0 < x) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z
Real.rpow_neg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹
Real.one_le_rpow {x z : ℝ} (hx : 1 ≤ x) (hz : 0 ≤ z) : 1 ≤ x ^ z
Real.rpow_le_rpow_of_exponent_le {x y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z) : x ^ y ≤ x ^ z
Real.strictMono_rpow_of_base_gt_one {b : ℝ} (hb : 1 < b) : StrictMono fun x => b ^ x
Real.rpow_pos_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : 0 < x ^ y
Nat.cast_pow.{u_1} {α : Type u_1} [Semiring α] (m n : ℕ) : ↑(m ^ n) = ↑m ^ n
Nat.cast_le.{u_1} {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [AddLeftMono α] [ZeroLEOneClass α] [CharZero α]
  {m n : ℕ} : ↑m ≤ ↑n ↔ m ≤ n
pow_le_pow_left₀.{u_2

$ /bin/bash -lc "cat > Query2.lean <<'EOF'
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#check Filter.tendsto_atTop
#check Filter.tendsto_atTop.1
#check Filter.tendsto_atTop.2
#check tendsto_natCast_atTop_atTop
#check Nat.ceil_le
#check Nat.le_ceil
#check Nat.cast_le
#check Nat.cast_pos
#check Real.rpow_mul_natCast
#check Real.rpow_natCast
#check Real.rpow_add
#check Real.rpow_sub
#check Real.rpow_neg
#check inv_le_inv₀
#check Real.rpow_le_rpow_of_exponent_le
#check Real.rpow_inv_natCast_pow
EOF
lake env lean Query2.lean 2>/dev/null"
Filter.tendsto_atTop.{u_3, u_4} {α : Type u_3} {β : Type u_4} [Preorder β] {m : α → β} {f : Filter α} :
  Filter.Tendsto m f Filter.atTop ↔ ∀ (b : β), ∀ᶠ (a : α) in f, b ≤ m a
Filter.tendsto_atTop.mp : Filter.Tendsto ?m.4 ?m.5 Filter.atTop → ∀ (b : ?m.2), ∀ᶠ (a : ?m.1) in ?m.5, b ≤ ?m.4 a
Filter.tendsto_atTop.mpr : (∀ (b : ?m.2), ∀ᶠ (a : ?m.1) in ?m.5, b ≤ ?m.4 a) → Filter.Tendsto ?m.4 ?m.5 Filter.atTop
tendsto_natCast_atTop_atTop.{u_2} {R : Type u_2} [Semiring R] [PartialOrder R] [IsOrderedRing R] [Archimedean R] :
  Filter.Tendsto Nat.cast Filter.atTop Filter.atTop
Nat.ceil_le.{u_2} {α : Type u_2} [Semiring α] [PartialOrder α] [FloorSemiring α] {a : α} {n : ℕ} : ⌈a⌉₊ ≤ n ↔ a ≤ ↑n
Nat.le_ceil.{u_1} {R : Type u_1} [Semiring R] [LinearOrder R] [FloorSemiring R] (a : R) : a ≤ ↑⌈a⌉₊
Nat.cast_le.{u_1} {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [AddLeftMono α] [ZeroLEOneClass α] [CharZero α]
  {m n : ℕ} : ↑m ≤ ↑n ↔ m ≤ n
Nat.cast_pos.{u_3} {α : Type u_3} [Semiring α] [PartialOrder α] [IsOrderedRing α] [Nontrivial α] {n : ℕ} :
  0 < ↑n ↔ 0 < n
Real.rpow_mul_natCast {x : ℝ} (hx : 0 ≤ x) (y : ℝ) (n : ℕ) : x ^ (y * ↑n) = (x ^ y) ^ n
Real.rpow_natCast (x : ℝ) (n : ℕ) : x ^ ↑n = x ^ n
Real.rpow_add {x : ℝ} (hx : 0 < x) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z
Real.rpow_sub {x : ℝ} (hx : 0 < x) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z
Real.rpow_neg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹
inv_le_inv₀.{u_3} {G₀ : Type u_3} [GroupWithZero G₀] [PartialOrder G₀] [PosMulReflectLT G₀] [MulPosReflectLT G₀]
  {a b : G₀} (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a
Real.rpow_le_rpow_of_exponent_le {x y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z) : x ^ y ≤ x ^ z
Real.rpow_inv_natCast_pow {x : ℝ} {n : ℕ} (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ (↑n)⁻¹) ^ n = x

$ /bin/bash -lc "sed -n '730,815p' SharpDistanceScratch.lean"
have hP : 0 < P := pow_pos hH _
  have hNG : N ≤ G := scaledNormal_norm_le σ B s
  have hprod : 1 ≤ A * P := sourceD_norm_product_lower σ B hst
  have hPinvA : 1 / P ≤ A := by
    apply (div_le_iff₀ hP).2
    simpa [A, P, H, mul_comm] using hprod
  have hgeom :
      A / (2 * N) ≤ pointLineDist (scaledPoint σ B t) (scaledLine σ B s) := by
    have h := pointLineDist_ge_inner_div_norm
      (scaledPoint σ B t) (scaledLine σ B s) (scaledNormal σ B s)
      (scaledNormal_ne_zero σ B s) (scaled_orthogonal σ B s)
    change
      |⟪scaledPoint σ B t - scaledPoint σ B s, scaledNormal σ B s⟫_ℝ| /
        ‖scaledNormal σ B s‖ ≤
          pointLineDist (scaledPoint σ B t) (scaledLine σ B s) at h
    rw [scaled_inner_eq] at h
    simpa [A, N, abs_div, div_div] using h
  calc
    (2 * G * P)⁻¹ = (1 / P) / (2 * G) := by
      field_simp [hG.ne', hP.ne']
      <;> ring
    _ ≤ A / (2 * G) := by
      exact div_le_div_of_nonneg_right hPinvA (by positivity)
    _ ≤ A / (2 * N) := by
      exact div_le_div_of_nonneg_left hA (by positivity) (by linarith)
    _ ≤ pointLineDist (scaledPoint σ B t) (scaledLine σ B s) := hgeom

structure PLConfiguration (n : ℕ) where
  points : Fin n → Point
  lines : Fin n → RealLine
  points_mem : ∀ i, InUnitSquare (points i)
  incident : ∀ i, points i ∈ (lines i).carrier

def PLConfiguration.Separated {n : ℕ} (C : PLConfiguration n) (δ : ℝ) : Prop :=
  ∀ i j, i ≠ j → δ ≤ pointLineDist (C.points i) (C.lines j)

lemma card_arithmeticSource_lower (B : TraceBasisData K) (m : ℕ)
    (hd : 0 < Module.finrank ℚ K) :
    m ^ (3 * Module.finrank ℚ K - 2) ≤
      Fintype.card (ArithmeticSource B m) := by
  let d := Module.finrank ℚ K
  rw [card_arithmeticSource]
  calc
    m ^ (3 * d - 2) = (m ^ 2) ^ (d - 1) * m ^ d := by
      rw [← pow_mul, ← pow_add]
      congr 1
      omega
    _ ≤ (m ^ 2 + 1) ^ (d - 1) * (m + 1) ^ d := by
      gcongr <;> omega

def sourceEmbedding (B : TraceBasisData K) (m n : ℕ)
    (hcard : n ≤ Fintype.card (ArithmeticSource B m)) :
 

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'rg -n "maximalRealSubfield|isCMField|finrank.*maximal|finrank.*two|Cyclotomic.*finrank|finrank.*totient|isTotallyReal" Mathlib/NumberTheory/NumberField/CMField.lean Mathlib/NumberTheory/NumberField/Cyclotomic | head -160'
Mathlib/NumberTheory/NumberField/Cyclotomic/Ideal.lean:383:    rw [finrank m, finrank (p ^ (k + 1)), finrank n, hn, mul_comm, Nat.totient_mul]
Mathlib/NumberTheory/NumberField/CMField.lean:44:* `IsCyclotomicExtension.Rat.isCMField`: A nontrivial cyclotomic extension of `ℚ` is CM.
Mathlib/NumberTheory/NumberField/CMField.lean:65:section maximalRealSubfield
Mathlib/NumberTheory/NumberField/CMField.lean:73:  [is_quadratic : IsQuadraticExtension (maximalRealSubfield K) K]
Mathlib/NumberTheory/NumberField/CMField.lean:81:local notation3 "K⁺" => maximalRealSubfield K
Mathlib/NumberTheory/NumberField/CMField.lean:93:    add_zero, ← IsTotallyReal.finrank, ← Nat.mul_left_cancel_iff zero_lt_two,
Mathlib/NumberTheory/NumberField/CMField.lean:95:    IsQuadraticExtension.finrank_eq_two _ K]
Mathlib/NumberTheory/NumberField/CMField.lean:132:    (IsQuadraticExtension.finrank_eq_two K⁺ K) ▸ IsGalois.card_aut_eq_finrank K⁺ K
Mathlib/NumberTheory/NumberField/CMField.lean:193:    IsQuadraticExtension.finrank_eq_two]
Mathlib/NumberTheory/NumberField/CMField.lean:451:end maximalRealSubfield
Mathlib/NumberTheory/NumberField/CMField.lean:458:theorem eq_maximalRealSubfield (E : Subfield K) [IsTotallyReal E] [IsQuadraticExtension E K] :
Mathlib/NumberTheory/NumberField/CMField.lean:459:    E = maximalRealSubfield K := by
Mathlib/NumberTheory/NumberField/CMField.lean:460:  refine le_antisymm (IsTotallyReal.le_maximalRealSubfield E) ?_
Mathlib/NumberTheory/NumberField/CMField.lean:462:  have h' : E ⊔ (maximalRealSubfield K) = ⊤ := by
Mathlib/NumberTheory/NumberField/CMField.lean:463:    let L : IntermediateField E K := (E ⊔ (maximalRealSubfield K)).toIntermediateField
Mathlib/NumberTheory/NumberField/CMField.lean:466:      (IsQuadraticExtension.finrank_eq_two E K ▸ Nat.prime_two)).eq_bot_or_eq_top L).resolve_left ?_
Mathlib/NumberTheory/NumberField/CMField.lean:472:  have : Algebra.IsAlgebraic (maximalRealSubfield K) K :=
Mathlib/NumberTheory/NumberField/CMField.lean:473:    Algebra.IsAlgebra

$ /bin/bash -lc "sed -n '60,110p' Mathlib/NumberTheory/NumberField/CMField.lean; sed -n '540,590p' Mathlib/NumberTheory/NumberField/CMField.lean; sed -n '1,60p' Mathlib/NumberTheory/NumberField/Cyclotomic/Basic.lean"
open scoped ComplexConjugate

namespace NumberField

section maximalRealSubfield

/--
A field `K` is `CM` if `K` is a totally complex quadratic extension of its maximal
real subfield `K⁺`.
-/
class IsCMField (K : Type*) [Field K] [CharZero K] : Prop where
  [to_isTotallyComplex : IsTotallyComplex K]
  [is_quadratic : IsQuadraticExtension (maximalRealSubfield K) K]

namespace IsCMField

open ComplexEmbedding

variable (K : Type*) [Field K] [CharZero K] [IsCMField K]

local notation3 "K⁺" => maximalRealSubfield K

instance isQuadraticExtension : IsQuadraticExtension K⁺ K :=
  IsCMField.is_quadratic

instance isTotallyComplex : IsTotallyComplex K :=
  IsCMField.to_isTotallyComplex

theorem card_infinitePlace_eq_card_infinitePlace [NumberField K] :
    Fintype.card (InfinitePlace K⁺) = Fintype.card (InfinitePlace K) := by
  rw [card_eq_nrRealPlaces_add_nrComplexPlaces, card_eq_nrRealPlaces_add_nrComplexPlaces,
    IsTotallyComplex.nrRealPlaces_eq_zero K, IsTotallyReal.nrComplexPlaces_eq_zero, zero_add,
    add_zero, ← IsTotallyReal.finrank, ← Nat.mul_left_cancel_iff zero_lt_two,
    ← IsTotallyComplex.finrank, ← Module.finrank_mul_finrank ℚ K⁺ K, mul_comm,
    IsQuadraticExtension.finrank_eq_two _ K]

/--
The equiv between the infinite places of `K` and the infinite places of `K⁺` induced by the
restriction to `K⁺`, see `equivInfinitePlace_apply`.
-/
noncomputable def equivInfinitePlace [NumberField K] : InfinitePlace K ≃ InfinitePlace K⁺ :=
  Equiv.ofBijective (fun w ↦ w.comap (algebraMap K⁺ K)) <|
    (Fintype.bijective_iff_surjective_and_card _).mpr
      ⟨comap_surjective, (card_infinitePlace_eq_card_infinitePlace K).symm⟩

@[simp]
theorem equivInfinitePlace_apply [NumberField K] (w : InfinitePlace K) :
    equivInfinitePlace K w = w.comap (algebraMap K⁺ K) := rfl

@[simp]
open scoped IsMulCommutative in
/--
A totally complex abelian extension of `ℚ` is CM.
-/
instance of_isAbelianGalois [IsAbelianGalois ℚ K] :
    IsCMField K := by
  let φ : K →+* ℂ := Classical.ch

$ /bin/bash -lc "rg -n \"maximalRealSubfield\" Mathlib/NumberTheory/NumberField/Basic.lean Mathlib/NumberTheory/NumberField/Embeddings.lean Mathlib/NumberTheory/NumberField/*.lean | head -100; rg -n \"instance.*NumberField.*IntermediateField|NumberField.*Subfield|numberField\" Mathlib/NumberTheory/NumberField -g '*.lean' | head -100"
rg: Mathlib/NumberTheory/NumberField/Embeddings.lean: No such file or directory (os error 2)
Mathlib/NumberTheory/NumberField/CMField.lean:65:section maximalRealSubfield
Mathlib/NumberTheory/NumberField/CMField.lean:73:  [is_quadratic : IsQuadraticExtension (maximalRealSubfield K) K]
Mathlib/NumberTheory/NumberField/CMField.lean:81:local notation3 "K⁺" => maximalRealSubfield K
Mathlib/NumberTheory/NumberField/CMField.lean:451:end maximalRealSubfield
Mathlib/NumberTheory/NumberField/CMField.lean:458:theorem eq_maximalRealSubfield (E : Subfield K) [IsTotallyReal E] [IsQuadraticExtension E K] :
Mathlib/NumberTheory/NumberField/CMField.lean:459:    E = maximalRealSubfield K := by
Mathlib/NumberTheory/NumberField/CMField.lean:460:  refine le_antisymm (IsTotallyReal.le_maximalRealSubfield E) ?_
Mathlib/NumberTheory/NumberField/CMField.lean:462:  have h' : E ⊔ (maximalRealSubfield K) = ⊤ := by
Mathlib/NumberTheory/NumberField/CMField.lean:463:    let L : IntermediateField E K := (E ⊔ (maximalRealSubfield K)).toIntermediateField
Mathlib/NumberTheory/NumberField/CMField.lean:472:  have : Algebra.IsAlgebraic (maximalRealSubfield K) K :=
Mathlib/NumberTheory/NumberField/CMField.lean:473:    Algebra.IsAlgebraic.tower_top (K := ℚ) (maximalRealSubfield K)
Mathlib/NumberTheory/NumberField/CMField.lean:482:    F ≃+* maximalRealSubfield K :=
Mathlib/NumberTheory/NumberField/CMField.lean:489:    exact eq_maximalRealSubfield K (algebraMap F K).fieldRange))
Mathlib/NumberTheory/NumberField/CMField.lean:496:theorem algebraMap_equivMaximalRealSubfield_symm_apply (x : maximalRealSubfield K) :
Mathlib/NumberTheory/NumberField/CMField.lean:498:      algebraMap (maximalRealSubfield K) K x := by
Mathlib/NumberTheory/NumberField/CMField.lean:26:* `NumberField.CMExtension.equivMaximalRealSubfield`: Any field `F` such that `K/F` is a
Mathlib/NumberTheory/NumberField/Discriminant/Defs.lean:75:theorem numberField_discr : discr ℚ = 1 := by
Mathlib/NumberTheory/NumberField/Discriminant/Defs.lean:89:

$ /bin/bash -lc "sed -n '110,160p' Mathlib/NumberTheory/NumberField/InfinitePlace/TotallyRealComplex.lean; sed -n '250,290p' Mathlib/NumberTheory/NumberField/Cyclotomic/Basic.lean; sed -n '335,385p' Mathlib/NumberTheory/NumberField/Cyclotomic/Basic.lean"
instance _root_.Subfield.isTotallyReal_bot [CharZero K] :
    IsTotallyReal (⊥ : Subfield K) := by
  rw [Subfield.bot_eq_of_charZero]
  exact IsTotallyReal.ofRingEquiv (algebraMap ℚ K).rangeRestrictFieldEquiv

section maximalRealSubfield

open ComplexEmbedding

/--
The maximal real subfield of `K`. It is totally real,
see `NumberField.isTotallyReal_maximalRealSubfield`, and contains all the other totally real
subfields of `K`, see `NumberField.IsTotallyReal.le_maximalRealSubfield`.
-/
def maximalRealSubfield : Subfield K where
  carrier := {x | ∀ φ : K →+* ℂ, star (φ x) = φ x}
  mul_mem' hx hy _ := by rw [map_mul, star_mul, hx, hy, mul_comm]
  one_mem' := by simp
  add_mem' hx hy _ := by rw [map_add, star_add, hx, hy]
  zero_mem' := by simp
  neg_mem' := by simp
  inv_mem' := by simp

variable {K}

theorem mem_maximalRealSubfield_iff (x : K) :
    x ∈ maximalRealSubfield K ↔ ∀ φ : K →+* ℂ, star (φ x) = φ x := .rfl

theorem IsTotallyReal.le_maximalRealSubfield (E : Subfield K) [IsTotallyReal E] :
    E ≤ maximalRealSubfield K := by
  intro x hx φ
  rw [show φ x = (φ.comp E.subtype) ⟨x, hx⟩ by simp, RCLike.star_def, ← conjugate_coe_eq]
  refine RingHom.congr_fun ?_ _
  exact ComplexEmbedding.isReal_iff.mp <| isReal_mk_iff.mp <| isReal _

@[simp]
theorem IsTotallyReal.maximalRealSubfield_eq_top [IsTotallyReal K] :
    maximalRealSubfield K = ⊤ :=
  top_unique <| NumberField.IsTotallyReal.le_maximalRealSubfield _

variable [CharZero K] [Algebra.IsAlgebraic ℚ K]

local instance (k : Subfield K) : Algebra.IsAlgebraic k K :=
  Algebra.IsAlgebraic.tower_top k (K := ℚ) (A := K)

instance isTotallyReal_maximalRealSubfield :
    IsTotallyReal (maximalRealSubfield K) where
  isReal w := by
    rw [InfinitePlace.isReal_iff, ComplexEmbedding.isReal_iff]
    ext x
    (hζ : IsPrimitiveRoot ζ (p ^ k)) :
    hζ.subOneIntegralPowerBasisOfPrimePow.gen =
      ⟨ζ - 1, Subalgebra.sub_mem _ (hζ.isIntegral (NeZero.pos _)) (Subalgebra.one_mem _)⟩ := by
  simp [subOneIntegralPowerBasisOfPri

$ /bin/bash -lc "rg -n \"def CyclotomicField|instance.*CyclotomicField|CyclotomicField.*IsCyclotomic|isCyclotomicExtension\" Mathlib/RingTheory/Polynomial/Cyclotomic -g '*.lean' Mathlib/NumberTheory -g '*.lean' | head -120; rg -n \"totient_two_pow|totient.*pow.*two|totient_prime_pow|totient_pow\" Mathlib | head -100"
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean:200:  have := IsPrimitiveRoot.adjoin_isCyclotomicExtension K this
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean:220:theorem dvd_of_isCyclotomicExtension [IsCyclotomicExtension {n} ℚ K] {ζ : K}
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean:258:  obtain ⟨a, ha⟩ := hlroot.dvd_of_isCyclotomicExtension n hlzero.1
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean:417:    convert! hη'.adjoin_isCyclotomicExtension K using 1
Mathlib/NumberTheory/Cyclotomic/Basic.lean:105:  simp [isCyclotomicExtension_iff, NeZero.ne]
Mathlib/NumberTheory/Cyclotomic/Basic.lean:108:  subsingleton_of_bot_eq_top <| by simpa [Algebra.eq_top_iff, isCyclotomicExtension_iff] using h
Mathlib/NumberTheory/Cyclotomic/Basic.lean:112:  simp [isCyclotomicExtension_iff, and_assoc]
Mathlib/NumberTheory/Cyclotomic/Basic.lean:117:    simpa [adjoin_singleton_one] using ((isCyclotomicExtension_iff _ _ _).1 h).2 x
Mathlib/NumberTheory/Cyclotomic/Basic.lean:126:theorem isCyclotomicExtension_zero_iff :
Mathlib/NumberTheory/Cyclotomic/Basic.lean:142:    · obtain ⟨b, hb⟩ := ((isCyclotomicExtension_iff _ _ _).1 hS).1 hn hn'
Mathlib/NumberTheory/Cyclotomic/Basic.lean:145:    · exact ((isCyclotomicExtension_iff _ _ _).1 hT).1 hn hn'
Mathlib/NumberTheory/Cyclotomic/Basic.lean:146:  · refine adjoin_induction (hx := ((isCyclotomicExtension_iff T B _).1 hT).2 x)
Mathlib/NumberTheory/Cyclotomic/Basic.lean:152:      ⟨b, ((isCyclotomicExtension_iff _ _ _).1 hS).2 b, rfl⟩
Mathlib/NumberTheory/Cyclotomic/Basic.lean:185:  refine ⟨fun hn => ((isCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => ?_⟩
Mathlib/NumberTheory/Cyclotomic/Basic.lean:186:  replace h := ((isCyclotomicExtension_iff _ _ _).1 h).2 b
Mathlib/NumberTheory/Cyclotomic/Basic.lean:195:  · obtain ⟨b, hb⟩ := ((isCyclotomicExtension_iff _ _ _).1 h).1 (hS hn) hn'
Mathlib/NumberTheory/Cyclotomic/Basic.lean:320:theorem _root_.Algebra.isCyclotomicExtension_adjoin_of_exists_isPrimitiveRoot

$ /bin/bash -lc "cat > Query3.lean <<'EOF'
import Mathlib.NumberTheory.NumberField.CMField
import Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
#check CyclotomicField
#check CyclotomicField.isCyclotomicExtension
#check IsCyclotomicExtension.Rat.finrank
#check IsCyclotomicExtension.Rat.isCMField
#check NumberField.IsCMField.isQuadraticExtension
#check IsQuadraticExtension.finrank_eq_two
#check Module.finrank_mul_finrank
#check Nat.totient_prime_pow
#check Nat.prime_two
#check NumberField.isTotallyReal_maximalRealSubfield
#check NumberField.maximalRealSubfield
#check NumberField.of_subfield
#check Module.finrank_pos
#check NumberField.IsTotallyReal.finrank
#check NumberField.IsTotallyReal.nrComplexPlaces_eq_zero
#check NumberField.ComplexEmbedding.isReal_iff
#check NumberField.IsTotallyReal.isReal
#check NumberField.IsTotallyReal.realAlgHom
#check NumberField.IsTotallyReal.toReal
#check NumberField.IsTotallyReal.embedding
EOF
lake env lean Query3.lean 2>/dev/null"
CyclotomicField.{w} (n : ℕ) (K : Type w) [Field K] : Type w
CyclotomicField.isCyclotomicExtension.{w} (n : ℕ) [NeZero n] (K : Type w) [Field K] [NeZero ↑n] :
  IsCyclotomicExtension {n} K (CyclotomicField n K)
IsCyclotomicExtension.Rat.finrank.{u} (k : ℕ) (K : Type u) [Field K] [CharZero K] [NeZero k]
  [IsCyclotomicExtension {k} ℚ K] : Module.finrank ℚ K = k.totient
IsCyclotomicExtension.Rat.isCMField.{u_1} (K : Type u_1) [Field K] [CharZero K] {S : Set ℕ} (hS : ∃ n ∈ S, 2 < n)
  [IsCyclotomicExtension S ℚ K] : NumberField.IsCMField K
NumberField.IsCMField.isQuadraticExtension.{u_1} (K : Type u_1) [Field K] [CharZero K] [NumberField.IsCMField K] :
  Algebra.IsQuadraticExtension (↥(NumberField.maximalRealSubfield K)) K
Query3.lean:8:7: error(lean.unknownIdentifier): Unknown identifier `IsQuadraticExtension.finrank_eq_two`
Module.finrank_mul_finrank.{u, v, w} (F : Type u) (K : Type v) (A : Type w) [Semiring F] [Semiring K] [AddCommMonoid A]
  [Module F K] [Module K A] [Module F A] [IsScalarTower F K A] [StrongRankCondition F] [StrongRankCondition K]
  [Module.Free F K] [Module.Free K A] : Module.finrank F K * Module.finrank K A = Module.finrank F A
Nat.totient_prime_pow {p : ℕ} (hp : Nat.Prime p) {n : ℕ} (hn : 0 < n) : (p ^ n).totient = p ^ (n - 1) * (p - 1)
Nat.prime_two : Nat.Prime 2
NumberField.isTotallyReal_maximalRealSubfield.{u_2} {K : Type u_2} [Field K] [CharZero K] [Algebra.IsAlgebraic ℚ K] :
  NumberField.IsTotallyReal ↥(NumberField.maximalRealSubfield K)
NumberField.maximalRealSubfield.{u_2} (K : Type u_2) [Field K] : Subfield K
NumberField.of_subfield.{u_1} {K : Type u_1} [Field K] [NumberField K] (E : Subfield K) : NumberField ↥E
Module.finrank_pos.{u_1, u_2} {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M]
  [StrongRankCondition R] [Module.Finite R M] [IsDomain R] [Module.IsTorsionFree R M] [h : Nontrivial M] :
  0 < Module.finrank R M
NumberField.IsTotallyReal.finrank.{u_2} (K : Type u_2) [Field K] [NumberField K] [h : NumberField.I

$ /bin/bash -lc "rg -n \"theorem finrank_eq_two|finrank_eq_two\" Mathlib | head -40; sed -n '650,715p' Mathlib/NumberTheory/Cyclotomic/Basic.lean"
Mathlib/NumberTheory/NumberField/Completion/Ramification.lean:48:theorem IsRamified.finrank_eq_two [w.LiesOver v] (h : w.IsRamified K) :
Mathlib/NumberTheory/NumberField/Completion/Ramification.lean:80:@[deprecated (since := "2026-07-10")] alias Completion.finrank_eq_two_of_isRamified :=
Mathlib/NumberTheory/NumberField/Completion/Ramification.lean:81:  IsRamified.finrank_eq_two
Mathlib/NumberTheory/NumberField/Completion/Ramification.lean:92:  · rw [h.finrank_eq_two v, hv, h.isReal.mult_eq_one, h.isComplex.mult_eq_two, one_mul]
Mathlib/NumberTheory/NumberField/Completion/Ramification.lean:119:  have := (Set.mem_ofPred.1 hw).1; hw.2.finrank_eq_two v ▸ inertiaDeg_eq_finrank v w
Mathlib/NumberTheory/NumberField/CMField.lean:95:    IsQuadraticExtension.finrank_eq_two _ K]
Mathlib/NumberTheory/NumberField/CMField.lean:132:    (IsQuadraticExtension.finrank_eq_two K⁺ K) ▸ IsGalois.card_aut_eq_finrank K⁺ K
Mathlib/NumberTheory/NumberField/CMField.lean:193:    IsQuadraticExtension.finrank_eq_two]
Mathlib/NumberTheory/NumberField/CMField.lean:466:      (IsQuadraticExtension.finrank_eq_two E K ▸ Nat.prime_two)).eq_bot_or_eq_top L).resolve_left ?_
Mathlib/NumberTheory/NumberField/CMField.lean:486:    { finrank_eq_two' :=
Mathlib/NumberTheory/NumberField/CMField.lean:487:        (IsQuadraticExtension.finrank_eq_two F K) ▸ Algebra.finrank_eq_of_equiv_equiv
Mathlib/NumberTheory/NumberField/CMField.lean:514:  is_quadratic := ⟨(IsQuadraticExtension.finrank_eq_two F K) ▸ finrank_eq_of_equiv_equiv
Mathlib/FieldTheory/IsAlgClosed/Basic.lean:359:theorem nonempty_algEquiv_or_of_finrank_eq_two {F F' : Type*} (E : Type*)
Mathlib/FieldTheory/Galois/Basic.lean:719:  have := finrank_eq_two F K ▸ AlgEquiv.card_le
Mathlib/FieldTheory/Normal/Basic.lean:295:    obtain h | h := le_iff_lt_or_eq.mp (finrank_eq_two F K ▸ minpoly.natDegree_le x)
Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean:157:lemma of_fact_finrank_eq_two [Fact (finrank K V = 2)] : FiniteDimensional K V :=
Mathlib/Geometry/Eucli

$ /bin/bash -lc "sed -i 's/#check IsQuadraticExtension.finrank_eq_two/#check Algebra.IsQuadraticExtension.finrank_eq_two/' Query3.lean; lake env lean Query3.lean 2>/dev/null | head -15"
CyclotomicField.{w} (n : ℕ) (K : Type w) [Field K] : Type w
CyclotomicField.isCyclotomicExtension.{w} (n : ℕ) [NeZero n] (K : Type w) [Field K] [NeZero ↑n] :
  IsCyclotomicExtension {n} K (CyclotomicField n K)
IsCyclotomicExtension.Rat.finrank.{u} (k : ℕ) (K : Type u) [Field K] [CharZero K] [NeZero k]
  [IsCyclotomicExtension {k} ℚ K] : Module.finrank ℚ K = k.totient
IsCyclotomicExtension.Rat.isCMField.{u_1} (K : Type u_1) [Field K] [CharZero K] {S : Set ℕ} (hS : ∃ n ∈ S, 2 < n)
  [IsCyclotomicExtension S ℚ K] : NumberField.IsCMField K
NumberField.IsCMField.isQuadraticExtension.{u_1} (K : Type u_1) [Field K] [CharZero K] [NumberField.IsCMField K] :
  Algebra.IsQuadraticExtension (↥(NumberField.maximalRealSubfield K)) K
Algebra.IsQuadraticExtension.finrank_eq_two.{u_2, u_3} (R : Type u_2) (S : Type u_3) [CommSemiring R]
  [StrongRankCondition R] [Semiring S] [Algebra R S] [Algebra.IsQuadraticExtension R S] : Module.finrank R S = 2
Module.finrank_mul_finrank.{u, v, w} (F : Type u) (K : Type v) (A : Type w) [Semiring F] [Semiring K] [AddCommMonoid A]
  [Module F K] [Module K A] [Module F A] [IsScalarTower F K A] [StrongRankCondition F] [StrongRankCondition K]
  [Module.Free F K] [Module.Free K A] : Module.finrank F K * Module.finrank K A = Module.finrank F A
Nat.totient_prime_pow {p : ℕ} (hp : Nat.Prime p) {n : ℕ} (hn : 0 < n) : (p ^ n).totient = p ^ (n - 1) * (p - 1)

$ /bin/bash -lc "cat > Query4.lean <<'EOF'
import Mathlib.NumberTheory.NumberField.CMField
import Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

open scoped Cyclotomic
open NumberField

noncomputable section

theorem test (k : ℕ) :
    let L := CyclotomicField (2 "'^ (k + 2)) ℚ
    let K := maximalRealSubfield L
    Module.finrank ℚ K = 2 ''^ k := by
  let L := CyclotomicField (2 ''^ (k + 2)) ℚ
  let N := 2 ''^ (k + 2)
  letI : NumberField.IsCMField L :=
    IsCyclotomicExtension.Rat.isCMField L (S := {N}) ⟨N, by simp, by
      dsimp [N]
      have : 4 ≤ 2 ''^ (k + 2) := by
        calc 4 = 2 ''^ 2 := by norm_num
          _ ≤ 2 ''^ (k + 2) := Nat.pow_le_pow_right (by omega) (by omega)
      omega⟩
  let K := maximalRealSubfield L
  have htower := Module.finrank_mul_finrank ℚ K L
  rw [Algebra.IsQuadraticExtension.finrank_eq_two] at htower
  have hL : Module.finrank ℚ L = N.totient :=
    IsCyclotomicExtension.Rat.finrank N L
  rw [hL, Nat.totient_prime_pow Nat.prime_two (by positivity : 0 < k + 2)] at htower
  dsimp [N] at htower
  norm_num at htower ⊢
  rw [pow_succ, pow_succ] at htower
  omega
EOF
lake env lean Query4.lean'
Query4.lean:16:4: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  IsCyclotomicExtension {N} ℚ L

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.
Query4.lean:24:6: error: failed to synthesize
  Algebra.IsQuadraticExtension ℚ ↥K

Hint: Additional diagnostic information may be available using the `set_option diagnostics true` command.

$ /bin/bash -lc 'lake env lean Query4.lean'
Query4.lean:16:4: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  IsCyclotomicExtension {2 ^ (k + 2)} ℚ L

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.
Query4.lean:25:4: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  IsCyclotomicExtension {N} ℚ L

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.
Query4.lean:29:6: error: Tactic `rewrite` failed: motive is not type correct:
  fun _a => Module.finrank ℚ ↥K * 2 = 2 ^ (k + 1)
Error: Application type mismatch: The argument
  CyclotomicField.instField _a ℚ
has type
  Field (CyclotomicField _a ℚ)
but is expected to have type
  Field L
in the application
  @Field.toDivisionRing L (CyclotomicField.instField _a ℚ)

Explanation: The rewrite tactic rewrites an expression 'e' using an equality 'a = b' by the following process. First, it looks for all 'a' in 'e'. Second, it tries to abstract these occurrences of 'a' to create a function 'm := fun _a => ...', called the *motive*, with the property that 'm a' is definitionally equal to 'e'. Third, we observe that 'congrArg' implies that 'm a = m b', which can be used with lemmas such as 'Eq.mpr' to change the goal. However, if 'e' depends on specific properties of 'a', then the motive 'm' might not typecheck.

Possible solutions: use rewrite's 'occs' configuration option to limit which occurrences are rewritten, or use 'simp' or 'conv' mode, which have strategies for certain kinds of dependencies (these tactics can handle proofs and 'Decidable' instances whose types depend on the rewritten term, and 'simp' can apply user-defined '@[congr]' theorems as well).

case refine_2
k : ℕ
L : Type := CyclotomicField (2 ^ (k + 2)) ℚ
N : ℕ := 2 ^ (k + 2)
this : IsCMField L :=
  IsCyclotomicExtension.Rat.isCMField L
    (Exists.intro (2 ^ (k + 2))
      ⟨of_eq_true (Eq.trans Set.mem

$ /bin/bash -lc 'lake env lean Query4.lean'
Query4.lean:16:4: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  IsCyclotomicExtension {2 ^ (k + 2)} ℚ L

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.
Query4.lean:25:4: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  IsCyclotomicExtension {2 ^ (k + 2)} ℚ L

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.
Query4.lean:28:6: error: Tactic `rewrite` failed: motive is not type correct:
  fun _a => Module.finrank ℚ ↥K * 2 = 2 ^ (k + 1)
Error: Application type mismatch: The argument
  CyclotomicField.instField _a ℚ
has type
  Field (CyclotomicField _a ℚ)
but is expected to have type
  Field L
in the application
  @Field.toDivisionRing L (CyclotomicField.instField _a ℚ)

Explanation: The rewrite tactic rewrites an expression 'e' using an equality 'a = b' by the following process. First, it looks for all 'a' in 'e'. Second, it tries to abstract these occurrences of 'a' to create a function 'm := fun _a => ...', called the *motive*, with the property that 'm a' is definitionally equal to 'e'. Third, we observe that 'congrArg' implies that 'm a = m b', which can be used with lemmas such as 'Eq.mpr' to change the goal. However, if 'e' depends on specific properties of 'a', then the motive 'm' might not typecheck.

Possible solutions: use rewrite's 'occs' configuration option to limit which occurrences are rewritten, or use 'simp' or 'conv' mode, which have strategies for certain kinds of dependencies (these tactics can handle proofs and 'Decidable' instances whose types depend on the rewritten term, and 'simp' can apply user-defined '@[congr]' theorems as well).

case refine_2
k : ℕ
L : Type := CyclotomicField (2 ^ (k + 2)) ℚ
this✝ : NeZero (2 ^ (k + 2)) :=
  {
    out :=
      pow_ne_zero (k + 2)
        (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.

$ /bin/bash -lc 'lake env lean Query4.lean'
Query4.lean:18:4: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  IsCyclotomicExtension {2 ^ (k + 2)} ℚ (CyclotomicField (2 ^ (k + 2)) ℚ)

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.

$ /bin/bash -lc 'lake env lean Query4.lean'
Query4.lean:19:4: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  IsCyclotomicExtension {2 ^ (k + 2)} ℚ (CyclotomicField (2 ^ (k + 2)) ℚ)

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.

$ /bin/bash -lc 'lake env lean Query4.lean'

$ /bin/bash -lc "cat > Query5.lean <<'EOF'
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#check tendsto_pow_atTop_atTop_of_one_lt
#check tendsto_pow_atTop_nhds_zero_of_lt_one
#check Nat.tendsto_pow_atTop_atTop_of_one_lt
#check Filter.tendsto_atTop
#check exists_nat_gt
#check exists_nat_one_div_lt
#check exists_pow_lt
#check Nat.le_two_pow
#check Nat.le_pow
#check Nat.lt_pow_self
#check Nat.pow_lt_pow_right
EOF
lake env lean Query5.lean 2>/dev/null"
tendsto_pow_atTop_atTop_of_one_lt.{u_1} {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α]
  [ExistsAddOfLE α] [Archimedean α] {r : α} (h : 1 < r) : Filter.Tendsto (fun n => r ^ n) Filter.atTop Filter.atTop
tendsto_pow_atTop_nhds_zero_of_lt_one.{u_4} {𝕜 : Type u_4} [Semifield 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
  [ExistsAddOfLE 𝕜] [Archimedean 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] {r : 𝕜} (h₁ : 0 ≤ r) (h₂ : r < 1) :
  Filter.Tendsto (fun n => r ^ n) Filter.atTop (nhds 0)
Query5.lean:4:7: error(lean.unknownIdentifier): Unknown constant `Nat.tendsto_pow_atTop_atTop_of_one_lt`
Filter.tendsto_atTop.{u_3, u_4} {α : Type u_3} {β : Type u_4} [Preorder β] {m : α → β} {f : Filter α} :
  Filter.Tendsto m f Filter.atTop ↔ ∀ (b : β), ∀ᶠ (a : α) in f, b ≤ m a
exists_nat_gt.{u_1} {R : Type u_1} [Semiring R] [PartialOrder R] [IsStrictOrderedRing R] [Archimedean R] (x : R) :
  ∃ n, x < ↑n
exists_nat_one_div_lt.{u_4} {K : Type u_4} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [Archimedean K] {ε : K}
  (hε : 0 < ε) : ∃ n, 1 / (↑n + 1) < ε
exists_pow_lt.{u_1} {R : Type u_1} [CommGroup R] [LinearOrder R] [IsOrderedMonoid R] [MulArchimedean R] {a : R}
  (ha : a < 1) (b : R) : ∃ n, a ^ n < b
Query5.lean:9:7: error(lean.unknownIdentifier): Unknown constant `Nat.le_two_pow`
Nat.le_pow {a b : ℕ} (h : 0 < b) : a ≤ a ^ b
Nat.lt_pow_self {n a : ℕ} (h : 1 < a) : n < a ^ n
Nat.pow_lt_pow_right {a m n : ℕ} (ha : 1 < a) (h : m < n) : a ^ m < a ^ n

$ /bin/bash -lc "cat > Query6.lean <<'EOF'
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics

lemma exponent_lt_of_large_degree {ε : ℝ} (hε : 0 < ε) {d : ℕ}
    (hd : 1 < d) (hlarge : 4 / ε + 6 < 9 * (d : ℝ)) :
    (2 * d : ℝ) / (3 * d - 2 : ℕ) < (2 : ℝ) / 3 + ε := by
  have h3d : 2 ≤ 3 * d := by omega
  have hcast : ((3 * d - 2 : ℕ) : ℝ) = 3 * (d : ℝ) - 2 := by
    rw [Nat.cast_sub h3d]
    norm_num
  rw [hcast]
  have hden : 0 < 3 * (d : ℝ) - 2 := by
    exact_mod_cast (by omega : 0 < 3 * d - 2)
  rw [div_lt_iff₀ hden]
  have hfour : 4 < (9 * (d : ℝ) - 6) * ε := by
    apply (div_lt_iff₀ hε).mp
    linarith
  nlinarith

lemma choose_degree (ε : ℝ) (hε : 0 < ε) :
    ∃ k : ℕ, 1 < 2 "'^ k ∧
      (2 * (2 ''^ k) : ℝ) / (3 * (2 ''^ k) - 2 : ℕ) < (2 : ℝ) / 3 + ε := by
  obtain ⟨q, hq⟩ := exists_nat_gt ((4 / ε + 6) / 9)
  refine ⟨q + 2, by positivity, ?_⟩
  apply exponent_lt_of_large_degree hε (by positivity)
  have hqd : (q : ℝ) < (2 ''^ (q + 2) : ℕ) := by
    exact_mod_cast (lt_trans (Nat.lt_pow_self (by omega : 1 < 2))
      (Nat.pow_lt_pow_right (by omega : 1 < 2) (by omega : q < q + 2)))
  have hq'"' : 4 / ε + 6 < 9 * (q : ℝ) := by linarith
  nlinarith
EOF
lake env lean Query6.lean"
Query6.lean:23:20: error: not a positivity goal
Query6.lean:24:2: error: Tactic `apply` failed: could not unify the conclusion of `exponent_lt_of_large_degree hε ?m.126`
  2 * ↑?m.125 / ↑(3 * ?m.125 - 2) < 2 / 3 + ε
with the goal
  2 * 2 ^ (q + 2) / ↑(3 * 2 ^ (q + 2) - 2) < 2 / 3 + ε

Note: The full type of `exponent_lt_of_large_degree hε ?m.126` is
  4 / ε + 6 < 9 * ↑?m.125 → 2 * ↑?m.125 / ↑(3 * ?m.125 - 2) < 2 / 3 + ε

ε : ℝ
hε : 0 < ε
q : ℕ
hq : (4 / ε + 6) / 9 < ↑q
⊢ 2 * 2 ^ (q + 2) / ↑(3 * 2 ^ (q + 2) - 2) < 2 / 3 + ε

$ /bin/bash -lc 'lake env lean Query6.lean'
Query6.lean:28:2: error: Tactic `apply` failed: could not unify the conclusion of `exponent_lt_of_large_degree hε ?m.207`
  2 * ↑(2 ^ (q + 2)) / ↑(3 * 2 ^ (q + 2) - 2) < 2 / 3 + ε
with the goal
  2 * 2 ^ (q + 2) / ↑(3 * 2 ^ (q + 2) - 2) < 2 / 3 + ε

Note: The full type of `exponent_lt_of_large_degree hε ?m.207` is
  4 / ε + 6 < 9 * ↑(2 ^ (q + 2)) → 2 * ↑(2 ^ (q + 2)) / ↑(3 * 2 ^ (q + 2) - 2) < 2 / 3 + ε

ε : ℝ
hε : 0 < ε
q : ℕ
hq : (4 / ε + 6) / 9 < ↑q
⊢ 2 * 2 ^ (q + 2) / ↑(3 * 2 ^ (q + 2) - 2) < 2 / 3 + ε

$ /bin/bash -lc 'lake env lean Query6.lean'
Query6.lean:29:62: error: omega could not prove the goal:
a possible counterexample may satisfy the constraints
  a ≤ 1
where
 a := ↑2 ^ (q + 2)

$ /bin/bash -lc 'lake env lean Query6.lean'

$ /bin/bash -lc "sed -n '1,110p' SharpDistanceScratch.lean"
import Mathlib.NumberTheory.NumberField.CMField
import Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.House
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics

open scoped BigOperators ComplexConjugate InnerProductSpace

namespace SharpDistance

open NumberField

noncomputable section

/-! The paper's naïve Euclidean objects. -/

abbrev Point := EuclideanSpace ℝ (Fin 2)

def point (x y : ℝ) : Point := !₂[x, y]

@[simp] lemma point_zero (x y : ℝ) : point x y 0 = x := by simp [point]
@[simp] lemma point_one (x y : ℝ) : point x y 1 = y := by simp [point]

structure RealLine where
  base : Point
  direction : Point
  direction_ne_zero : direction ≠ 0

def RealLine.carrier (ℓ : RealLine) : Set Point :=
  Set.range fun t : ℝ ↦ ℓ.base + t • ℓ.direction

def pointLineDist (p : Point) (ℓ : RealLine) : ℝ :=
  Metric.infDist p ℓ.carrier

lemma RealLine.base_mem (ℓ : RealLine) : ℓ.base ∈ ℓ.carrier :=
  ⟨0, by simp [RealLine.carrier]⟩

lemma RealLine.carrier_nonempty (ℓ : RealLine) : ℓ.carrier.Nonempty :=
  ⟨ℓ.base, ℓ.base_mem⟩

lemma pointLineDist_ge_inner_div_norm (p : Point) (ℓ : RealLine) (normal : Point)
    (hnormal : normal ≠ 0) (horth : ⟪(ℓ.direction), normal⟫_ℝ = 0) :
    |⟪p - ℓ.base, normal⟫_ℝ| / ‖normal‖ ≤ pointLineDist p ℓ := by
  rw [pointLineDist, Metric.le_infDist ℓ.carrier_nonempty]
  intro q hq
  obtain ⟨t, rfl⟩ := hq
  have hinner :
      ⟪p - (ℓ.base + t • ℓ.direction), normal⟫_ℝ =
        ⟪p - ℓ.base, normal⟫_ℝ := by
    rw [sub_add_eq_sub_sub, inner_sub_left, real_inner_smul_left, horth, mul_zero, sub_zero]
  rw [← hinner, dist_eq_norm]
  exact (div_le_iff₀ (norm_pos_iff.mpr hnormal)).2
    (abs_real_inner_le_norm (p - (ℓ.base + t • ℓ.direction)) normal)

lemma pointLineDist_self (ℓ : RealLine) : pointLineDist ℓ.base ℓ = 0 := by
  apply le_antisymm
  · exact Metric.infDist_le_dist_of_mem ℓ.base_mem |>.trans_eq (dist_self

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistanceScratch.lean'
SharpDistanceScratch.lean:37:15: warning: This simp argument is unused:
  RealLine.carrier

Hint: Omit it from the simp argument list.
  [apply] simp

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:113:2: warning: `push_neg` has been deprecated. Prefer using `push Not` instead.
If you'd rather continue using `push_neg` in your project, you can implement it as follows:
```
open Lean.Parser.Tactic in
macro "push_neg" cfg:optConfig loc:(location)? : tactic =>
  `(tactic| push $cfg:optConfig Not $[$loc]?)
```
SharpDistanceScratch.lean:109:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.exists_integerBasis_trace_ne_zero`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:151:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.trace_traceGenerator`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:154:8: warning: This simp argument is unused:
  smul_eq_mul

Hint: Omit it from the simp argument list.
  [apply] simp [mul_comm]

Note: This linter can be disabled with `set_option linter.unusedSimpArgs false`
SharpDistanceScratch.lean:159:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.linearCombo_integerBasis_injective`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_op

$ /bin/bash -lc 'rg -n "def traceBasisData" SharpDistanceScratch.lean -A6'
139:def traceBasisData (hd : 0 < Module.finrank ℚ K) : TraceBasisData K :=
140-  ⟨(exists_integerBasis_trace_ne_zero hd).choose,
141-    (exists_integerBasis_trace_ne_zero hd).choose_spec⟩
142-
143-abbrev TraceIndex (B : TraceBasisData K) := {i : IntegerIndex K // i ≠ B.pivot}
144-
145-def traceGenerator (B : TraceBasisData K) (i : TraceIndex B) : 𝓞 K :=

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistanceScratch.lean >/tmp/sharp.log 2>&1; code=$?; tail -50 /tmp/sharp.log; exit $code'
Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:348:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.house_linearCombo_le`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:492:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.abs_embedding_le_house`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:497:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.one_le_abs_norm_of_integer`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:653:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.scaled_orthogonal`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistanceScratch.lean:750:10: warning: 'ring' tactic does nothing

Note: This linter can be disabled with `set_option linter.unusedTactic false`
SharpDistanceScratch.lean:750:10: warning: this tactic is never executed

Note: This linter can be disabled with `set_option linter.unreachableTactic false`
Sh

$ /bin/bash -lc "sed -n '1,220p' /work/paper.txt"
THE SHARP EXPONENT FOR THE MINIMAL DISTANCE PROBLEM
COSMIN POHOATA
Abstract. We show that for every fixed ε > 0, there exist arbitrarily large families of
point–line pairs ( x1 , ℓ1 ), . . . , ( xn , ℓn ) in [0, 1]2 , with xi ∈ ℓi for all i, and such that
dist( xi , ℓ j ) ≥ n−2/3−ε

( i ̸ = j ).

Combined with a previous result of Cohen, the author, and Zakharov, this solves the
minimal distance problem.

arXiv:2607.20422v1 [math.CO] 22 Jul 2026

1. Introduction
Choose points x1 , . . . , xn ∈ [0, 1]2 and, for each i = 1, . . . , n, let ℓi be a line through xi .
The minimal distance problem asks how far every point can be kept from every line to
which it is not assigned. More precisely, we define
∆PL (n) := sup min dist( xi , ℓ j ),
i̸= j

(1)

where the supremum is over all configurations of points x1 , . . . , xn ∈ [0, 1]2 and lines
ℓ1 , . . . , ℓn ⊂ R2 such that xi ∈ ℓi . For convenience, let us call these point-line incidence
configurations of size n. The quantity ∆PL (n) can therefore be regarded as the largest
scale at which one can realize n prescribed point–line incidences while avoiding every
nontrivial incidence. In [7], Cohen, the author, and Zakharov introduced the problem
of determining the asymptotics of ∆PL (n) as n grows, in connection with the Heilbronn
triangle problem.
Trivially, taking xi = (i/n, 0) and ℓi = xi + (0, 1)R, for i = 1, . . . n, gives a point-line
incidence configuration with dist( xi , ℓ j ) ≥ 1/n for all i ̸= j. Thus, ∆PL (n) ≥ 1/n. In the
other direction, it is not difficult to see that in any point-line incidence configuration of
size n, we must have dist( xi , ℓ j ) ≤ | xi − x j | ≲ n−1/2 for some i ̸= j. Hence ∆PL (n) ≲ n−1/2 .
In [7], Cohen-Pohoata-Zakharov improved this upper bound to
∆PL (n) ≤ n−2/3+o(1) ,

(2)

and used this estimate to establish the latest record for the Heilbronn triangle problem.
For more history and context, see this recent survey by Zakharov [24].
The connection itself between the Heilbronn triangle pr

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistance.lean >/tmp/sharp-final.log 2>&1; code=$?; tail -25 /tmp/sharp-final.log; rg -n "'"\\bsorry\\b|\\badmit\\b|"'^axiom'"\\b|"'^constant'"\\b\" SharpDistance.lean README.md 2>/dev/null || true; git status --short; exit "'$code'
Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:752:10: warning: 'ring' tactic does nothing

Note: This linter can be disabled with `set_option linter.unusedTactic false`
SharpDistance.lean:752:10: warning: this tactic is never executed

Note: This linter can be disabled with `set_option linter.unreachableTactic false`
SharpDistance.lean:793:13: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
SharpDistance.lean:794:11: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
Try this:
  [apply] ring_nf
  
  The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.
    
  Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead.
SharpDistance.lean:1060:cardinalities admit `n` incident point-line pairs in the unit square whose
?? SharpDistance.lean

$ /bin/bash -lc "nl -ba SharpDistance.lean | sed -n '730,760p'; nl -ba SharpDistance.lean | sed -n '780,800p'; rg -n \"sorry|admit|axiom|constant\" SharpDistance.lean | tail -20"
730	        (mul_nonneg (by norm_num) (sq_nonneg _))
   731	    · positivity
   732	  have hP : 0 < P := pow_pos hH _
   733	  have hNG : N ≤ G := scaledNormal_norm_le σ B s
   734	  have hprod : 1 ≤ A * P := sourceD_norm_product_lower σ B hst
   735	  have hPinvA : 1 / P ≤ A := by
   736	    apply (div_le_iff₀ hP).2
   737	    simpa [A, P, H, mul_comm] using hprod
   738	  have hgeom :
   739	      A / (2 * N) ≤ pointLineDist (scaledPoint σ B t) (scaledLine σ B s) := by
   740	    have h := pointLineDist_ge_inner_div_norm
   741	      (scaledPoint σ B t) (scaledLine σ B s) (scaledNormal σ B s)
   742	      (scaledNormal_ne_zero σ B s) (scaled_orthogonal σ B s)
   743	    change
   744	      |⟪scaledPoint σ B t - scaledPoint σ B s, scaledNormal σ B s⟫_ℝ| /
   745	        ‖scaledNormal σ B s‖ ≤
   746	          pointLineDist (scaledPoint σ B t) (scaledLine σ B s) at h
   747	    rw [scaled_inner_eq] at h
   748	    simpa [A, N, abs_div, div_div] using h
   749	  calc
   750	    (2 * G * P)⁻¹ = (1 / P) / (2 * G) := by
   751	      field_simp [hG.ne', hP.ne']
   752	      <;> ring
   753	    _ ≤ A / (2 * G) := by
   754	      exact div_le_div_of_nonneg_right hPinvA (by positivity)
   755	    _ ≤ A / (2 * N) := by
   756	      exact div_le_div_of_nonneg_left hA (by positivity) (by linarith)
   757	    _ ≤ pointLineDist (scaledPoint σ B t) (scaledLine σ B s) := hgeom
   758	
   759	structure PLConfiguration (n : ℕ) where
   760	  points : Fin n → Point
   780	      gcongr <;> omega
   781	
   782	def sourceEmbedding (B : TraceBasisData K) (m n : ℕ)
   783	    (hcard : n ≤ Fintype.card (ArithmeticSource B m)) :
   784	    Fin n ↪ ArithmeticSource B m :=
   785	  Classical.choice <| Function.Embedding.nonempty_of_card_le <| by
   786	    simpa using hcard
   787	
   788	def selectedConfiguration (σ : K →+* ℝ) (B : TraceBasisData K)
   789	    (m n : ℕ) (hcard : n ≤ Fintype.card (ArithmeticSource B m)) :
   790	    PLConfiguration n where
   791	  points i := scaledPoint σ 

$ /bin/bash -lc "sed -n '105,330p' SharpDistance.lean; sed -n '330,570p' SharpDistance.lean; sed -n '570,735p' SharpDistance.lean"
· rintro rfl
    simp

abbrev IntegerIndex (K : Type*) [Field K] [NumberField K] :=
  Module.Free.ChooseBasisIndex ℤ (𝓞 K)

lemma exists_integerBasis_trace_ne_zero (hd : 0 < Module.finrank ℚ K) :
    ∃ j : IntegerIndex K,
      Algebra.trace ℤ (𝓞 K) (NumberField.RingOfIntegers.basis K j) ≠ 0 := by
  by_contra h
  push_neg at h
  have ht : Algebra.trace ℤ (𝓞 K) = 0 := by
    apply (NumberField.RingOfIntegers.basis K).ext
    intro i
    simp [h i]
  have h1 := LinearMap.congr_fun ht (1 : 𝓞 K)
  have hrank : 0 < Module.finrank ℤ (𝓞 K) := by
    rwa [NumberField.RingOfIntegers.rank]
  have hone : Algebra.trace ℤ (𝓞 K) (1 : 𝓞 K) =
      (Module.finrank ℤ (𝓞 K) : ℤ) := by
    rw [Algebra.trace]
    have hlmul : (Algebra.lmul ℤ (𝓞 K)) 1 = 1 := by
      ext x
      simp
    rw [LinearMap.comp_apply]
    change LinearMap.trace ℤ (𝓞 K) ((Algebra.lmul ℤ (𝓞 K)) 1) = _
    rw [hlmul, LinearMap.trace_one]
  rw [hone] at h1
  simp only [LinearMap.zero_apply] at h1
  omega

structure TraceBasisData (K : Type*) [Field K] [NumberField K] where
  pivot : IntegerIndex K
  pivot_trace_ne_zero :
    Algebra.trace ℤ (𝓞 K) (NumberField.RingOfIntegers.basis K pivot) ≠ 0

def traceBasisData (hd : 0 < Module.finrank ℚ K) : TraceBasisData K :=
  ⟨(exists_integerBasis_trace_ne_zero hd).choose,
    (exists_integerBasis_trace_ne_zero hd).choose_spec⟩

abbrev TraceIndex (B : TraceBasisData K) := {i : IntegerIndex K // i ≠ B.pivot}

def traceGenerator (B : TraceBasisData K) (i : TraceIndex B) : 𝓞 K :=
  (Algebra.trace ℤ (𝓞 K) (NumberField.RingOfIntegers.basis K B.pivot)) •
      NumberField.RingOfIntegers.basis K i.1 -
    (Algebra.trace ℤ (𝓞 K) (NumberField.RingOfIntegers.basis K i.1)) •
      NumberField.RingOfIntegers.basis K B.pivot

lemma trace_traceGenerator (B : TraceBasisData K) (i : TraceIndex B) :
    Algebra.trace ℤ (𝓞 K) (traceGenerator B i) = 0 := by
  rw [traceGenerator, map_sub, map_zsmul, map_zsmul]
  simp [smul_eq_mul, mul_comm]

def linearCombo {ι : Type*} [Fintype ι] (g : ι → 𝓞 

$ /bin/bash -lc "ls -la | head -30; test -f README.md && sed -n '1,160p' README.md || true"
total 640
drwxr-xr-x 17 root root   4096 Jul 23 22:22 .
drwxr-xr-x  6 root root   4096 Jul 23 22:23 ..
drwxr-xr-x  2 root root   4096 Jul 23 21:41 .devcontainer
drwxr-xr-x  5 root root   4096 Jul 23 21:41 .docker
drwxr-xr-x  8 root root   4096 Jul 23 22:23 .git
-rw-r--r--  1 root root    401 Jul 23 21:41 .git-blame-ignore-revs
drwxr-xr-x  4 root root   4096 Jul 23 21:41 .github
-rw-r--r--  1 root root    440 Jul 23 21:41 .gitignore
-rw-r--r--  1 root root    155 Jul 23 21:41 .gitpod.yml
drwxr-xr-x  5 root root   4096 Jul 23 21:41 .lake
-rw-r--r--  1 root root    419 Jul 23 21:41 .pre-commit-config.yaml
drwxr-xr-x  2 root root   4096 Jul 23 21:41 .vscode
drwxr-xr-x  7 root root   4096 Jul 23 21:41 Archive
-rw-r--r--  1 root root   2901 Jul 23 21:41 Archive.lean
-rw-r--r--  1 root root    677 Jul 23 21:41 CITATION.md
-rw-r--r--  1 root root   5319 Jul 23 21:41 CODE_OF_CONDUCT.md
drwxr-xr-x  2 root root   4096 Jul 23 21:41 Cache
drwxr-xr-x  2 root root   4096 Jul 23 21:41 Counterexamples
-rw-r--r--  1 root root   1261 Jul 23 21:41 Counterexamples.lean
drwxr-xr-x  2 root root   4096 Jul 23 21:41 DownstreamTest
-rw-r--r--  1 root root  11357 Jul 23 21:41 LICENSE
drwxr-xr-x 33 root root   4096 Jul 23 21:41 Mathlib
-rw-r--r--  1 root root 452322 Jul 23 21:41 Mathlib.lean
drwxr-xr-x 23 root root   4096 Jul 23 21:41 MathlibTest
-rw-r--r--  1 root root   8936 Jul 23 21:41 README.md
-rw-r--r--  1 root root  43537 Jul 23 22:23 SharpDistance.lean
-rw-r--r--  1 root root   1857 Jul 23 21:41 bors.toml
drwxr-xr-x  3 root root   4096 Jul 23 21:41 docs
-rw-r--r--  1 root root     53 Jul 23 21:41 docs.lean
# mathlib4

![GitHub CI](https://github.com/leanprover-community/mathlib4/actions/workflows/build.yml/badge.svg?branch=master)
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[![project chat](https://img.shields.io/badge/zulip-join_chat-brightgreen

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistance.lean >/tmp/sharp-audit.log 2>&1; code=$?; tail -35 /tmp/sharp-audit.log; exit $code'
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:499:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.one_le_abs_norm_of_integer`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:655:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.scaled_orthogonal`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:792:13: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
SharpDistance.lean:793:11: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
Try this:
  [apply] ring_nf
  
  The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.
    
  Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead.
'SharpDistance.sharp

$ /bin/bash -lc 'git diff --stat && git status --short && git diff --check && rg -n "(''^|[''^[:alpha:]])(sorry|admit)([''^[:alpha:]]|$)|''^[[:space:]]*(axiom|constant)[[:space:]]" SharpDistance.lean || true; tail -30 SharpDistance.lean; sed -n '"'1,60p' README.md"
README.md | 330 ++++++++++++++++++++++++++++++++++++--------------------------
 1 file changed, 193 insertions(+), 137 deletions(-)
 M README.md
?? SharpDistance.lean
1059:cardinalities admit `n` incident point-line pairs in the unit square whose
  apply exponent_lt_of_large_degree hε (d := 2 ^ (q + 2)) hdpow
  have hqd : (q : ℝ) < (2 ^ (q + 2) : ℕ) := by
    exact_mod_cast (lt_trans (Nat.lt_pow_self (by omega : 1 < 2))
      (Nat.pow_lt_pow_right (by omega : 1 < 2) (by omega : q < q + 2)))
  have hq' : 4 / ε + 6 < 9 * (q : ℝ) := by linarith
  nlinarith

/-- The central result: for every loss `ε > 0`, all sufficiently large
cardinalities admit `n` incident point-line pairs in the unit square whose
off-diagonal point-to-line distances are at least `n ^ (-(2/3 + ε))`. -/
theorem sharp_exponent_for_minimal_distance (ε : ℝ) (hε : 0 < ε) :
    ∃ n₀ : ℕ, ∀ n ≥ n₀, ∃ C : PLConfiguration n,
      C.Separated ((n : ℝ) ^ (-((2 : ℝ) / 3 + ε))) := by
  obtain ⟨k, hkdeg, hkexp⟩ := exists_cyclotomic_degree_for_exponent ε hε
  let L := CyclotomicField (2 ^ (k + 2)) ℚ
  let K := maximalRealSubfield L
  letI : NumberField K := NumberField.of_subfield K
  letI : NumberField.IsTotallyReal K :=
    NumberField.isTotallyReal_maximalRealSubfield
  have hdeg : Module.finrank ℚ K = 2 ^ k := by
    simpa [L, K] using realCyclotomicField_finrank k
  let φ : K →ₐ[ℚ] ℂ :=
    (Classical.choice (inferInstance : Nonempty (K →+* ℂ))).toRatAlgHom
  let σ : K →+* ℝ := realEmbedding φ
  let B : TraceBasisData K := traceBasisData (by rw [hdeg]; omega)
  apply eventually_configuration_of_fixed_field σ B ((2 : ℝ) / 3 + ε)
  · simpa [hdeg] using hkdeg
  · simpa [hdeg] using hkexp

end SharpDistance
# Reproduction of “The sharp exponent for the minimal distance problem”

This directory contains a complete Lean 4/mathlib proof of Theorem 1.1 of
arXiv:2607.20422v1. The development is in `SharpDistance.lean`.

## Result and verification

The final theorem is:

```lean
theorem SharpDistance.sharp_exponent_fo

$ /bin/bash -lc 'lake env lean -DmaxHeartbeats=1500000 SharpDistance.lean >/tmp/sharp-final.log 2>&1; code=$?; tail -35 /tmp/sharp-final.log; exit $code'
[IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:505:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.one_le_abs_norm_of_integer`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:661:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.scaled_orthogonal`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:820:13: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
SharpDistance.lean:821:11: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
Try this:
  [apply] ring_nf
  
  The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.
    
  Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead.

$ /bin/bash -lc 'grep -n -B8 -A4 "Try this" /tmp/sharp-final.log'
132-
133-Note: This linter can be disabled with `set_option linter.unusedVariables false`
134-SharpDistance.lean:821:11: warning: Variable name `i` is not explicitly referenced.
135-
136-Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
137-  [apply] _i
138-
139-Note: This linter can be disabled with `set_option linter.unusedVariables false`
140:Try this:
141-  [apply] ring_nf
142-  
143-  The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.
144-

$ /bin/bash -lc "sed -n '100,155p' /tmp/sharp-final.log"
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:356:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.house_linearCombo_le`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:500:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.abs_embedding_le_house`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:505:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.one_le_abs_norm_of_integer`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:661:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.scaled_orthogonal`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:820:13: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively,

$ /bin/bash -lc 'lake env lean SharpDistance.lean >/tmp/sharp-default.log 2>&1; code=$?; tail -30 /tmp/sharp-default.log; exit $code'
SharpDistance.lean:505:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.one_le_abs_norm_of_integer`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:661:0: warning: automatically included section variable(s) unused in theorem `SharpDistance.scaled_orthogonal`:
  [IsTotallyReal K]
consider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:
  omit [IsTotallyReal K] in theorem ...

Note: This linter can be disabled with `set_option linter.unusedSectionVars false`
SharpDistance.lean:820:13: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
SharpDistance.lean:821:11: warning: Variable name `i` is not explicitly referenced.

Hint: The binding can be removed (if unused) or named `_` (if used implicitly). Alternatively, prefix the name with `_` to silence this warning:
  [apply] _i

Note: This linter can be disabled with `set_option linter.unusedVariables false`
Try this:
  [apply] ring_nf
  
  The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.
    
  Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead.

$ /bin/bash -lc 'git diff --check && if rg -n "(''^|[''^[:alpha:]])(sorry|admit)([''^[:alpha:]]|$)|''^[[:space:]]*(axiom|constant)[[:space:]]" SharpDistance.lean; then exit 1; else true; fi; wc -l SharpDistance.lean README.md; git status --short'
1108 SharpDistance.lean
  226 README.md
 1334 total
 M README.md
?? SharpDistance.lean