From 321aa6b651c3fd300e74c9976b9857812e48f202 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Tue, 11 Aug 2026 07:23:56 +0200 Subject: [PATCH] First chapters --- .gitignore | 2 + AlgebraInLean.lean | 4 +- AlgebraInLean/Basics.lean | 95 + AlgebraInLean/Cauchy.lean | 359 +- AlgebraInLean/LittleFermat.lean | 114 +- AlgebraInLean/Quotients.lean | 141 + Book.lean | 65 + Book/Basics.lean | 100 + Book/Cauchy.lean | 328 + Book/LittleFermat.lean | 147 + Book/Quotients.lean | 148 + Main.lean | 67 + README.md | 20 + deploy.sh | 10 + lake-manifest.json | 62 +- lakefile.lean | 39 + lakefile.toml | 10 - static/README.txt | 1 + static/colors.css | 32 + static/fonts/source-code-pro/LICENSE.md | 93 + .../OTF/SourceCodePro-Black.otf | Bin 0 -> 133768 bytes .../OTF/SourceCodePro-BlackIt.otf | Bin 0 -> 112052 bytes .../OTF/SourceCodePro-Bold.otf | Bin 0 -> 134020 bytes .../OTF/SourceCodePro-BoldIt.otf | Bin 0 -> 110288 bytes .../OTF/SourceCodePro-ExtraLight.otf | Bin 0 -> 125412 bytes 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static/lean_logo.svg create mode 100644 static/math.js create mode 100644 static/print.css create mode 100644 static/print.js create mode 100644 static/theme.css diff --git a/.gitignore b/.gitignore index 01f8cdb..3266446 100644 --- a/.gitignore +++ b/.gitignore @@ -1 +1,3 @@ .lake/ +_out/ +public/ diff --git a/AlgebraInLean.lean b/AlgebraInLean.lean index e9ee4f3..cebbd62 100644 --- a/AlgebraInLean.lean +++ b/AlgebraInLean.lean @@ -1,3 +1,5 @@ --- Root module: imports all chapters of "Algebra in Lean". +-- Wurzelmodul: importiert alle Übungsdateien. +import AlgebraInLean.Basics +import AlgebraInLean.Quotients import AlgebraInLean.LittleFermat import AlgebraInLean.Cauchy diff --git a/AlgebraInLean/Basics.lean b/AlgebraInLean/Basics.lean new file mode 100644 index 0000000..8391720 --- /dev/null +++ b/AlgebraInLean/Basics.lean @@ -0,0 +1,95 @@ +/- +Algebra in Lean — Kapitel 2 des Skripts +======================================= + +Diese Datei begleitet das Skript „Algebra und Zahlentheorie“ +(Stefan Kebekus, CC-BY 4.0). Sie ist der Einstieg: erste +Taktiken, erste Beweise, und die Frage, wie Mathlib den +Begriff „Gruppe“ aus Definition 2.1.1 des Skripts darstellt. +-/ +import Mathlib + +namespace AlgebraInLean + +/-! +# Erste Schritte + +## Beweise lesen und schreiben + +Ein Beweis in Lean beginnt mit `by` und besteht aus +*Taktiken*. Setzen Sie den Cursor in einen Beweis und +beobachten Sie das Infoview-Panel: Es zeigt zu jedem +Zeitpunkt die Hypothesen und das noch zu zeigende Ziel. + +Vier Taktiken reichen für den Anfang erstaunlich weit: + +* `norm_num` verrechnet konkrete Zahlen, +* `ring` beweist Gleichungen, die in jedem kommutativen + Ring gelten, +* `omega` löst lineare Arithmetik über ℕ und ℤ, +* `rw [h]` schreibt mit einer Gleichung `h` um. +-/ + +example : 2 + 3 = 5 := by norm_num + +example (a b : ℤ) : + (a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2 := by + ring + +example (n : ℕ) (h : n < 5) : n ≤ 7 := by omega + +example (x y : ℚ) (h : x = y) : x + 1 = y + 1 := by + rw [h] + +/-! +## Gruppen in Mathlib + +Definition 2.1.1 des Skripts: + +**Definition (Gruppe).** *Eine Gruppe ist eine nicht-leere +Menge G mit einer Abbildung m : G ⨯ G → G, sodass folgende +Eigenschaften gelten. Assoziativität: … Neutrales Element: +Es gibt genau ein Element e aus G, sodass für alle a aus G +gilt: m(e,a) = m(a,e) = a. Inverse Elemente: Für alle a aus +G gibt es genau ein Element b aus G, sodass +m(a,b) = m(b,a) = e ist.* + +In Mathlib ist eine Gruppe eine Typklasse `Group G`. Die +Verknüpfung schreibt sich `a * b`, das neutrale Element `1`, +das Inverse `a⁻¹`; die Axiome heißen `mul_assoc`, `one_mul`, +`mul_one`, `inv_mul_cancel`, `mul_inv_cancel`. + +Ein Unterschied fällt auf: Das Skript *fordert* die +Eindeutigkeit von neutralem Element und Inversen, Mathlib +nicht. Das ist kein Widerspruch — die Eindeutigkeit folgt +aus den übrigen Axiomen. Genau das beweisen wir jetzt, mit +dem Standardargument aus der linearen Algebra: + +*Es sei e ein weiteres Element, das von links neutral wirkt. +Weil 1 neutral ist, gilt e·1 = e. Weil e neutral wirkt, +gilt e·1 = 1. Zusammen folgt e = e·1 = 1.* +-/ + +theorem neutral_eindeutig {G : Type*} [Group G] (e : G) + (he : ∀ a : G, e * a = a) : e = 1 := by + -- „Weil 1 neutral ist, gilt e·1 = e. Weil e neutral + -- wirkt, gilt e·1 = 1. Zusammen folgt e = e·1 = 1.“ + calc e = e * 1 := (mul_one e).symm + _ = 1 := he 1 + +/-! +## Übungsaufgabe + +Zeigen Sie ebenso die Eindeutigkeit des Inversen: Wenn +a·b = 1 ist, dann ist b bereits *das* Inverse a⁻¹. Das +Standardargument: b = 1·b = (a⁻¹·a)·b = a⁻¹·(a·b) = a⁻¹·1 += a⁻¹. Ersetzen Sie das `sorry` durch einen Beweis — die +Taktik `calc` aus dem Beweis oben und die Lemmata `one_mul`, +`inv_mul_cancel`, `mul_assoc`, `mul_one` genügen. +-/ + +theorem inverses_eindeutig {G : Type*} [Group G] (a b : G) + (h : a * b = 1) : b = a⁻¹ := by + sorry + +end AlgebraInLean diff --git a/AlgebraInLean/Cauchy.lean b/AlgebraInLean/Cauchy.lean index a245a52..c0fc8b3 100644 --- a/AlgebraInLean/Cauchy.lean +++ b/AlgebraInLean/Cauchy.lean @@ -1,13 +1,13 @@ /- -Algebra in Lean — Chapter 18 +Algebra in Lean — Kapitel 18 ============================ -This file accompanies the lecture notes "Algebra und Zahlentheorie" -(Stefan Kebekus, CC-BY 4.0). It translates the central key lemma -(`lem:zsl`) and Cauchy's theorem (`Satz_von_Cauchy`) from Chapter 18 -into Lean, following the proof of the lecture notes sentence by -sentence. The German original of every sentence is quoted as a comment -directly above the Lean code that implements it. +Diese Datei begleitet das Skript „Algebra und Zahlentheorie“ +(Stefan Kebekus, CC-BY 4.0). Sie übersetzt das zentrale +Schlüssellemma (`lem:zsl`) und den Satz von Cauchy +(`Satz_von_Cauchy`) aus Kapitel 18 Satz für Satz nach Lean. +Das deutsche Original jedes Beweissatzes steht als Kommentar +direkt über dem Lean-Code, der ihn umsetzt. -/ import Mathlib @@ -16,252 +16,295 @@ namespace AlgebraInLean open Equiv.Perm open Equiv.Perm.VectorsProdEqOne open MulAction +open Multiplicative /-! -# The key lemma and Cauchy's theorem +# Das Schlüssellemma und der Satz von Cauchy -## The key lemma +## Das zentrale Schlüssellemma -**Lemma (Zentrales Schlüssellemma).** *Es sei m ∈ ℕ und es sei p eine -Primzahl. Weiter sei G eine Gruppe der Ordnung p^m, die auf einer -endlichen Menge M operiert. Weiter sei M₀ = { m ∈ M : ∀ g ∈ G: g·m = m } -die Menge der Fixpunkte. Dann ist |M| ≡ |M₀| (mod p).* +**Lemma (Zentrales Schlüssellemma).** *Es sei m ∈ ℕ und es +sei p eine Primzahl. Weiter sei G eine Gruppe der Ordnung +p^m, die auf einer endlichen Menge M operiert. Weiter sei +M₀ = { m ∈ M : ∀ g ∈ G: g·m = m } die Menge der Fixpunkte. +Dann ist |M| ≡ |M₀| (mod p).* -How does Mathlib say all this? +Wie sagt Mathlib das alles? -* "eine Gruppe der Ordnung p^m": Mathlib has a predicate `IsPGroup p G` - for this ("the order of every element is a power of p" — for finite - groups this is equivalent, by Cauchy's theorem below!). The lemma - `IsPGroup.of_card` converts our hypothesis `Nat.card G = p ^ m` into - it. +* „eine Gruppe der Ordnung p^m“: Dafür hat Mathlib ein + Prädikat `IsPGroup p G` („die Ordnung jedes Elements ist + eine p-Potenz“ — für endliche Gruppen ist das äquivalent, + nach dem Satz von Cauchy unten!). Das Lemma + `IsPGroup.of_card` übersetzt unsere Voraussetzung + `Nat.card G = p ^ m` dorthin. -* "die auf einer endlichen Menge M operiert": an action of `G` on `M` is - a typeclass, `[MulAction G M]`; finiteness of `M` is the typeclass - `[Finite M]`. +* „die auf einer endlichen Menge M operiert“: Eine Wirkung von + `G` auf `M` ist eine Typklasse, `[MulAction G M]`; die + Endlichkeit von `M` ist die Typklasse `[Finite M]`. -* The set of fixed points is `MulAction.fixedPoints G M`, and the - congruence `|M| ≡ |M₀| (mod p)` is written +* Die Fixpunktmenge heißt `MulAction.fixedPoints G M`, und die + Kongruenz |M| ≡ |M₀| (mod p) schreibt sich `Nat.card M ≡ Nat.card (fixedPoints G M) [MOD p]`. -The proof in the lecture notes decomposes M into orbits and quotes the -Bahnengleichung. This is *exactly* how Mathlib proves the statement -`IsPGroup.card_modEq_card_fixedPoints` — so here, just as the lecture -notes quote Satz 17.2.6, we simply cite the library. +Der Beweis im Skript zerlegt M in Bahnen und zitiert die +Bahnengleichung. Genau so beweist Mathlib die Aussage +`IsPGroup.card_modEq_card_fixedPoints` — wir zitieren hier +also einfach die Bibliothek, so wie das Skript an dieser +Stelle Satz 17.2.6 zitiert. -/ -theorem key_lemma {p m : ℕ} (hp : p.Prime) {G : Type*} [Group G] - (hG : Nat.card G = p ^ m) (M : Type*) [Finite M] [MulAction G M] : +theorem key_lemma {p m : ℕ} (hp : p.Prime) {G : Type*} + [Group G] (hG : Nat.card G = p ^ m) (M : Type*) + [Finite M] [MulAction G M] : Nat.card M ≡ Nat.card (fixedPoints G M) [MOD p] := by have : Fact p.Prime := ⟨hp⟩ exact (IsPGroup.of_card hG).card_modEq_card_fixedPoints M /-! -## Cauchy's theorem +## Der Satz von Cauchy -**Satz (Satz von Cauchy).** *Wenn die Ordnung einer endlichen Gruppe -durch p teilbar ist, dann existiert ein Element von Ordnung p.* +**Satz (Satz von Cauchy).** *Wenn die Ordnung einer endlichen +Gruppe durch p teilbar ist, dann existiert ein Element von +Ordnung p.* -We follow the proof of the lecture notes sentence by sentence. The -proof constructs a clever auxiliary set with a clever group action, so -this time there is real work to do *before* the final theorem: we set up -the set and the action first. +Wir folgen dem Beweis des Skripts Satz für Satz. Der Beweis +konstruiert eine raffinierte Hilfsmenge mit einer raffinierten +Gruppenwirkung — diesmal gibt es also vor dem eigentlichen +Satz echte Arbeit: Wir bauen erst die Menge und die Wirkung. -„Betrachte die Menge M = { (a₁, …, a_p) ∈ G ⨯ ⋯ ⨯ G : a₁·a₂ ⋯ a_p = e }.“ +„Betrachte die Menge +M = { (a₁, …, a_p) ∈ G ⨯ ⋯ ⨯ G : a₁·a₂ ⋯ a_p = e }.“ -Mathlib knows this set. A p-tuple of group elements is a -`List.Vector G p` — a list of length p — and M is the set -`Equiv.Perm.vectorsProdEqOne G p` of all vectors whose entries multiply -to 1. +Mathlib kennt diese Menge. Ein p-Tupel von Gruppenelementen +ist ein `List.Vector G p` — eine Liste der Länge p —, und M +ist die Menge `Equiv.Perm.vectorsProdEqOne G p` aller +Vektoren, deren Einträge sich zu 1 multiplizieren. -„Gegeben ein Tupel (a₁, …, a_p) ∈ M, dann stellen wir erst einmal fest, -dass der letzte Eintrag des Tupels durch die ersten Einträge eindeutig -bestimmt ist, a_p = (a₁ ⋯ a_{p-1})⁻¹. Wir erhalten die folgende -Gleichung: |M| = |G^{p-1}| = |G|^{p-1}.“ +„Gegeben ein Tupel (a₁, …, a_p) ∈ M, dann stellen wir erst +einmal fest, dass der letzte Eintrag des Tupels durch die +ersten Einträge eindeutig bestimmt ist, +a_p = (a₁ ⋯ a_{p-1})⁻¹. Wir erhalten die folgende Gleichung: +|M| = |G^{p-1}| = |G|^{p-1}.“ -This, too, is already in Mathlib: the bijection -(a₁, …, a_{p-1}) ↦ (a₁, …, a_{p-1}, (a₁ ⋯ a_{p-1})⁻¹) is -`VectorsProdEqOne.vectorEquiv`, and the resulting counting formula is -`VectorsProdEqOne.card`: --/ +Auch das steht schon in Mathlib: Die Bijektion +(a₁, …, a_{p-1}) ↦ (a₁, …, a_{p-1}, (a₁ ⋯ a_{p-1})⁻¹) heißt +`VectorsProdEqOne.vectorEquiv`, die resultierende Zählformel +`VectorsProdEqOne.card`. -#check @Equiv.Perm.VectorsProdEqOne.card --- ∀ (G : Type u_1) [inst : Group G] (n : ℕ) [inst_1 : Fintype G], --- Fintype.card ↥(vectorsProdEqOne G n) = Fintype.card G ^ (n - 1) +„Als Nächstes brauchen wir eine schicke Gruppenwirkung, denn +wir wollen das zentrale Schlüssellemma anwenden. Dazu lassen +wir die zyklische Gruppe ℤ/(p) auf M durch zyklisches +Vertauschen wirken.“ -/-! -„Als Nächstes brauchen wir eine schicke Gruppenwirkung, denn wir wollen -das zentrale Schlüssellemma anwenden. Dazu lassen wir die zyklische -Gruppe ℤ/(p) auf M durch zyklisches Vertauschen wirken.“ +Der zyklische Shift eines Vektors `v ∈ vectorsProdEqOne G p` +um k Stellen ist `VectorsProdEqOne.rotate v k`. Die Fußnote +des Skripts — der Shift bildet M auf sich ab, „weil in jeder +Gruppe aus a·b = e auch b·a = e gilt“ — ist das Mathlib-Lemma +`List.prod_rotate_eq_one_of_prod_eq_one`, das schon in der +Definition von `rotate` steckt. -The cyclic shift of a vector `v ∈ vectorsProdEqOne G p` by k places is -`VectorsProdEqOne.rotate v k`. The footnote of the lecture notes — the -shift maps M to itself „weil in jeder Gruppe aus a·b = e auch b·a = e -gilt“ — is the Mathlib lemma `List.prod_rotate_eq_one_of_prod_eq_one`, -which is used in the very definition of `rotate`. - -To let ℤ/(p) act *as a group* we must check the action axioms: rotating -by 0 does nothing, and rotating by j + k is the same as rotating by k -and then by j. Mathlib provides `rotate_zero`, `rotate_rotate` and -`rotate_length` (rotating by the full length p does nothing); from the -last one we first derive that rotation only depends on the shift -*modulo p* — this is why ℤ/(p), and not just ℕ, acts on M. +Damit ℤ/(p) *als Gruppe* wirkt, müssen wir die +Wirkungsaxiome nachprüfen: Der Shift um 0 tut nichts, und der +Shift um j + k ist der Shift um k gefolgt vom Shift um j. +Mathlib stellt `rotate_zero`, `rotate_rotate` und +`rotate_length` bereit (der Shift um die volle Länge p tut +nichts); daraus leiten wir zuerst ab, dass der Shift nur von +der Verschiebung *modulo p* abhängt — genau deshalb wirkt +ℤ/(p) und nicht bloß ℕ. -/ theorem rotate_mul {G : Type*} [Group G] {p : ℕ} - (v : vectorsProdEqOne G p) (q : ℕ) : rotate v (p * q) = v := by + (v : vectorsProdEqOne G p) (q : ℕ) : + rotate v (p * q) = v := by induction q with | zero => rw [Nat.mul_zero, rotate_zero] - | succ q ih => rw [Nat.mul_succ, ← rotate_rotate, ih, rotate_length] + | succ q ih => + rw [Nat.mul_succ, ← rotate_rotate, ih, rotate_length] theorem rotate_mod {G : Type*} [Group G] {p : ℕ} - (v : vectorsProdEqOne G p) (k : ℕ) : rotate v (k % p) = rotate v k := by + (v : vectorsProdEqOne G p) (k : ℕ) : + rotate v (k % p) = rotate v k := by calc rotate v (k % p) - = rotate (rotate v (k % p)) (p * (k / p)) := (rotate_mul _ _).symm - _ = rotate v (k % p + p * (k / p)) := rotate_rotate _ _ _ + = rotate (rotate v (k % p)) (p * (k / p)) := + (rotate_mul _ _).symm + _ = rotate v (k % p + p * (k / p)) := + rotate_rotate _ _ _ _ = rotate v k := by rw [Nat.mod_add_div] -/-- „Dazu lassen wir die zyklische Gruppe ℤ/(p) auf M durch zyklisches -Vertauschen wirken.“ — An element k of ℤ/(p) acts by rotating k places. +/-! +Nun können wir die Wirkung definieren. Zwei technische +Anmerkungen. Mathlibs `MulAction` erwartet eine +multiplikativ geschriebene Gruppe; ℤ/(p) = `ZMod p` ist aber +additiv geschrieben. Der Wrapper `Multiplicative` wechselt +die Notation. Die Voraussetzung `[NeZero p]` schließt p = 0 +aus, wo „Shift um eine Restklasse“ sinnlos wäre. +-/ -Two technical remarks. Mathlib's `MulAction` wants a multiplicatively -written group, while ℤ/(p) = `ZMod p` is written additively; the wrapper -`Multiplicative` performs the change of notation. The assumption -`[NeZero p]` excludes p = 0, where "rotation by a residue class" would -make no sense. -/ -instance rotateAction {G : Type*} [Group G] {p : ℕ} [NeZero p] : - MulAction (Multiplicative (ZMod p)) (vectorsProdEqOne G p) where - smul k v := rotate v (Multiplicative.toAdd k).val +/-- „Dazu lassen wir die zyklische Gruppe ℤ/(p) auf M durch +zyklisches Vertauschen wirken.“ — Ein Element k von ℤ/(p) +wirkt durch den Shift um k Stellen. -/ +instance rotateAction {G : Type*} [Group G] {p : ℕ} + [NeZero p] : + MulAction (Multiplicative (ZMod p)) + (vectorsProdEqOne G p) where + smul k v := rotate v (toAdd k).val one_smul v := by show rotate v (ZMod.val 0) = v rw [ZMod.val_zero, rotate_zero] mul_smul j k v := by - show rotate v ((Multiplicative.toAdd j + Multiplicative.toAdd k).val) = - rotate (rotate v (Multiplicative.toAdd k).val) (Multiplicative.toAdd j).val - rw [ZMod.val_add, rotate_mod, rotate_rotate, Nat.add_comm] + show rotate v ((toAdd j + toAdd k).val) = + rotate (rotate v (toAdd k).val) (toAdd j).val + rw [ZMod.val_add, rotate_mod, rotate_rotate, + Nat.add_comm] /-! -„Die Fixpunktmenge dieser Wirkung ist M₀ = { (a, …, a) ∈ G^p : a^p = e }.“ +„Die Fixpunktmenge dieser Wirkung ist +M₀ = { (a, …, a) ∈ G^p : a^p = e }.“ -In other words: a tuple is a fixed point if and only if it is constant, -i.e. its underlying list is `List.replicate p a` — the list (a, …, a) — -for some a. (The condition a^p = e then holds automatically, because -the entries of a tuple in M multiply to e.) The key step is the lemma -`List.rotate_one_eq_self_iff_eq_replicate`: a list that is unchanged by -the cyclic shift by *one* place is constant. +Mit anderen Worten: Ein Tupel ist genau dann ein Fixpunkt, +wenn es konstant ist, seine Liste also für ein a die Liste +`List.replicate p a` — das ist (a, …, a) — ist. (Die +Bedingung a^p = e gilt dann automatisch, weil sich die +Einträge eines Tupels in M zu e multiplizieren.) Der +Schlüsselschritt ist das Lemma +`List.rotate_one_eq_self_iff_eq_replicate`: Eine Liste, die +der zyklische Shift um *eine* Stelle nicht ändert, ist +konstant. -/ -theorem mem_fixedPoints_iff_replicate {G : Type*} [Group G] {p : ℕ} - [Fact (1 < p)] (v : vectorsProdEqOne G p) : - v ∈ fixedPoints (Multiplicative (ZMod p)) (vectorsProdEqOne G p) ↔ - ∃ a : G, (v : List.Vector G p).toList = List.replicate p a := by +theorem mem_fixedPoints_iff_replicate {G : Type*} [Group G] + {p : ℕ} [Fact (1 < p)] (v : vectorsProdEqOne G p) : + v ∈ fixedPoints (Multiplicative (ZMod p)) + (vectorsProdEqOne G p) ↔ + ∃ a : G, + (v : List.Vector G p).toList + = List.replicate p a := by rw [mem_fixedPoints] constructor - · -- A fixed point is in particular fixed by 1 ∈ ℤ/(p), so it is - -- unchanged by the cyclic shift by one place … + · -- Ein Fixpunkt wird insbesondere von 1 ∈ ℤ/(p) fixiert, + -- bleibt also unter dem Shift um eine Stelle + -- unverändert … intro hv have h1 : rotate v 1 = v := by - have h := hv (Multiplicative.ofAdd (1 : ZMod p)) - rwa [show Multiplicative.ofAdd (1 : ZMod p) • v - = rotate v (1 : ZMod p).val from rfl, ZMod.val_one] at h - -- … and hence constant. - obtain ⟨a, ha⟩ := List.rotate_one_eq_self_iff_eq_replicate.mp - (Subtype.ext_iff.mp (Subtype.ext_iff.mp h1)) + have h := hv (ofAdd (1 : ZMod p)) + rwa [show ofAdd (1 : ZMod p) • v + = rotate v (1 : ZMod p).val from rfl, + ZMod.val_one] at h + -- … und ist daher konstant. + obtain ⟨a, ha⟩ := + List.rotate_one_eq_self_iff_eq_replicate.mp + (Subtype.ext_iff.mp (Subtype.ext_iff.mp h1)) refine ⟨a, ha.trans ?_⟩ congr 1 exact (v : List.Vector G p).2 - · -- Conversely, a constant tuple is unchanged by every cyclic shift. + · -- Umgekehrt ändert kein zyklischer Shift ein konstantes + -- Tupel. rintro ⟨a, ha⟩ g apply Subtype.ext apply Subtype.ext - show (v : List.Vector G p).toList.rotate (Multiplicative.toAdd g).val = + show (v : List.Vector G p).toList.rotate (toAdd g).val = (v : List.Vector G p).toList rw [ha, List.rotate_replicate] /-! -Now we can put the pieces together, following the lecture notes line by -line. +Jetzt setzen wir die Teile zusammen und folgen dem Skript +Zeile für Zeile. -/ -theorem cauchy {G : Type*} [Group G] [Fintype G] {p : ℕ} (hp : p.Prime) - (hdvd : p ∣ Fintype.card G) : ∃ a : G, orderOf a = p := by - -- Register the consequences of primality that instance search needs: +theorem cauchy {G : Type*} [Group G] [Fintype G] {p : ℕ} + (hp : p.Prime) (hdvd : p ∣ Fintype.card G) : + ∃ a : G, orderOf a = p := by + -- Die Konsequenzen der Primalität, die die Instanzsuche + -- braucht: have : Fact p.Prime := ⟨hp⟩ have : NeZero p := ⟨hp.ne_zero⟩ have : Fact (1 < p) := ⟨hp.one_lt⟩ - -- „Wir erhalten die folgende Gleichung: |M| = |G^{p-1}| = |G|^{p-1}.“ - have hM : Nat.card (vectorsProdEqOne G p) = Fintype.card G ^ (p - 1) := by + -- Abkürzung: M₀ ist die Fixpunktmenge der Wirkung. + set M₀ := fixedPoints (Multiplicative (ZMod p)) + (vectorsProdEqOne G p) + -- „Wir erhalten die folgende Gleichung: + -- |M| = |G^{p-1}| = |G|^{p-1}.“ + have hM : Nat.card (vectorsProdEqOne G p) + = Fintype.card G ^ (p - 1) := by rw [Nat.card_eq_fintype_card, VectorsProdEqOne.card] - -- „Die zyklische Gruppe ℤ/(p)“ has order p = p¹, so the key lemma - -- applies to the rotation action and gives |M| ≡ |M₀| (mod p): - have hZp : Nat.card (Multiplicative (ZMod p)) = p ^ 1 := by + -- Die zyklische Gruppe ℤ/(p) hat Ordnung p = p¹; das + -- Schlüssellemma, angewandt auf die Rotationswirkung, + -- liefert |M| ≡ |M₀| (mod p): + have hZp : + Nat.card (Multiplicative (ZMod p)) = p ^ 1 := by simp [Nat.card_eq_fintype_card] have hcong : Nat.card (vectorsProdEqOne G p) ≡ - Nat.card (fixedPoints (Multiplicative (ZMod p)) - (vectorsProdEqOne G p)) [MOD p] := + Nat.card M₀ [MOD p] := key_lemma hp hZp (vectorsProdEqOne G p) - -- „Auf der anderen Seite folgt aus dem zentralen Schlüssellemma, dass + -- „Auf der anderen Seite folgt aus dem zentralen + -- Schlüssellemma, dass -- |M₀| ≡ |M| ≡ |G|^{p-1} ≡ 0 (mod p) ist.“ - have hdvdM0 : p ∣ Nat.card (fixedPoints (Multiplicative (ZMod p)) - (vectorsProdEqOne G p)) := by + have hdvdM0 : p ∣ Nat.card M₀ := by have hp1 : p - 1 ≠ 0 := by have := hp.one_lt; omega - have h0 : (0 : ℕ) ≡ Nat.card (fixedPoints (Multiplicative (ZMod p)) - (vectorsProdEqOne G p)) [MOD p] := + have h0 : (0 : ℕ) ≡ Nat.card M₀ [MOD p] := calc (0 : ℕ) ≡ Fintype.card G ^ (p - 1) [MOD p] := - (Nat.modEq_zero_iff_dvd.mpr (dvd_pow hdvd hp1)).symm + (Nat.modEq_zero_iff_dvd.mpr + (dvd_pow hdvd hp1)).symm _ = Nat.card (vectorsProdEqOne G p) := hM.symm - _ ≡ _ [MOD p] := hcong + _ ≡ Nat.card M₀ [MOD p] := hcong exact Nat.modEq_zero_iff_dvd.mp h0.symm - -- „Wegen (e, …, e) ∈ M₀ ist schon einmal klar, dass M₀ ≠ ∅ ist.“ + -- „Wegen (e, …, e) ∈ M₀ ist schon einmal klar, dass + -- M₀ ≠ ∅ ist.“ let v₀ : vectorsProdEqOne G p := - ⟨List.Vector.replicate p 1, (List.prod_replicate p 1).trans (one_pow p)⟩ - have hv₀ : v₀ ∈ fixedPoints (Multiplicative (ZMod p)) - (vectorsProdEqOne G p) := + ⟨List.Vector.replicate p 1, + (List.prod_replicate p 1).trans (one_pow p)⟩ + have hv₀ : v₀ ∈ M₀ := (mem_fixedPoints_iff_replicate v₀).mpr ⟨1, rfl⟩ - -- M₀ is nonempty and its cardinality is divisible by p ≥ 2, so - -- |M₀| ≥ p > 1 … - have := Fintype.ofFinite (fixedPoints (Multiplicative (ZMod p)) - (vectorsProdEqOne G p)) - have hlt : 1 < Fintype.card (fixedPoints (Multiplicative (ZMod p)) - (vectorsProdEqOne G p)) := by + -- M₀ ist nichtleer, und |M₀| ist durch p ≥ 2 teilbar, + -- also ist |M₀| ≥ p > 1 … + have := Fintype.ofFinite M₀ + have hlt : 1 < Fintype.card M₀ := by rw [← Nat.card_eq_fintype_card] - have hpos : 0 < Nat.card (fixedPoints (Multiplicative (ZMod p)) - (vectorsProdEqOne G p)) := Nat.card_pos_iff.mpr ⟨⟨⟨v₀, hv₀⟩⟩, inferInstance⟩ + have hpos : 0 < Nat.card M₀ := + Nat.card_pos_iff.mpr ⟨⟨⟨v₀, hv₀⟩⟩, inferInstance⟩ have := Nat.le_of_dvd hpos hdvdM0 have := hp.one_lt omega -- „Also existiert mindestens ein a ≠ e mit a^p = e.“ - obtain ⟨w, hw⟩ := Fintype.exists_ne_of_one_lt_card hlt ⟨v₀, hv₀⟩ - obtain ⟨a, ha⟩ := (mem_fixedPoints_iff_replicate w.1).mp w.2 - -- The tuple w lies in M, so its entries multiply to e; being constant, - -- this says exactly a^p = e: + obtain ⟨w, hw⟩ := + Fintype.exists_ne_of_one_lt_card hlt ⟨v₀, hv₀⟩ + obtain ⟨a, ha⟩ := + (mem_fixedPoints_iff_replicate w.1).mp w.2 + -- Das Tupel w liegt in M, sein Produkt ist also e; weil + -- das Tupel konstant ist, heißt das genau a^p = e: have hpow : a ^ p = 1 := by - have hprod : (w.1 : List.Vector G p).toList.prod = 1 := w.1.2 + have hprod : (w.1 : List.Vector G p).toList.prod = 1 := + w.1.2 rwa [ha, List.prod_replicate] at hprod - -- and a ≠ e, because otherwise w would be the tuple (e, …, e) = v₀: + -- Und a ≠ e, denn sonst wäre w das Tupel (e, …, e) = v₀: have hne : a ≠ 1 := by rintro rfl exact hw (Subtype.ext (Subtype.ext (Subtype.ext ha))) - -- „Nach Satz 17.4.11 hat a dann automatisch die Ordnung p.“ + -- „Nach Satz 17.4.11 hat a dann automatisch die + -- Ordnung p.“ exact ⟨a, orderOf_eq_prime hpow hne⟩ /-! -## Remarks +## Bemerkungen -1. Mathlib's version of Cauchy's theorem is - `exists_prime_orderOf_dvd_card`; its proof is the same - counting argument, organized slightly differently. +1. Mathlibs Fassung des Satzes von Cauchy heißt + `exists_prime_orderOf_dvd_card`; ihr Beweis ist dasselbe + Zählargument, nur etwas anders organisiert. -2. **Exercise** (Satz 18.2.3 of the lecture notes): *Es sei p eine - Primzahl und G eine nichttriviale Gruppe, deren Ordnung eine p-Potenz - ist. Dann ist das Zentrum von G nicht trivial.* Prove this in Lean: - apply `key_lemma` to the conjugation action of G on itself — or find - the statement in Mathlib. Replace the `sorry` below by a proof. +2. **Übungsaufgabe** (Satz 18.2.3 im Skript): *Es sei p eine + Primzahl und G eine nichttriviale Gruppe, deren Ordnung + eine p-Potenz ist. Dann ist das Zentrum von G nicht + trivial.* Beweisen Sie das in Lean: Wenden Sie + `key_lemma` auf die Konjugationswirkung von G auf sich + selbst an — oder finden Sie die Aussage in Mathlib. + Ersetzen Sie dazu das `sorry` durch einen Beweis. -/ -theorem center_nontrivial {p m : ℕ} (hp : p.Prime) {G : Type*} [Group G] - [Finite G] (hm : m ≠ 0) (hG : Nat.card G = p ^ m) : +theorem center_nontrivial {p m : ℕ} (hp : p.Prime) + {G : Type*} [Group G] [Finite G] (hm : m ≠ 0) + (hG : Nat.card G = p ^ m) : Nontrivial (Subgroup.center G) := by sorry diff --git a/AlgebraInLean/LittleFermat.lean b/AlgebraInLean/LittleFermat.lean index 553cec1..9231c17 100644 --- a/AlgebraInLean/LittleFermat.lean +++ b/AlgebraInLean/LittleFermat.lean @@ -1,104 +1,120 @@ /- -Algebra in Lean — pilot file +Algebra in Lean — Kapitel 17 ============================ -This file accompanies the lecture notes "Algebra und Zahlentheorie" -(Stefan Kebekus, CC-BY 4.0). It translates one theorem of the notes — -Fermat's little theorem, `satz:kleinerFermat` in Chapter 17 — into Lean, -following the proof of the lecture notes *sentence by sentence*. The -German original of every sentence is quoted as a comment directly above -the Lean code that implements it, so you can see how the standard -phrases of a lecture-style proof translate into Lean/Mathlib. +Diese Datei begleitet das Skript „Algebra und Zahlentheorie“ +(Stefan Kebekus, CC-BY 4.0). Sie übersetzt einen Satz des +Skripts — den kleinen Satz von Fermat, `satz:kleinerFermat` in +Kapitel 17 — Satz für Satz nach Lean. Das deutsche Original +jedes Beweissatzes steht als Kommentar direkt über dem +Lean-Code, der ihn umsetzt: So sieht man, wie sich die +Standardphrasen eines Vorlesungsbeweises nach Lean/Mathlib +übersetzen. -/ import Mathlib namespace AlgebraInLean /-! -# Fermat's little theorem +# Der kleine Satz von Fermat -**Satz (Kleiner Satz von Fermat).** *Es sei p ∈ ℕ eine Primzahl und es -sei a ∈ ℤ irgendeine Zahl. Dann ist a^p ≡ a (mod p).* +**Satz (Kleiner Satz von Fermat).** *Es sei p ∈ ℕ eine +Primzahl und es sei a ∈ ℤ irgendeine Zahl. Dann ist +a^p ≡ a (mod p).* -Before we can state this in Lean, we need to know how Mathlib speaks -about the objects involved. +Bevor wir das in Lean formulieren können, müssen wir wissen, +wie Mathlib über die beteiligten Objekte spricht. -* The ring ℤ/(p) of residue classes is called `ZMod p`. The residue - class of an integer `a : ℤ` is written `(a : ZMod p)` — Lean inserts - the canonical ring morphism ℤ → ℤ/(p) automatically ("coercion"). +* Der Restklassenring ℤ/(p) heißt `ZMod p`. Die Restklasse + einer ganzen Zahl `a : ℤ` schreibt sich `(a : ZMod p)` — + Lean fügt den kanonischen Ringmorphismus ℤ → ℤ/(p) + automatisch ein („Koerzion“). -* The congruence `a ≡ b (mod p)` for integers is written +* Die Kongruenz a ≡ b (mod p) für ganze Zahlen schreibt sich `a ≡ b [ZMOD p]`. -* The multiplicative group 𝔽_p^* is the group of *units* of the ring - `ZMod p`, written `(ZMod p)ˣ`. A unit `u : (ZMod p)ˣ` remembers its - inverse; the underlying ring element is again written `(u : ZMod p)`. +* Die multiplikative Gruppe 𝔽_p^* ist die Gruppe der + *Einheiten* des Rings `ZMod p`, geschrieben `(ZMod p)ˣ`. + Eine Einheit `u : (ZMod p)ˣ` kennt ihr Inverses; das + zugrundeliegende Ringelement schreibt sich wieder + `(u : ZMod p)`. -* The lecture notes say "es sei p eine Primzahl". In Lean we carry the - primality of `p` as a hypothesis `hp : p.Prime`. Some facts — - for instance that ℤ/(p) is a field — are found by Lean's automation - only if the hypothesis is registered as an *instance*; this is what - the first line `have : Fact p.Prime := ⟨hp⟩` of the proof does. +* Das Skript sagt „es sei p eine Primzahl“. In Lean führen + wir die Primalität von `p` als Hypothese `hp : p.Prime` mit. + Manche Tatsachen — etwa dass ℤ/(p) ein Körper ist — findet + Leans Automatisierung nur, wenn die Hypothese als *Instanz* + registriert ist; genau das leistet die erste Beweiszeile + `have : Fact p.Prime := ⟨hp⟩`. -/ theorem little_fermat (p : ℕ) (hp : p.Prime) (a : ℤ) : a ^ p ≡ a [ZMOD p] := by have : Fact p.Prime := ⟨hp⟩ - -- The congruence „a^p ≡ a (mod p)“ means precisely that a^p and a have - -- the same residue class in ℤ/(p). So we may prove an *equation* in - -- the ring `ZMod p` instead; the goal becomes + -- Die Kongruenz „a^p ≡ a (mod p)“ bedeutet gerade, dass + -- a^p und a dieselbe Restklasse in ℤ/(p) haben. Wir + -- dürfen also stattdessen eine *Gleichung* im Ring + -- `ZMod p` zeigen; das Beweisziel wird -- (a : ZMod p) ^ p = (a : ZMod p). rw [← ZMod.intCast_eq_intCast_iff] push_cast -- „Falls a ein Vielfaches von p ist, ist die Sache klar.“ by_cases ha : (a : ZMod p) = 0 · rw [ha, zero_pow hp.ne_zero] - -- „Ansonsten liefert die Restklasse von a ein nicht-verschwindendes - -- Element ā ∈ ℤ/(p) = 𝔽_p, also ein Element der multiplikativen - -- Gruppe 𝔽_p^*, …“ - · obtain ⟨u, hu⟩ : IsUnit (a : ZMod p) := isUnit_iff_ne_zero.mpr ha + -- „Ansonsten liefert die Restklasse von a ein + -- nicht-verschwindendes Element ā ∈ ℤ/(p) = 𝔽_p, also + -- ein Element der multiplikativen Gruppe 𝔽_p^*, …“ + · obtain ⟨u, hu⟩ : IsUnit (a : ZMod p) := + isUnit_iff_ne_zero.mpr ha rw [← hu] -- „… welche p−1 Elemente hat.“ have card_units : Nat.card (ZMod p)ˣ = p - 1 := by rw [Nat.card_eq_fintype_card, ZMod.card_units p] - -- „Nach Satz 17.3.6 («Satz von Lagrange») ist die Ordnung von ā, also - -- die Größe der von ā erzeugten Untergruppe, ein Teiler von - -- |𝔽_p^*| = p−1.“ - have lagrange : Nat.card (Subgroup.zpowers u) ∣ Nat.card (ZMod p)ˣ := + -- „Nach Satz 17.3.6 («Satz von Lagrange») ist die + -- Ordnung von ā, also die Größe der von ā erzeugten + -- Untergruppe, ein Teiler von |𝔽_p^*| = p−1.“ + have lagrange : + Nat.card (Subgroup.zpowers u) ∣ + Nat.card (ZMod p)ˣ := Subgroup.card_subgroup_dvd_card (Subgroup.zpowers u) have ord_dvd : orderOf u ∣ p - 1 := by rw [← Nat.card_zpowers, ← card_units] exact lagrange -- „Es gilt also ā^(p−1) = 1 ∈ 𝔽_p^* …“ - have pow_eq_one : u ^ (p - 1) = 1 := orderOf_dvd_iff_pow_eq_one.mp ord_dvd + have pow_eq_one : u ^ (p - 1) = 1 := + orderOf_dvd_iff_pow_eq_one.mp ord_dvd -- „… oder äquivalent a^p ≡ a (mod p).“ have key : u ^ p = u := by - calc u ^ p = u ^ (p - 1 + 1) := by rw [Nat.sub_add_cancel hp.one_lt.le] + calc u ^ p + = u ^ (p - 1 + 1) := by + rw [Nat.sub_add_cancel hp.one_lt.le] _ = u ^ (p - 1) * u := pow_succ u (p - 1) _ = u := by rw [pow_eq_one, one_mul] exact_mod_cast congrArg Units.val key /-! -## Remarks +## Bemerkungen -1. Mathlib of course already contains Fermat's little theorem; the - statement about residue classes is `ZMod.pow_card`. You can find - such lemmas yourself with the tactic `exact?`, or by searching on - https://leansearch.net or https://loogle.lean-lang.org. +1. Mathlib enthält den kleinen Satz von Fermat natürlich + längst; die Aussage über Restklassen heißt `ZMod.pow_card`. + Solche Lemmata findet man selbst mit der Taktik `exact?` + oder über die Suchmaschinen https://leansearch.net und + https://loogle.lean-lang.org. -/ example (p : ℕ) [Fact p.Prime] (a : ZMod p) : a ^ p = a := ZMod.pow_card a /-! -2. **Exercise** (this is `bem:kleinerFermat` of the lecture notes). - In applications one often uses the equivalent formulation - a^(p−1) ≡ 1 (mod p) for a not divisible by p. Derive it from - `little_fermat` — or give a direct proof following the ideas above. - Replace the `sorry` below by a proof. +2. **Übungsaufgabe** (das ist `bem:kleinerFermat` im Skript). + In Anwendungen verwendet man häufig die äquivalente + Formulierung a^(p−1) ≡ 1 (mod p) für nicht durch p + teilbare a. Leiten Sie sie aus `little_fermat` ab — oder + geben Sie einen direkten Beweis nach den Ideen oben. + Ersetzen Sie dazu das `sorry` durch einen Beweis. -/ -theorem little_fermat' (p : ℕ) (hp : p.Prime) (a : ℤ) (ha : ¬ (p : ℤ) ∣ a) : +theorem little_fermat' (p : ℕ) (hp : p.Prime) (a : ℤ) + (ha : ¬ (p : ℤ) ∣ a) : a ^ (p - 1) ≡ 1 [ZMOD p] := by sorry diff --git a/AlgebraInLean/Quotients.lean b/AlgebraInLean/Quotients.lean new file mode 100644 index 0000000..a0d8a2c --- /dev/null +++ b/AlgebraInLean/Quotients.lean @@ -0,0 +1,141 @@ +/- +Algebra in Lean — Kapitel 2 und 17 des Skripts +============================================== + +Diese Datei begleitet das Skript „Algebra und Zahlentheorie“ +(Stefan Kebekus, CC-BY 4.0). Sie übersetzt zwei Beweise +rund um normale Untergruppen Satz für Satz nach Lean: das +Argument aus Erklärvideo 1-1, dass Kerne von +Gruppenmorphismen abgeschlossen unter Konjugation sind, und +die Beobachtung aus Kapitel 17.3, dass U·N für normales N +wieder eine Untergruppe ist. +-/ +import Mathlib + +namespace AlgebraInLean + +/-! +# Gruppen und Restklassengruppen + +## Das Mathlib-Wörterbuch + +* Eine Untergruppe von `G` ist ein Term `U : Subgroup G`; + die Mitgliedschaft schreibt sich `g ∈ U`. Dass `U` unter + Verknüpfung und Inversen abgeschlossen ist, sagen + `U.mul_mem` und `U.inv_mem`. + +* Ein Gruppenmorphismus ist ein Term `φ : G →* Q` — ein + „Bündel“ aus der Abbildung und den Nachweisen + `map_mul : φ (a·b) = φ a · φ b` (daraus folgen + `map_one` und `map_inv`). Der Kern heißt `φ.ker`; + Mitgliedschaft entpackt `MonoidHom.mem_ker : g ∈ φ.ker ↔ + φ g = 1`. + +* „N ist Normalteiler“ ist das Prädikat `N.Normal`; sein + Feld `conj_mem` ist wörtlich die Bedingung des Skripts: + für alle n ∈ N und alle g ist g·n·g⁻¹ ∈ N. + +## Kerne sind abgeschlossen unter Konjugation + +Aus Erklärvideo 1-1 des Skripts: „Die Antwort lautet: Nein! +Nicht jede Untergruppe kann Kern eines Gruppenmorphismus +sein. Wenn es nämlich einen Gruppenmorphismus φ : G → Q +mit H = ker(φ) gibt, dann gilt für alle h ∈ H und alle +g ∈ G die Gleichung φ(g·h·g⁻¹) = φ(g)·φ(h)·φ(g)⁻¹ = e, +wobei e das neutrale Element von Q bezeichnet. Also ist +stets g·h·g⁻¹ ∈ H.“ +-/ + +theorem kern_konjugation {G Q : Type*} [Group G] [Group Q] + (φ : G →* Q) {h : G} (hh : h ∈ φ.ker) (g : G) : + g * h * g⁻¹ ∈ φ.ker := by + -- Mitgliedschaft im Kern heißt: φ bildet auf 1 ab. + rw [MonoidHom.mem_ker] at hh ⊢ + -- „… dann gilt für alle h ∈ H und alle g ∈ G die + -- Gleichung φ(g·h·g⁻¹) = φ(g)·φ(h)·φ(g)⁻¹ = e …“ + calc φ (g * h * g⁻¹) + = φ g * φ h * (φ g)⁻¹ := by + rw [map_mul, map_mul, map_inv] + _ = φ g * 1 * (φ g)⁻¹ := by rw [hh] + _ = 1 := by rw [mul_one, mul_inv_cancel] + +/-! +Mathlib kennt diese Aussage als Instanz +`MonoidHom.normal_ker : φ.ker.Normal`. Das Skript fährt +fort: Untergruppen mit dieser Eigenschaft heißen *normale +Untergruppen* oder *Normalteiler*, und die Bedingung ist +nicht nur notwendig, sondern auch hinreichend — zu jedem +Normalteiler existiert eine Restklassengruppe. + +## Die Restklassengruppe und ihre universelle Eigenschaft + +Das Skript definiert die Restklassengruppe in Kapitel 17.3 +über ihre universelle Eigenschaft. Mathlib stellt beides +bereit: den Quotienten `G ⧸ N` mit der Quotientenabbildung +`QuotientGroup.mk' N : G →* G ⧸ N`, und die universelle +Eigenschaft als `QuotientGroup.lift` — gegeben ein +Morphismus `α : G →* H` mit `N ≤ α.ker` erhält man den +induzierten Morphismus auf dem Quotienten: +-/ + +example {G H : Type*} [Group G] [Group H] (N : Subgroup G) + [N.Normal] (α : G →* H) (hα : N ≤ α.ker) : + G ⧸ N →* H := + QuotientGroup.lift N α + fun _g hg => MonoidHom.mem_ker.mp (hα hg) + +/-! +## Die Beobachtung: U·N ist eine Untergruppe + +Beobachtung aus Kapitel 17.3 des Skripts: „Es sei G eine +Gruppe und es sei N ⊂ G eine normale Untergruppe. Weiter +sei U ⊂ G irgendeine Untergruppe. Dann ist +U·N = {u·n : u ∈ U, n ∈ N} wieder eine Untergruppe. Zum +Beweis müssen wir lediglich zeigen, dass U·N abgeschlossen +unter der Gruppenoperation ist. … Wir wissen, dass N +normal ist. Also ist ñ₁ := u₂⁻¹·n₁·u₂ ∈ N und es gilt +n₁·u₂ = u₂·ñ₁. Demnach ist u₁n₁u₂n₂ = u₁u₂·ñ₁n₂ ∈ U·N. +Fertig ist der Beweis.“ + +Neu ist hier die Taktik `group`: Sie verrechnet Ausdrücke, +die allein mit den Gruppenaxiomen ineinander umgeformt +werden können. (Das ñ₁ des Skripts heißt im Code `n₁'`.) +-/ + +theorem mul_mem_mul {G : Type*} [Group G] (U N : Subgroup G) + (hN : N.Normal) {u₁ n₁ u₂ n₂ : G} + (hu₁ : u₁ ∈ U) (hn₁ : n₁ ∈ N) + (hu₂ : u₂ ∈ U) (hn₂ : n₂ ∈ N) : + ∃ u ∈ U, ∃ n ∈ N, u₁ * n₁ * (u₂ * n₂) = u * n := by + -- „Wir wissen, dass N normal ist. Also ist + -- ñ₁ := u₂⁻¹·n₁·u₂ ∈ N …“ + set n₁' := u₂⁻¹ * n₁ * u₂ with hn₁'_def + have hn₁' : n₁' ∈ N := hN.conj_mem' n₁ hn₁ u₂ + -- „… und es gilt n₁·u₂ = u₂·ñ₁.“ + have hswap : n₁ * u₂ = u₂ * n₁' := by + rw [hn₁'_def]; group + -- „Demnach ist u₁n₁u₂n₂ = u₁u₂·ñ₁n₂ ∈ U·N.“ + refine ⟨u₁ * u₂, U.mul_mem hu₁ hu₂, + n₁' * n₂, N.mul_mem hn₁' hn₂, ?_⟩ + calc u₁ * n₁ * (u₂ * n₂) + = u₁ * (n₁ * u₂) * n₂ := by group + _ = u₁ * (u₂ * n₁') * n₂ := by rw [hswap] + _ = u₁ * u₂ * (n₁' * n₂) := by group + +/-! +## Übungsaufgabe + +Das Skript sagt, es sei „lediglich“ die Abgeschlossenheit +unter der Verknüpfung zu zeigen — prüfen Sie den +verschwiegenen Teil selbst nach: U·N ist auch abgeschlossen +unter Inversen. Tipp: (u·n)⁻¹ = u⁻¹·(u·n⁻¹·u⁻¹), und der +zweite Faktor liegt nach `conj_mem` in N. Ersetzen Sie das +`sorry` durch einen Beweis. +-/ + +theorem inv_mem_mul {G : Type*} [Group G] (U N : Subgroup G) + (hN : N.Normal) {u n : G} (hu : u ∈ U) (hn : n ∈ N) : + ∃ u' ∈ U, ∃ n' ∈ N, (u * n)⁻¹ = u' * n' := by + sorry + +end AlgebraInLean diff --git a/Book.lean b/Book.lean new file mode 100644 index 0000000..e7d7a34 --- /dev/null +++ b/Book.lean @@ -0,0 +1,65 @@ +-- Wurzeldokument der Kursnotizen: `lake exe algebrainlean +-- --output _out` rendert sie nach `_out/html-multi`. +import VersoManual + +import Book.Basics +import Book.Quotients +import Book.LittleFermat +import Book.Cauchy + +open Verso.Genre Manual + +#doc (Manual) "Algebra in Lean" => +%%% +tag := "algebra-in-lean" +authors := ["Stefan Kebekus"] +%%% + +Dies sind Kursnotizen zum Formalisieren von Algebra mit dem +interaktiven Theorembeweiser [Lean](https://lean-lang.org) und seiner +Mathematikbibliothek [Mathlib](https://leanprover-community.github.io). +Sie begleiten die Vorlesung _Algebra und Zahlentheorie_ (Universität +Freiburg, Winter 2026/27) mit ihrem deutschsprachigen Skript und sind +dafür gedacht, parallel zum Kurs +[_Interactive Theorem Proving using Lean_](https://pfaffelh.github.io/leancourse/) +von Peter Pfaffelhuber gelesen zu werden, der Lean selbst einführt. +Wir gehen hier den komplementären Weg: Wir nehmen an, dass Sie gerade +die _Algebra_ lernen, und zeigen an Beispielen, dass sich der Stoff der +Vorlesung mit vertretbarem Aufwand formalisieren lässt — und dass das +sogar richtig Spaß machen kann. + +*Wie diese Notizen funktionieren.* Ausgewählte Beweise aus dem Skript +werden _Satz für Satz_ nach Lean übersetzt: Jeder deutsche Originalsatz +steht als Kommentar direkt über dem Lean-Code, der ihn umsetzt. Sie +werden sehen, dass die Standardphrasen eines Vorlesungsbeweises — +„Ansonsten liefert die Restklasse …“, „Nach dem Satz von Lagrange …“, +„Also existiert mindestens ein …“ — wiedererkennbaren Zügen in Lean +entsprechen. Jedes Kapitel endet mit einer Übungsaufgabe, deren +`sorry` Sie durch einen Beweis ersetzen sollen. + +*Woher bekomme ich das Material?* Die Übungsdateien liegen im selben +Repository wie diese Notizen, im Ordner `AlgebraInLean/`: + +``` +git clone https://git.cplx.vm.uni-freiburg.de/kebekus/AlgebraInLean.git +cd AlgebraInLean +lake exe cache get +code . +``` + +Öffnen Sie dann zum Beispiel `AlgebraInLean/LittleFermat.lean`. +Warten Sie, bis die orangefarbenen Balken verschwinden; setzen Sie den +Cursor in einen Beweis und beobachten Sie, wie das _Infoview_-Panel +den aktuellen Beweiszustand anzeigt. Für die Installation von Lean +und VS Code selbst folgen Sie der +[Anleitung der Lean-Community](https://leanprover-community.github.io/get_started.html) +oder der ausführlichen Anleitung am Anfang der +[Notizen von Peter Pfaffelhuber](https://pfaffelh.github.io/leancourse/). + +{include 0 Book.Basics} + +{include 0 Book.Quotients} + +{include 0 Book.LittleFermat} + +{include 0 Book.Cauchy} diff --git a/Book/Basics.lean b/Book/Basics.lean new file mode 100644 index 0000000..7415889 --- /dev/null +++ b/Book/Basics.lean @@ -0,0 +1,100 @@ +import VersoManual +import Manual.Meta +-- Minimal-Imports statt `import Mathlib`: ermittelt mit +-- `#min_imports`, ergänzt um Taktik-Module und ℚ. +import Mathlib.Algebra.Group.Basic +import Mathlib.Data.Rat.Defs +import Mathlib.Tactic.Ring +import Mathlib.Tactic.NormNum + +open Verso.Genre Manual +open Verso.Genre.Manual.InlineLean + +set_option pp.rawOnError true +set_option verso.docstring.allowMissing true + +#doc (Manual) "Erste Schritte" => +%%% +htmlSplit := .never +tag := "erste-schritte" +%%% + +Ein Beweis in Lean beginnt mit `by` und besteht aus _Taktiken_. +Öffnen Sie die Übungsdatei `AlgebraInLean/Basics.lean`, setzen Sie den +Cursor in einen Beweis und beobachten Sie das _Infoview_-Panel: Es +zeigt zu jedem Zeitpunkt die Hypothesen und das noch zu zeigende Ziel. + +# Vier Taktiken für den Anfang + +* `norm_num` verrechnet konkrete Zahlen, +* `ring` beweist Gleichungen, die in jedem kommutativen Ring gelten, +* `omega` löst lineare Arithmetik über `ℕ` und `ℤ`, +* `rw [h]` schreibt mit einer Gleichung `h` um. + +```lean +example : 2 + 3 = 5 := by norm_num + +example (a b : ℤ) : + (a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2 := by + ring + +example (n : ℕ) (h : n < 5) : n ≤ 7 := by omega + +example (x y : ℚ) (h : x = y) : x + 1 = y + 1 := by + rw [h] +``` + +# Gruppen in Mathlib + +Definition 2.1.1 des Skripts: + +> *Definition (Gruppe).* Eine Gruppe ist eine nicht-leere Menge `G` +> mit einer Abbildung `m : G ⨯ G → G`, sodass folgende Eigenschaften +> gelten. _Assoziativität:_ … _Neutrales Element:_ Es gibt genau ein +> Element `e` aus `G`, sodass für alle `a` aus `G` gilt: +> `m(e,a) = m(a,e) = a`. _Inverse Elemente:_ Für alle `a` aus `G` +> gibt es genau ein Element `b` aus `G`, sodass `m(a,b) = m(b,a) = e` +> ist. + +In Mathlib ist eine Gruppe eine Typklasse `Group G`. Die Verknüpfung +schreibt sich `a * b`, das neutrale Element `1`, das Inverse `a⁻¹`; +die Axiome heißen `mul_assoc`, `one_mul`, `mul_one`, +`inv_mul_cancel`, `mul_inv_cancel`. + +Ein Unterschied fällt auf: Das Skript _fordert_ die Eindeutigkeit von +neutralem Element und Inversen, Mathlib nicht. Das ist kein +Widerspruch — die Eindeutigkeit folgt aus den übrigen Axiomen. Genau +das beweisen wir jetzt, mit dem Standardargument aus der linearen +Algebra: + +> Es sei `e` ein weiteres Element, das von links neutral wirkt. Weil +> `1` neutral ist, gilt `e·1 = e`. Weil `e` neutral wirkt, gilt +> `e·1 = 1`. Zusammen folgt `e = e·1 = 1`. + +Der Lean-Beweis benutzt die Taktik `calc`, die eine Kette von +Gleichungen Schritt für Schritt abarbeitet — das Lean-Gegenstück zur +Gleichungskette am Ende des Textbeweises: + +```lean +theorem neutral_eindeutig {G : Type*} [Group G] (e : G) + (he : ∀ a : G, e * a = a) : e = 1 := by + -- „Weil 1 neutral ist, gilt e·1 = e. Weil e neutral + -- wirkt, gilt e·1 = 1. Zusammen folgt e = e·1 = 1.“ + calc e = e * 1 := (mul_one e).symm + _ = 1 := he 1 +``` + +# Übungsaufgabe + +Zeigen Sie ebenso die Eindeutigkeit des Inversen: Wenn `a·b = 1` ist, +dann ist `b` bereits _das_ Inverse `a⁻¹`. Das Standardargument: +`b = 1·b = (a⁻¹·a)·b = a⁻¹·(a·b) = a⁻¹·1 = a⁻¹`. Öffnen Sie +`AlgebraInLean/Basics.lean` und ersetzen Sie das `sorry` durch einen +Beweis — `calc` und die Lemmata `one_mul`, `inv_mul_cancel`, +`mul_assoc`, `mul_one` genügen. + +```lean +theorem inverses_eindeutig {G : Type*} [Group G] (a b : G) + (h : a * b = 1) : b = a⁻¹ := by + sorry +``` diff --git a/Book/Cauchy.lean b/Book/Cauchy.lean new file mode 100644 index 0000000..cfe448e --- /dev/null +++ b/Book/Cauchy.lean @@ -0,0 +1,328 @@ +import VersoManual +import Manual.Meta +-- Minimal-Imports statt `import Mathlib`: lädt nur den +-- benötigten Teil der Bibliothek und beschleunigt den +-- Buch-Build erheblich (ermittelt mit `#min_imports`). +import Mathlib.Algebra.Field.ZMod +import Mathlib.Algebra.Torsor.Defs +import Mathlib.Data.Finite.Vector +import Mathlib.GroupTheory.PGroup +import Mathlib.GroupTheory.Perm.Cycle.Type + +open Verso.Genre Manual +open Verso.Genre.Manual.InlineLean + +set_option pp.rawOnError true +set_option verso.docstring.allowMissing true + +#doc (Manual) "Das Schlüssellemma und der Satz von Cauchy" => +%%% +htmlSplit := .never +tag := "cauchy" +%%% + +Dieses Kapitel übersetzt den Anfang von Kapitel 18 des Skripts: das +_zentrale Schlüssellemma_ über Fixpunkte von p-Gruppen-Wirkungen und +seine erste Anwendung, den Satz von Cauchy. Unterwegs werden wir zum +ersten Mal selbst etwas in Lean _definieren_: eine Gruppenwirkung. + +# Das zentrale Schlüssellemma + +> *Lemma (Zentrales Schlüssellemma).* Es sei `m ∈ ℕ` und es sei `p` +> eine Primzahl. Weiter sei `G` eine Gruppe der Ordnung `p^m`, die auf +> einer endlichen Menge `M` operiert. Weiter sei +> `M₀ = { m ∈ M : ∀ g ∈ G: g·m = m }` die Menge der Fixpunkte. Dann +> ist `|M| ≡ |M₀| (mod p)`. + +Wie sagt Mathlib das alles? + +* „eine Gruppe der Ordnung `p^m`“: Dafür hat Mathlib ein Prädikat + `IsPGroup p G` („die Ordnung jedes Elements ist eine p-Potenz“ — für + endliche Gruppen ist das äquivalent, nach dem Satz von Cauchy + unten!). Das Lemma `IsPGroup.of_card` übersetzt unsere + Voraussetzung `Nat.card G = p ^ m` dorthin. + +* „die auf einer endlichen Menge `M` operiert“: Eine Wirkung von `G` + auf `M` ist eine Typklasse, `[MulAction G M]`; die Endlichkeit von + `M` ist die Typklasse `[Finite M]`. + +* Die Fixpunktmenge heißt `MulAction.fixedPoints G M`, und die + Kongruenz `|M| ≡ |M₀| (mod p)` schreibt sich + `Nat.card M ≡ Nat.card (fixedPoints G M) [MOD p]`. + +Der Beweis im Skript zerlegt `M` in Bahnen und zitiert die +Bahnengleichung. Genau so beweist Mathlib die Aussage +`IsPGroup.card_modEq_card_fixedPoints` — wir zitieren hier also +einfach die Bibliothek, so wie das Skript an dieser Stelle Satz 17.2.6 +zitiert. + +```lean +open Equiv.Perm Equiv.Perm.VectorsProdEqOne +open MulAction Multiplicative + +theorem key_lemma {p m : ℕ} (hp : p.Prime) {G : Type*} + [Group G] (hG : Nat.card G = p ^ m) (M : Type*) + [Finite M] [MulAction G M] : + Nat.card M ≡ Nat.card (fixedPoints G M) [MOD p] := by + have : Fact p.Prime := ⟨hp⟩ + exact (IsPGroup.of_card hG).card_modEq_card_fixedPoints M +``` + +(Die `open`-Zeilen holen die Namensräume, die wir in diesem Kapitel +brauchen, in den Geltungsbereich; `Equiv.Perm.vectorsProdEqOne` tritt +gleich auf.) + +# Der Satz von Cauchy + +> *Satz (Satz von Cauchy).* Wenn die Ordnung einer endlichen Gruppe +> durch `p` teilbar ist, dann existiert ein Element von Ordnung `p`. + +Wir folgen dem Beweis des Skripts Satz für Satz. Der Beweis +konstruiert eine raffinierte Hilfsmenge mit einer raffinierten +Gruppenwirkung — diesmal gibt es also vor dem eigentlichen Satz echte +Arbeit: Wir bauen erst die Menge und die Wirkung. + +## Die Menge M + +„Betrachte die Menge +`M = { (a₁, …, a_p) ∈ G ⨯ ⋯ ⨯ G : a₁·a₂ ⋯ a_p = e }`.“ + +Mathlib kennt diese Menge. Ein p-Tupel von Gruppenelementen ist ein +`List.Vector G p` — eine Liste der Länge `p` —, und `M` ist die Menge +`Equiv.Perm.vectorsProdEqOne G p` aller Vektoren, deren Einträge sich +zu 1 multiplizieren. + +„Gegeben ein Tupel `(a₁, …, a_p) ∈ M`, dann stellen wir erst einmal +fest, dass der letzte Eintrag des Tupels durch die ersten Einträge +eindeutig bestimmt ist, `a_p = (a₁ ⋯ a_{p-1})⁻¹`. Wir erhalten die +folgende Gleichung: `|M| = |G^{p-1}| = |G|^{p-1}`.“ + +Auch das steht schon in Mathlib: Die Bijektion +`(a₁, …, a_{p-1}) ↦ (a₁, …, a_{p-1}, (a₁ ⋯ a_{p-1})⁻¹)` heißt +`VectorsProdEqOne.vectorEquiv`, und die resultierende Zählformel ist +`VectorsProdEqOne.card`: + +``` +VectorsProdEqOne.card (G : Type) [Group G] (n : ℕ) [Fintype G] : + Fintype.card ↑(vectorsProdEqOne G n) = Fintype.card G ^ (n - 1) +``` + +## Die Wirkung von ℤ/(p) + +„Als Nächstes brauchen wir eine schicke Gruppenwirkung, denn wir +wollen das zentrale Schlüssellemma anwenden. Dazu lassen wir die +zyklische Gruppe `ℤ/(p)` auf `M` durch zyklisches Vertauschen wirken.“ + +Der zyklische Shift eines Vektors `v ∈ vectorsProdEqOne G p` um `k` +Stellen ist `VectorsProdEqOne.rotate v k`. Die Fußnote des Skripts — +der Shift bildet `M` auf sich ab, „weil in jeder Gruppe aus `a·b = e` +auch `b·a = e` gilt“ — ist das Mathlib-Lemma +`List.prod_rotate_eq_one_of_prod_eq_one`, das schon in der Definition +von `rotate` steckt. + +Damit `ℤ/(p)` _als Gruppe_ wirkt, müssen wir die Wirkungsaxiome +nachprüfen: Der Shift um 0 tut nichts, und der Shift um `j + k` ist +der Shift um `k` gefolgt vom Shift um `j`. Mathlib stellt +`rotate_zero`, `rotate_rotate` und `rotate_length` bereit (der Shift +um die volle Länge `p` tut nichts); daraus leiten wir zuerst ab, dass +der Shift nur von der Verschiebung _modulo p_ abhängt — genau deshalb +wirkt `ℤ/(p)` und nicht bloß `ℕ`. + +```lean +theorem rotate_mul {G : Type*} [Group G] {p : ℕ} + (v : vectorsProdEqOne G p) (q : ℕ) : + rotate v (p * q) = v := by + induction q with + | zero => rw [Nat.mul_zero, rotate_zero] + | succ q ih => + rw [Nat.mul_succ, ← rotate_rotate, ih, rotate_length] + +theorem rotate_mod {G : Type*} [Group G] {p : ℕ} + (v : vectorsProdEqOne G p) (k : ℕ) : + rotate v (k % p) = rotate v k := by + calc rotate v (k % p) + = rotate (rotate v (k % p)) (p * (k / p)) := + (rotate_mul _ _).symm + _ = rotate v (k % p + p * (k / p)) := + rotate_rotate _ _ _ + _ = rotate v k := by rw [Nat.mod_add_div] +``` + +Nun können wir die Wirkung definieren. Zwei technische Anmerkungen. +Mathlibs `MulAction` erwartet eine multiplikativ geschriebene Gruppe; +`ℤ/(p)` = `ZMod p` ist aber additiv geschrieben. Der Wrapper +`Multiplicative` wechselt die Notation (dank +`open Multiplicative` oben schreiben sich seine Übergänge kurz +`toAdd` und `ofAdd`). Die Voraussetzung `[NeZero p]` schließt +`p = 0` aus, wo „Shift um eine Restklasse“ sinnlos wäre. + +```lean +instance rotateAction {G : Type*} [Group G] {p : ℕ} + [NeZero p] : + MulAction (Multiplicative (ZMod p)) + (vectorsProdEqOne G p) where + smul k v := rotate v (toAdd k).val + one_smul v := by + show rotate v (ZMod.val 0) = v + rw [ZMod.val_zero, rotate_zero] + mul_smul j k v := by + show rotate v ((toAdd j + toAdd k).val) = + rotate (rotate v (toAdd k).val) (toAdd j).val + rw [ZMod.val_add, rotate_mod, rotate_rotate, + Nat.add_comm] +``` + +## Die Fixpunkte + +„Die Fixpunktmenge dieser Wirkung ist +`M₀ = { (a, …, a) ∈ G^p : a^p = e }`.“ + +Mit anderen Worten: Ein Tupel ist genau dann ein Fixpunkt, wenn es +konstant ist, seine Liste also für ein `a` die Liste +`List.replicate p a` — das ist `(a, …, a)` — ist. (Die Bedingung +`a^p = e` gilt dann automatisch, weil sich die Einträge eines Tupels +in `M` zu `e` multiplizieren.) Der Schlüsselschritt ist das Lemma +`List.rotate_one_eq_self_iff_eq_replicate`: Eine Liste, die der +zyklische Shift um _eine_ Stelle nicht ändert, ist konstant. + +```lean +theorem mem_fixedPoints_iff_replicate {G : Type*} [Group G] + {p : ℕ} [Fact (1 < p)] (v : vectorsProdEqOne G p) : + v ∈ fixedPoints (Multiplicative (ZMod p)) + (vectorsProdEqOne G p) ↔ + ∃ a : G, + (v : List.Vector G p).toList + = List.replicate p a := by + rw [mem_fixedPoints] + constructor + · -- Ein Fixpunkt wird insbesondere von 1 ∈ ℤ/(p) fixiert, + -- bleibt also unter dem Shift um eine Stelle + -- unverändert … + intro hv + have h1 : rotate v 1 = v := by + have h := hv (ofAdd (1 : ZMod p)) + rwa [show ofAdd (1 : ZMod p) • v + = rotate v (1 : ZMod p).val from rfl, + ZMod.val_one] at h + -- … und ist daher konstant. + obtain ⟨a, ha⟩ := + List.rotate_one_eq_self_iff_eq_replicate.mp + (Subtype.ext_iff.mp (Subtype.ext_iff.mp h1)) + refine ⟨a, ha.trans ?_⟩ + congr 1 + exact (v : List.Vector G p).2 + · -- Umgekehrt ändert kein zyklischer Shift ein konstantes + -- Tupel. + rintro ⟨a, ha⟩ g + apply Subtype.ext + apply Subtype.ext + show (v : List.Vector G p).toList.rotate (toAdd g).val = + (v : List.Vector G p).toList + rw [ha, List.rotate_replicate] +``` + +## Der Beweis + +Jetzt setzen wir die Teile zusammen und folgen dem Skript Zeile für +Zeile. + +```lean +theorem cauchy {G : Type*} [Group G] [Fintype G] {p : ℕ} + (hp : p.Prime) (hdvd : p ∣ Fintype.card G) : + ∃ a : G, orderOf a = p := by + -- Die Konsequenzen der Primalität, die die Instanzsuche + -- braucht: + have : Fact p.Prime := ⟨hp⟩ + have : NeZero p := ⟨hp.ne_zero⟩ + have : Fact (1 < p) := ⟨hp.one_lt⟩ + -- Abkürzung: M₀ ist die Fixpunktmenge der Wirkung. + set M₀ := fixedPoints (Multiplicative (ZMod p)) + (vectorsProdEqOne G p) + -- „Wir erhalten die folgende Gleichung: + -- |M| = |G^{p-1}| = |G|^{p-1}.“ + have hM : Nat.card (vectorsProdEqOne G p) + = Fintype.card G ^ (p - 1) := by + rw [Nat.card_eq_fintype_card, VectorsProdEqOne.card] + -- Die zyklische Gruppe ℤ/(p) hat Ordnung p = p¹; das + -- Schlüssellemma, angewandt auf die Rotationswirkung, + -- liefert |M| ≡ |M₀| (mod p): + have hZp : + Nat.card (Multiplicative (ZMod p)) = p ^ 1 := by + simp [Nat.card_eq_fintype_card] + have hcong : + Nat.card (vectorsProdEqOne G p) ≡ + Nat.card M₀ [MOD p] := + key_lemma hp hZp (vectorsProdEqOne G p) + -- „Auf der anderen Seite folgt aus dem zentralen + -- Schlüssellemma, dass + -- |M₀| ≡ |M| ≡ |G|^{p-1} ≡ 0 (mod p) ist.“ + have hdvdM0 : p ∣ Nat.card M₀ := by + have hp1 : p - 1 ≠ 0 := by have := hp.one_lt; omega + have h0 : (0 : ℕ) ≡ Nat.card M₀ [MOD p] := + calc (0 : ℕ) + ≡ Fintype.card G ^ (p - 1) [MOD p] := + (Nat.modEq_zero_iff_dvd.mpr + (dvd_pow hdvd hp1)).symm + _ = Nat.card (vectorsProdEqOne G p) := hM.symm + _ ≡ Nat.card M₀ [MOD p] := hcong + exact Nat.modEq_zero_iff_dvd.mp h0.symm + -- „Wegen (e, …, e) ∈ M₀ ist schon einmal klar, dass + -- M₀ ≠ ∅ ist.“ + let v₀ : vectorsProdEqOne G p := + ⟨List.Vector.replicate p 1, + (List.prod_replicate p 1).trans (one_pow p)⟩ + have hv₀ : v₀ ∈ M₀ := + (mem_fixedPoints_iff_replicate v₀).mpr ⟨1, rfl⟩ + -- M₀ ist nichtleer, und |M₀| ist durch p ≥ 2 teilbar, + -- also ist |M₀| ≥ p > 1 … + have := Fintype.ofFinite M₀ + have hlt : 1 < Fintype.card M₀ := by + rw [← Nat.card_eq_fintype_card] + have hpos : 0 < Nat.card M₀ := + Nat.card_pos_iff.mpr ⟨⟨⟨v₀, hv₀⟩⟩, inferInstance⟩ + have := Nat.le_of_dvd hpos hdvdM0 + have := hp.one_lt + omega + -- „Also existiert mindestens ein a ≠ e mit a^p = e.“ + obtain ⟨w, hw⟩ := + Fintype.exists_ne_of_one_lt_card hlt ⟨v₀, hv₀⟩ + obtain ⟨a, ha⟩ := + (mem_fixedPoints_iff_replicate w.1).mp w.2 + -- Das Tupel w liegt in M, sein Produkt ist also e; weil + -- das Tupel konstant ist, heißt das genau a^p = e: + have hpow : a ^ p = 1 := by + have hprod : (w.1 : List.Vector G p).toList.prod = 1 := + w.1.2 + rwa [ha, List.prod_replicate] at hprod + -- Und a ≠ e, denn sonst wäre w das Tupel (e, …, e) = v₀: + have hne : a ≠ 1 := by + rintro rfl + exact hw (Subtype.ext (Subtype.ext (Subtype.ext ha))) + -- „Nach Satz 17.4.11 hat a dann automatisch die + -- Ordnung p.“ + exact ⟨a, orderOf_eq_prime hpow hne⟩ +``` + +# Bemerkungen + +Mathlibs Fassung des Satzes von Cauchy heißt +`exists_prime_orderOf_dvd_card`; ihr Beweis ist dasselbe Zählargument, +nur etwas anders organisiert. + +# Übungsaufgabe + +Das ist Satz 18.2.3 im Skript: _Es sei `p` eine Primzahl und `G` eine +nichttriviale Gruppe, deren Ordnung eine p-Potenz ist. Dann ist das +Zentrum von `G` nicht trivial._ Beweisen Sie das in Lean: Wenden Sie +`key_lemma` auf die Konjugationswirkung von `G` auf sich selbst an — +oder finden Sie die Aussage in Mathlib. Öffnen Sie +`AlgebraInLean/Cauchy.lean` im Übungs-Repository und ersetzen Sie das +`sorry` durch einen Beweis. + +```lean +theorem center_nontrivial {p m : ℕ} (hp : p.Prime) + {G : Type*} [Group G] [Finite G] (hm : m ≠ 0) + (hG : Nat.card G = p ^ m) : + Nontrivial (Subgroup.center G) := by + sorry +``` diff --git a/Book/LittleFermat.lean b/Book/LittleFermat.lean new file mode 100644 index 0000000..d627aba --- /dev/null +++ b/Book/LittleFermat.lean @@ -0,0 +1,147 @@ +import VersoManual +import Manual.Meta +-- Minimal-Import statt `import Mathlib`: lädt nur den +-- benötigten Teil der Bibliothek und beschleunigt den +-- Buch-Build erheblich (ermittelt mit `#min_imports`). +import Mathlib.FieldTheory.Finite.Basic + +open Verso.Genre Manual +open Verso.Genre.Manual.InlineLean + +set_option pp.rawOnError true +set_option verso.docstring.allowMissing true + +#doc (Manual) "Der kleine Satz von Fermat" => +%%% +htmlSplit := .never +tag := "little-fermat" +%%% + +Unser erster Beweis übersetzt Satz 17.5.2 des Skripts. Hier das +deutsche Original, Aussage und Beweis: + +> *Satz (Kleiner Satz von Fermat).* Es sei `p ∈ ℕ` eine Primzahl und es +> sei `a ∈ ℤ` irgendeine Zahl. Dann ist `a^p ≡ a (mod p)`. +> +> _Beweis._ Falls `a` ein Vielfaches von `p` ist, ist die Sache klar. +> Ansonsten liefert die Restklasse von `a` ein nicht-verschwindendes +> Element `ā ∈ ℤ/(p) = 𝔽_p`, also ein Element der multiplikativen +> Gruppe `𝔽_p^*`, welche `p−1` Elemente hat. Nach dem Satz von +> Lagrange ist die Ordnung von `ā`, also die Größe der von `ā` +> erzeugten Untergruppe, ein Teiler von `|𝔽_p^*| = p−1`. Es gilt also +> `ā^(p−1) = 1 ∈ 𝔽_p^*`, oder äquivalent `a^p ≡ a (mod p)`. ∎ + +Fünf Sätze. Unser Ziel ist ein Lean-Beweis mit genau dieser Struktur — +jeder deutsche Satz kehrt als Kommentar über dem Lean-Code wieder, der +ihn umsetzt. + +# Das Mathlib-Wörterbuch + +Bevor wir den Satz überhaupt formulieren können, müssen wir wissen, wie +Mathlib über die beteiligten Objekte spricht. + +* Der Restklassenring `ℤ/(p)` heißt `ZMod p`. Die Restklasse einer + ganzen Zahl `a : ℤ` schreibt sich `(a : ZMod p)` — Lean fügt den + kanonischen Ringmorphismus `ℤ → ℤ/(p)` automatisch ein (eine + _Koerzion_). + +* Die Kongruenz `a ≡ b (mod p)` für ganze Zahlen schreibt sich + `a ≡ b [ZMOD p]`. + +* Die multiplikative Gruppe `𝔽_p^*` ist die Gruppe der _Einheiten_ des + Rings `ZMod p`, geschrieben `(ZMod p)ˣ`. Eine Einheit + `u : (ZMod p)ˣ` kennt ihr Inverses; das zugrundeliegende Ringelement + schreibt sich wieder `(u : ZMod p)`. + +* Das Skript sagt „es sei `p` eine Primzahl“. In Lean führen wir die + Primalität von `p` als Hypothese `hp : p.Prime` mit. Manche + Tatsachen — etwa dass `ℤ/(p)` ein Körper ist — findet Leans + Automatisierung nur, wenn die Hypothese als _Instanz_ registriert + ist; genau das leistet die erste Beweiszeile + `have : Fact p.Prime := ⟨hp⟩`. + +# Der Beweis, Satz für Satz + +```lean +theorem little_fermat (p : ℕ) (hp : p.Prime) (a : ℤ) : + a ^ p ≡ a [ZMOD p] := by + have : Fact p.Prime := ⟨hp⟩ + -- Die Kongruenz „a^p ≡ a (mod p)“ bedeutet gerade, dass + -- a^p und a dieselbe Restklasse in ℤ/(p) haben. Wir + -- dürfen also stattdessen eine *Gleichung* im Ring + -- `ZMod p` zeigen; das Beweisziel wird + -- (a : ZMod p) ^ p = (a : ZMod p). + rw [← ZMod.intCast_eq_intCast_iff] + push_cast + -- „Falls a ein Vielfaches von p ist, ist die Sache klar.“ + by_cases ha : (a : ZMod p) = 0 + · rw [ha, zero_pow hp.ne_zero] + -- „Ansonsten liefert die Restklasse von a ein + -- nicht-verschwindendes Element ā ∈ ℤ/(p) = 𝔽_p, also + -- ein Element der multiplikativen Gruppe 𝔽_p^*, …“ + · obtain ⟨u, hu⟩ : IsUnit (a : ZMod p) := + isUnit_iff_ne_zero.mpr ha + rw [← hu] + -- „… welche p−1 Elemente hat.“ + have card_units : Nat.card (ZMod p)ˣ = p - 1 := by + rw [Nat.card_eq_fintype_card, ZMod.card_units p] + -- „Nach Satz 17.3.6 («Satz von Lagrange») ist die + -- Ordnung von ā, also die Größe der von ā erzeugten + -- Untergruppe, ein Teiler von |𝔽_p^*| = p−1.“ + have lagrange : + Nat.card (Subgroup.zpowers u) ∣ + Nat.card (ZMod p)ˣ := + Subgroup.card_subgroup_dvd_card (Subgroup.zpowers u) + have ord_dvd : orderOf u ∣ p - 1 := by + rw [← Nat.card_zpowers, ← card_units] + exact lagrange + -- „Es gilt also ā^(p−1) = 1 ∈ 𝔽_p^* …“ + have pow_eq_one : u ^ (p - 1) = 1 := + orderOf_dvd_iff_pow_eq_one.mp ord_dvd + -- „… oder äquivalent a^p ≡ a (mod p).“ + have key : u ^ p = u := by + calc u ^ p + = u ^ (p - 1 + 1) := by + rw [Nat.sub_add_cancel hp.one_lt.le] + _ = u ^ (p - 1) * u := pow_succ u (p - 1) + _ = u := by rw [pow_eq_one, one_mul] + exact_mod_cast congrArg Units.val key +``` + +Man beachte, wie treu die Übersetzung ist. Die Fallunterscheidung +„Falls … ist die Sache klar / Ansonsten …“ wird zu `by_cases`; das +Zitat des Satzes von Lagrange wird zum Bibliothekslemma +`Subgroup.card_subgroup_dvd_card`; und die Formulierung „die Ordnung +von `ā`, also die Größe der von `ā` erzeugten Untergruppe“ ist genau +das Umschreiben mit `Nat.card_zpowers`, das `orderOf u` mit der +Mächtigkeit der von `u` erzeugten Untergruppe `Subgroup.zpowers u` +identifiziert. + +# Bemerkungen + +Mathlib enthält den kleinen Satz von Fermat natürlich längst; die +Aussage über Restklassen heißt `ZMod.pow_card`. Solche Lemmata findet +man selbst mit der Taktik `exact?` oder über die Suchmaschinen +[leansearch.net](https://leansearch.net) und +[loogle.lean-lang.org](https://loogle.lean-lang.org). + +```lean +example (p : ℕ) [Fact p.Prime] (a : ZMod p) : a ^ p = a := + ZMod.pow_card a +``` + +# Übungsaufgabe + +Das ist die Bemerkung nach dem Satz im Skript: In Anwendungen +verwendet man häufig die äquivalente Formulierung +`a^(p−1) ≡ 1 (mod p)` für nicht durch `p` teilbare `a`. Leiten Sie +sie aus `little_fermat` ab — oder geben Sie einen direkten Beweis nach +den Ideen oben. Öffnen Sie `AlgebraInLean/LittleFermat.lean` im +Übungs-Repository und ersetzen Sie das `sorry` durch einen Beweis. + +```lean +theorem little_fermat' (p : ℕ) (hp : p.Prime) (a : ℤ) + (ha : ¬ (p : ℤ) ∣ a) : + a ^ (p - 1) ≡ 1 [ZMOD p] := by + sorry +``` diff --git a/Book/Quotients.lean b/Book/Quotients.lean new file mode 100644 index 0000000..ea24993 --- /dev/null +++ b/Book/Quotients.lean @@ -0,0 +1,148 @@ +import VersoManual +import Manual.Meta +-- Minimal-Imports statt `import Mathlib`: ermittelt mit +-- `#min_imports`, ergänzt um QuotientGroup und die +-- Taktik `group`. +import Mathlib.Algebra.Group.Int.Defs +import Mathlib.Algebra.Group.Subgroup.Ker +import Mathlib.GroupTheory.QuotientGroup.Defs +import Mathlib.Tactic.Group + +open Verso.Genre Manual +open Verso.Genre.Manual.InlineLean + +set_option pp.rawOnError true +set_option verso.docstring.allowMissing true + +#doc (Manual) "Gruppen und Restklassengruppen" => +%%% +htmlSplit := .never +tag := "quotients" +%%% + +Dieses Kapitel übersetzt zwei Beweise rund um normale Untergruppen +Satz für Satz nach Lean: das Argument aus Erklärvideo 1-1 des +Skripts, dass Kerne von Gruppenmorphismen abgeschlossen unter +Konjugation sind, und die Beobachtung aus Kapitel 17.3, dass `U·N` +für normales `N` wieder eine Untergruppe ist. + +# Das Mathlib-Wörterbuch + +* Eine Untergruppe von `G` ist ein Term `U : Subgroup G`; die + Mitgliedschaft schreibt sich `g ∈ U`. Dass `U` unter Verknüpfung + und Inversen abgeschlossen ist, sagen `U.mul_mem` und `U.inv_mem`. + +* Ein Gruppenmorphismus ist ein Term `φ : G →* Q` — ein „Bündel“ aus + der Abbildung und den Nachweisen `map_mul : φ (a·b) = φ a · φ b` + (daraus folgen `map_one` und `map_inv`). Der Kern heißt `φ.ker`; + die Mitgliedschaft entpackt + `MonoidHom.mem_ker : g ∈ φ.ker ↔ φ g = 1`. + +* „`N` ist Normalteiler“ ist das Prädikat `N.Normal`; sein Feld + `conj_mem` ist wörtlich die Bedingung des Skripts: für alle + `n ∈ N` und alle `g` ist `g·n·g⁻¹ ∈ N`. + +# Kerne sind abgeschlossen unter Konjugation + +Aus Erklärvideo 1-1 des Skripts: + +> Die Antwort lautet: Nein! Nicht jede Untergruppe kann Kern eines +> Gruppenmorphismus sein. Wenn es nämlich einen Gruppenmorphismus +> `φ : G → Q` mit `H = ker(φ)` gibt, dann gilt für alle `h ∈ H` und +> alle `g ∈ G` die Gleichung `φ(g·h·g⁻¹) = φ(g)·φ(h)·φ(g)⁻¹ = e`, +> wobei `e` das neutrale Element von `Q` bezeichnet. Also ist stets +> `g·h·g⁻¹ ∈ H`. + +```lean +theorem kern_konjugation {G Q : Type*} [Group G] [Group Q] + (φ : G →* Q) {h : G} (hh : h ∈ φ.ker) (g : G) : + g * h * g⁻¹ ∈ φ.ker := by + -- Mitgliedschaft im Kern heißt: φ bildet auf 1 ab. + rw [MonoidHom.mem_ker] at hh ⊢ + -- „… dann gilt für alle h ∈ H und alle g ∈ G die + -- Gleichung φ(g·h·g⁻¹) = φ(g)·φ(h)·φ(g)⁻¹ = e …“ + calc φ (g * h * g⁻¹) + = φ g * φ h * (φ g)⁻¹ := by + rw [map_mul, map_mul, map_inv] + _ = φ g * 1 * (φ g)⁻¹ := by rw [hh] + _ = 1 := by rw [mul_one, mul_inv_cancel] +``` + +Mathlib kennt diese Aussage als Instanz +`MonoidHom.normal_ker : φ.ker.Normal`. Das Skript fährt fort: +Untergruppen mit dieser Eigenschaft heißen _normale Untergruppen_ +oder _Normalteiler_, und die Bedingung ist nicht nur notwendig, +sondern auch hinreichend — zu jedem Normalteiler existiert eine +Restklassengruppe. + +# Die Restklassengruppe und ihre universelle Eigenschaft + +Das Skript definiert die Restklassengruppe in Kapitel 17.3 über ihre +universelle Eigenschaft. Mathlib stellt beides bereit: den +Quotienten `G ⧸ N` mit der Quotientenabbildung +`QuotientGroup.mk' N : G →* G ⧸ N`, und die universelle Eigenschaft +als `QuotientGroup.lift` — gegeben ein Morphismus `α : G →* H` mit +`N ≤ α.ker` erhält man den induzierten Morphismus auf dem +Quotienten: + +```lean +example {G H : Type*} [Group G] [Group H] (N : Subgroup G) + [N.Normal] (α : G →* H) (hα : N ≤ α.ker) : + G ⧸ N →* H := + QuotientGroup.lift N α + fun _g hg => MonoidHom.mem_ker.mp (hα hg) +``` + +# Die Beobachtung: U·N ist eine Untergruppe + +Beobachtung aus Kapitel 17.3 des Skripts: + +> Es sei `G` eine Gruppe und es sei `N ⊂ G` eine normale +> Untergruppe. Weiter sei `U ⊂ G` irgendeine Untergruppe. Dann ist +> `U·N = {u·n : u ∈ U, n ∈ N}` wieder eine Untergruppe. Zum Beweis +> müssen wir lediglich zeigen, dass `U·N` abgeschlossen unter der +> Gruppenoperation ist. … Wir wissen, dass `N` normal ist. Also +> ist `ñ₁ := u₂⁻¹·n₁·u₂ ∈ N` und es gilt `n₁·u₂ = u₂·ñ₁`. Demnach +> ist `u₁n₁u₂n₂ = u₁u₂·ñ₁n₂ ∈ U·N`. Fertig ist der Beweis. + +Neu ist hier die Taktik `group`: Sie verrechnet Ausdrücke, die +allein mit den Gruppenaxiomen ineinander umgeformt werden können. +(Das `ñ₁` des Skripts heißt im Code `n₁'`.) + +```lean +theorem mul_mem_mul {G : Type*} [Group G] (U N : Subgroup G) + (hN : N.Normal) {u₁ n₁ u₂ n₂ : G} + (hu₁ : u₁ ∈ U) (hn₁ : n₁ ∈ N) + (hu₂ : u₂ ∈ U) (hn₂ : n₂ ∈ N) : + ∃ u ∈ U, ∃ n ∈ N, u₁ * n₁ * (u₂ * n₂) = u * n := by + -- „Wir wissen, dass N normal ist. Also ist + -- ñ₁ := u₂⁻¹·n₁·u₂ ∈ N …“ + set n₁' := u₂⁻¹ * n₁ * u₂ with hn₁'_def + have hn₁' : n₁' ∈ N := hN.conj_mem' n₁ hn₁ u₂ + -- „… und es gilt n₁·u₂ = u₂·ñ₁.“ + have hswap : n₁ * u₂ = u₂ * n₁' := by + rw [hn₁'_def]; group + -- „Demnach ist u₁n₁u₂n₂ = u₁u₂·ñ₁n₂ ∈ U·N.“ + refine ⟨u₁ * u₂, U.mul_mem hu₁ hu₂, + n₁' * n₂, N.mul_mem hn₁' hn₂, ?_⟩ + calc u₁ * n₁ * (u₂ * n₂) + = u₁ * (n₁ * u₂) * n₂ := by group + _ = u₁ * (u₂ * n₁') * n₂ := by rw [hswap] + _ = u₁ * u₂ * (n₁' * n₂) := by group +``` + +# Übungsaufgabe + +Das Skript sagt, es sei „lediglich“ die Abgeschlossenheit unter der +Verknüpfung zu zeigen — prüfen Sie den verschwiegenen Teil selbst +nach: `U·N` ist auch abgeschlossen unter Inversen. Tipp: +`(u·n)⁻¹ = u⁻¹·(u·n⁻¹·u⁻¹)`, und der zweite Faktor liegt nach +`conj_mem` in `N`. Öffnen Sie `AlgebraInLean/Quotients.lean` und +ersetzen Sie das `sorry` durch einen Beweis. + +```lean +theorem inv_mem_mul {G : Type*} [Group G] (U N : Subgroup G) + (hN : N.Normal) {u n : G} (hu : u ∈ U) (hn : n ∈ N) : + ∃ u' ∈ U, ∃ n' ∈ N, (u * n)⁻¹ = u' * n' := by + sorry +``` diff --git a/Main.lean b/Main.lean new file mode 100644 index 0000000..6b4fd2c --- /dev/null +++ b/Main.lean @@ -0,0 +1,67 @@ +import Book +import Manual.Meta + +open Verso.Genre Manual +open Verso.Genre.Manual + +-- The static assets (theme, fonts, KaTeX) follow the setup of +-- Peter Pfaffelhuber's course notes, https://github.com/pfaffelh/leancourse. + +open Verso.Output.Html in +def staticCss := {{ + + + + + + + }} + +open Verso.Output.Html in +def staticJs := {{ + + + + }} + +def KaTeXLicense : LicenseInfo where + identifier := "MIT" + dependency := "KaTeX" + howUsed := "KaTeX is used to render mathematical notation." + link := "https://katex.org/" + text := #[(some "The MIT License", text)] +where + text := r#" +Copyright (c) 2013-2020 Khan Academy and other contributors + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in all +copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +SOFTWARE. +"# + +def main := + manualMain (%doc Book) (config := config) +where + config := { + extraFiles := [("static", "static")], + extraHead := #[staticCss, staticJs], + emitTeX := false, + -- Deployt wird nur die Multi-Page-Ausgabe; die + -- Single-Page-Ausgabe wäre doppelte Generierungsarbeit. + emitHtmlSingle := .no, + emitHtmlMulti := .immediately, + } diff --git a/README.md b/README.md index b79753b..b17de5a 100644 --- a/README.md +++ b/README.md @@ -32,9 +32,29 @@ Lean/Mathlib. | File | Lecture notes | Topic | |------|---------------|-------| +| `AlgebraInLean/Basics.lean` | Kapitel 2 | First steps: tactics, groups in Mathlib | +| `AlgebraInLean/Quotients.lean` | Kapitel 2, 17 | Kernels, normal subgroups, quotient groups | | `AlgebraInLean/LittleFermat.lean` | Kapitel 17 | Fermat's little theorem via Lagrange | | `AlgebraInLean/Cauchy.lean` | Kapitel 18 | The key lemma on fixed points and Cauchy's theorem | +## The rendered course notes + +The folder `Book/` contains the same material as a +[Verso](https://github.com/leanprover/verso) book (the technical setup +follows Peter Pfaffelhuber's +[leancourse](https://github.com/pfaffelh/leancourse)). To render it: + +```bash +lake build +lake exe algebrainlean --output _out +python3 -m http.server 8000 -d _out/html-multi # then open http://localhost:8000 +``` + +All Lean code in the book is elaborated during the build, so the +rendered proofs are guaranteed to compile. `./deploy.sh` builds the +site and copies it to `public/`, which is synced to the CPLX web +server as for the lecture notes. + ## License CC-BY 4.0, like the lecture notes. diff --git a/deploy.sh b/deploy.sh new file mode 100755 index 0000000..111deda --- /dev/null +++ b/deploy.sh @@ -0,0 +1,10 @@ +#!/bin/bash +# Baut die Kursnotizen und legt sie in public/ ab — analog zum +# deploy.sh des Vorlesungsskripts. Der Abgleich von public/ mit dem +# CPLX-Server läuft wie dort. +set -e + +lake build +lake exe algebrainlean --output _out +mkdir -p public +rsync -a --delete _out/html-multi/ public/ diff --git a/lake-manifest.json b/lake-manifest.json index f029945..842fbf3 100644 --- a/lake-manifest.json +++ b/lake-manifest.json @@ -11,6 +11,16 @@ "inputRev": "87adeaebd370a3b6a41ac4f044fddd4bf81803ad", "inherited": false, "configFile": "lakefile.lean"}, + {"url": "https://github.com/leanprover/reference-manual.git", + "type": "git", + "subDir": null, + "scope": "", + "rev": "527c864cfd8da8b3d119ee09382749b6b6204eb4", + "name": "«verso-manual»", + "manifestFile": "lake-manifest.json", + "inputRev": "v4.33.0-rc2", + "inherited": false, + "configFile": "lakefile.lean"}, {"url": "https://github.com/leanprover-community/plausible", "type": "git", "subDir": null, @@ -81,6 +91,36 @@ "inputRev": "main", "inherited": true, "configFile": "lakefile.toml"}, + {"url": "https://github.com/leanprover/verso.git", + "type": "git", + "subDir": null, + "scope": "", + "rev": "f62e380bcf8e11a2df70697916c8333bafdf3540", + "name": "verso", + "manifestFile": "lake-manifest.json", + "inputRev": "nightly-testing", + "inherited": true, + "configFile": "lakefile.lean"}, + {"url": "https://github.com/leanprover/illuminate", + "type": "git", + "subDir": null, + "scope": "", + "rev": "006dc1d1db18c5dc73d637c926cf132e88df05b5", + "name": "illuminate", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.lean"}, + {"url": "https://github.com/leanprover/verso-web-components", + "type": "git", + "subDir": null, + "scope": "", + "rev": "61182a9187f497946b1a9170018e241c9500e6da", + "name": "versowebcomponents", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.lean"}, {"url": "https://github.com/leanprover/lean4-cli", "type": "git", "subDir": null, @@ -90,7 +130,27 @@ "manifestFile": "lake-manifest.json", "inputRev": "v4.33.0-rc2", "inherited": true, - "configFile": "lakefile.toml"}], + "configFile": "lakefile.toml"}, + {"url": "https://github.com/acmepjz/md4lean", + "type": "git", + "subDir": null, + "scope": "", + "rev": "31907cc18f48a95384f99cee5582c00fb39e0f67", + "name": "MD4Lean", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.lean"}, + {"url": "https://github.com/leanprover/subverso", + "type": "git", + "subDir": null, + "scope": "", + "rev": "859ab80c32c5851151919a7d757d7c0c0b6e39d2", + "name": "subverso", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.lean"}], "name": "AlgebraInLean", "lakeDir": ".lake", "fixedToolchain": false} diff --git a/lakefile.lean b/lakefile.lean new file mode 100644 index 0000000..f3b108e --- /dev/null +++ b/lakefile.lean @@ -0,0 +1,39 @@ +import Lake +open Lake DSL + +-- The manual genre of the Lean reference manual is pinned to the tag +-- matching the toolchain; verso is NOT required explicitly but resolved +-- through the reference manual's own pin (its v4.33.0-rc2 tag needs a +-- verso nightly, not the verso tag of the same name). Mathlib is +-- pinned to the same commit as before (a master snapshot on toolchain +-- v4.33.0-rc2). Mathlib comes LAST so that its pins of shared +-- dependencies (batteries, …) take precedence — otherwise +-- `lake exe cache get` refuses to work. +require «verso-manual» from git + "https://github.com/leanprover/reference-manual.git"@"v4.33.0-rc2" +require mathlib from git + "https://github.com/leanprover-community/mathlib4"@"87adeaebd370a3b6a41ac4f044fddd4bf81803ad" + +package «AlgebraInLean» where + -- building the C code costs much more than the optimizations save + moreLeancArgs := #["-O0"] + -- work around clang emitting invalid linker optimization hints that lld rejects + moreLinkArgs := + if System.Platform.isOSX then + #["-Wl,-ignore_optimization_hints"] + else #[] + +/-- The exercise files that students open in VS Code. -/ +@[default_target] +lean_lib «AlgebraInLean» where + +/-- The Verso sources of the course notes. -/ +@[default_target] +lean_lib «Book» where + +/-- The generator executable: `lake exe algebrainlean --output _out` +renders the course notes to `_out/html-multi`. -/ +@[default_target] +lean_exe «algebrainlean» where + srcDir := "./" + root := `Main diff --git a/lakefile.toml b/lakefile.toml deleted file mode 100644 index d272fdf..0000000 --- a/lakefile.toml +++ /dev/null @@ -1,10 +0,0 @@ -name = "AlgebraInLean" -defaultTargets = ["AlgebraInLean"] - -[[require]] -name = "mathlib" -git = "https://github.com/leanprover-community/mathlib4" -rev = "87adeaebd370a3b6a41ac4f044fddd4bf81803ad" - -[[lean_lib]] -name = "AlgebraInLean" diff --git a/static/README.txt b/static/README.txt new file mode 100644 index 0000000..a78b98c --- /dev/null +++ b/static/README.txt @@ -0,0 +1 @@ +The directory `katex` contains KaTeX v0.16.11 (MIT license) diff --git a/static/colors.css b/static/colors.css new file mode 100644 index 0000000..3b03b5e --- /dev/null +++ b/static/colors.css @@ -0,0 +1,32 @@ +/* CSS variables for each Lean branding color */ +:root { + /* Main color palette */ + --lean-black: #000000; + --lean-white: #ffffff; + --lean-blue: #0073a3; + + /* Accent color palette */ + /* This light blue should only be used as a background, and any text + on it should be black to ensure sufficient contrast. */ + --lean-accent-light-blue: #b0c5cd; + /* This orange should be used only as an accent in small amounts. No + small type. Bold subhead or larger for typography.*/ + --lean-accent-orange: #f15732; + + /* Non-branding colors */ + /* These colors should never be used as actual brand colors, but they + are complementary to our colors and can be used for elements in + presentations or charts or other peripheral materials. They are + only to be used as an accent or a complementary design element, + never as Lean branding. */ + --lean-compl-orangered: #F15732; + --lean-compl-yellow: #FDC05F; + --lean-compl-green: #699E88; + --lean-compl-bluegray: #A9C7C9; + --lean-compl-darkgray: #354140; + --lean-compl-gray: #96A0A5; + --lean-compl-pink: #F05665; + --lean-compl-lightpink: #F7A8B0; + --lean-compl-darkblue: #005E7D; + --lean-compl-turquoise: #48C6E2; +} diff --git a/static/fonts/source-code-pro/LICENSE.md b/static/fonts/source-code-pro/LICENSE.md new file mode 100644 index 0000000..70288a8 --- /dev/null +++ b/static/fonts/source-code-pro/LICENSE.md @@ -0,0 +1,93 @@ +© 2023 Adobe (http://www.adobe.com/), with Reserved Font Name 'Source'. All Rights Reserved. Source is a trademark of Adobe in the United States and/or other countries. + +This Font Software is licensed under the SIL Open Font License, Version 1.1. + +This license is copied below, and is also available with a FAQ at: http://scripts.sil.org/OFL + + +----------------------------------------------------------- +SIL OPEN FONT LICENSE Version 1.1 - 26 February 2007 +----------------------------------------------------------- + +PREAMBLE +The goals of the Open Font License (OFL) are to stimulate worldwide +development of collaborative font projects, to support the font creation +efforts of academic and linguistic communities, and to provide a free and +open framework in which fonts may be shared and improved in partnership +with others. + +The OFL allows the licensed fonts to be used, studied, modified and +redistributed freely as long as they are not sold by themselves. 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The requirement for fonts to +remain under this license does not apply to any document created +using the Font Software. + +TERMINATION +This license becomes null and void if any of the above conditions are +not met. + +DISCLAIMER +THE FONT SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, +EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTIES OF +MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT +OF COPYRIGHT, PATENT, TRADEMARK, OR OTHER RIGHT. IN NO EVENT SHALL THE +COPYRIGHT HOLDER BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, +INCLUDING ANY GENERAL, SPECIAL, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL +DAMAGES, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING +FROM, OUT OF THE USE OR INABILITY TO USE THE FONT SOFTWARE OR FROM +OTHER DEALINGS IN THE FONT SOFTWARE. diff --git a/static/fonts/source-code-pro/OTF/SourceCodePro-Black.otf b/static/fonts/source-code-pro/OTF/SourceCodePro-Black.otf new file mode 100644 index 0000000000000000000000000000000000000000..fb6858c6ee854ca0046ebad31c9f358f2d06ac5f GIT binary patch literal 133768 zcmbTf2Vhi1_clJa?51q8={~0z&LK+|_ND&Z0LK2cdNJ0u#1Odg4iiiaj zL{vnYfTD;LDHc=|d!;D$UKU-1?>TdB2)_Q_-}n9h=(Eqk5RQ?9hMwaYW-yRt!u$muQ3ZSAMu6kgQ-cHDQN=BQ@q z&TH;}I{2!Hp1%uC`wB7Q@X(?uoqW3csUBLMPUUGrrC9vIbK3i-e#FBgPj^4vUCWa> z@jd=UiowF_!iA=e)D{Q}$wM89>jBKlSm9C5MXW72MAJ1@)m=jMtD;!r420-YUOEV2 z*Xh&h52vkEq7pSuC=bzniqFm{2hV7z< z(qx7sM5^+*8IFV;_L$+Qv+5a*G=G@!F=z4#LHlk|qh)*uq~2Y9&uD)c3HPKIf 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