example.typ

#import "/lib.typ": *
// #import cosmos.simple: *
#import cosmos.fancy: *
// #import cosmos.rainbow: *
// #import cosmos.clouds: *
#show: show-theorion

#set page(height: auto)
#set heading(numbering: "1.1")
#set text(lang: "en")
#set par(first-line-indent: 1em, justify: true)

/// 1. Change the counters and numbering:
// #set-inherited-levels(1)
// #set-zero-fill(true)
// #set-leading-zero(true)
// #set-theorion-numbering("1.1")

/// 2. Other options:
// #set-indent-mode(none)  // auto (default), none, length or dictionary
// #set-result("noanswer")
// #set-qed-symbol[#math.qed]

/// 3. Custom theorem environment for yourself
// #let (theorem-counter, theorem-box, theorem, show-theorem) = make-frame(
//   "theorem",
//   "Theorem",  // supplement, string or dictionary like `(en: "Theorem")`, or `theorion-i18n-map.at("theorem")` for built-in i18n support
//   counter: theorem-counter,  // inherit the counter, `none` by default
//   inherited-levels: 2,  // useful when you need a new counter
//   inherited-from: heading,  // heading or another counter
//   render: (prefix: none, title: "", full-title: auto, body) => [#strong[#full-title.]#sym.space#emph(body)],
// )
// #show: show-theorem

/// 4. Just use it.
// #theorem[Euclid's Theorem][
//   There are infinitely many prime numbers.
// ] <thm:euclid>
// #theorem-box[Theorem without numbering][
//   This theorem is not numbered.
// ]

/// 5. Example of appendix
// #counter(heading).update(0)
// #set heading(numbering: "A.1")
// #set-theorion-numbering("A.1")

/// 6. Table of contents
// #outline(title: none, target: figure.where(kind: "theorem"))

/// 7. Specify a number or supplement
// #theorem(title: "Euclid's Theorem", number: "233", supplement: [Theorion])[
//   There are infinitely many prime numbers.
// ]

/// 8. Counter continuation: use an array number to continue numbering from a specific value
// #theorem(number: (2, 3))[
//   This theorem is explicitly numbered 2.3.
//   The counter continues from here, so the next auto-numbered theorem is 2.4.
// ]

/// 9. Flexible References via specific supplements. A reference without the title using @label[-]; or one with title and number using @label[!!]
// A reference without the title: @thm:euclid[-]; or one with title and number: @thm:euclid[!!]

/// 10. Custom full-title: override the auto-generated "Prefix Number (Title)" format
// #theorem(full-title: [Fundamental Theorem])[
//   There is a fundamental result.
// ]

= Theorion Environments

== Quick Start

```typst
#import "@preview/theorion:0.6.0": *
#import cosmos.fancy: *
// #import cosmos.rainbow: *
// #import cosmos.clouds: *
#show: show-theorion

#theorem[Euclid's Theorem][
  There are infinitely many prime numbers.
] <thm:euclid>

#theorem-box(outlined: false)[Theorem without numbering][
  This theorem is not numbered.
]
```

== Customization

```typst
// 1. Change the counters and numbering:
#set-inherited-levels(1)
#set-zero-fill(true)
#set-leading-zero(true)
#set-theorion-numbering("1.1")

// 2. Other options:
#set-result("noanswer")
#set-qed-symbol[#math.qed]

// 3. Custom theorem environment for yourself
#let (theorem-counter, theorem-box, theorem, show-theorem) = make-frame(
  "theorem",
  "Theorem",  // supplement, string or dictionary like `(en: "Theorem")`, or `theorion-i18n-map.at("theorem")` for built-in i18n support
  counter: theorem-counter,  // inherit the old counter, `none` by default
  inherited-levels: 2,  // useful when you need a new counter
  inherited-from: heading,  // heading or just another counter
  render: (prefix: none, title: "", full-title: auto, body) => [#strong[#full-title.]#sym.space#emph(body)],
)
#show: show-theorem

// 4. Just use it.
#theorem[Euclid's Theorem][
  There are infinitely many prime numbers.
] <thm:euclid>
#theorem-box(outlined: false)[Theorem without numbering][
  This theorem is not numbered.
]

// 5. Example of appendix
#counter(heading).update(0)
#set heading(numbering: "A.1")
#set-theorion-numbering("A.1")

// 6. Table of contents
#outline(title: none, target: figure.where(kind: "theorem"))


// 7. Specify a number or supplement
#theorem(title: "Euclid's Theorem", number: "233", supplement: [Theorion])[
  There are infinitely many prime numbers.
]

// 8. Counter continuation: use an array number to continue numbering from a specific value
#theorem(number: (2, 3))[
  This theorem is explicitly numbered 2.3.
  The counter continues from here, so the next auto-numbered theorem is 2.4.
]

// 9. Flexible References via specific supplements. A reference without the title using @label[-]; or one with title and number using @label[!!]
A reference without the title: @thm:euclid[-]; or one with title and number: @thm:euclid[!!]

// 10. Custom full-title: override the auto-generated "Prefix Number (Title)" format
#theorem(full-title: [Fundamental Theorem])[
  There is a fundamental result.
]
```

== Table of Theorems

#outline(title: none, target: figure.where(kind: "theorem"))

== Basic Theorem Environments

Let's start with the most fundamental definition.

#definition[
  A natural number is called a #highlight[_prime number_] if it is greater than 1
  and cannot be written as the product of two smaller natural numbers.
] <def:prime>

#example[
  The numbers $2$, $3$, and $17$ are prime. As proven in @cor:infinite-prime,
  this list is far from complete! See @thm:euclid for the full proof.
]

#assumption[
  For all $n in NN$, assume $n$ is even if $n = 2k$ for some $k in NN$.
]

#property[
  The sum of two even numbers is always even.
]

#conjecture[Twin Prime Conjecture][
  There are infinitely many primes $p$ such that $p+2$ is also prime.
]

#theorem[Euclid's Theorem][
  There are infinitely many prime numbers.
] <thm:euclid>

#proof[Proof of @thm:euclid][
  By contradiction: Suppose $p_1, p_2, dots, p_n$ is a finite enumeration of all primes.
  Let $P = p_1 p_2 dots p_n$. Since $P + 1$ is not in our list,
  it cannot be prime. Thus, some prime $p_j$ divides $P + 1$.
  Since $p_j$ also divides $P$, it must divide their difference $(P + 1) - P = 1$,
  a contradiction.
]

#corollary[
  There is no largest prime number.
] <cor:infinite-prime>

#lemma[
  There are infinitely many composite numbers.
]

== Functions and Continuity

#theorem[Continuity Theorem][
  If a function $f$ is differentiable at every point, then $f$ is continuous.
] <thm:continuous>

#tip-block[
  @thm:continuous tells us that differentiability implies continuity,
  but not vice versa. For example, $f(x) = |x|$ is continuous but not differentiable at $x = 0$.
  For a deeper understanding of continuous functions, see @thm:max-value in the appendix.
]

== Geometric Theorems

#theorem[Pythagorean Theorem][
  In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides:
  $x^2 + y^2 = z^2$
] <thm:pythagoras>

#important-block[
  @thm:pythagoras is one of the most fundamental and important theorems in plane geometry,
  bridging geometry and algebra.
]

#corollary[
  There exists no right triangle with sides measuring 3cm, 4cm, and 6cm.
  This directly follows from @thm:pythagoras.
] <cor:pythagoras>

#lemma[
  Given two line segments of lengths $a$ and $b$, there exists a real number $r$
  such that $b = r a$.
] <lem:proportion>

== Algebraic Structures

#definition[Ring][
  Let $R$ be a non-empty set with two binary operations $+$ and $dot$, satisfying:
  1. $(R, +)$ is an abelian group
  2. $(R, dot)$ is a semigroup
  3. The distributive laws hold
  Then $(R, +, dot)$ is called a ring.
] <def:ring>

#proposition[
  Every field is a ring, but not every ring is a field. This concept builds upon @def:ring.
] <prop:ring-field>

#example[
  Consider @def:ring. The ring of integers $ZZ$ is not a field, as no elements except $plus.minus 1$
  have multiplicative inverses.
]

/// Appendix
#counter(heading).update(0)
#set heading(numbering: "A.1")
#set-theorion-numbering("A.1")

= Theorion Appendices

== Advanced Analysis

#theorem[Maximum Value Theorem][
  A continuous function on a closed interval must attain both a maximum and a minimum value.
] <thm:max-value>

#warning-block[
  Both conditions of this theorem are essential:
  - The function must be continuous
  - The domain must be a closed interval
]

== Advanced Algebra Supplements

#axiom[Group Axioms][
  A group $(G, \cdot)$ must satisfy:
  1. Closure
  2. Associativity
  3. Identity element exists
  4. Inverse elements exist
] <axiom:group>

#postulate[Fundamental Theorem of Algebra][
  Every non-zero polynomial with complex coefficients has a complex root.
] <post:fta>

#remark-block[
  This theorem is also known as Gauss's theorem, as it was first rigorously proved by Gauss.
]

== Common Problems and Solutions

#problem[
  Prove: For any integer $n > 1$, there exists a sequence of $n$ consecutive composite numbers.
]

#solution(qed: auto)[
  Consider the sequence: $n! + 2, n! + 3, ..., n! + n$

  For any $2 <= k <= n$, $n! + k$ is divisible by $k$ because:
  $n! + k = k(n! / k + 1)$

  Thus, this forms a sequence of $n-1$ consecutive composite numbers.
]

#exercise[
  1. Prove: The twin prime conjecture remains unproven.
  2. Try to explain why this problem is so difficult.
]

#conclusion[
  Number theory contains many unsolved problems that appear deceptively simple
  yet are profoundly complex.
]

== Important Notes

#note-block[
  Remember that mathematical proofs should be both rigorous and clear.
  Clarity without rigor is insufficient, and rigor without clarity is ineffective.
]

#caution-block[
  When dealing with infinite series, always verify convergence before discussing other properties.
]

#quote-block[
  Mathematics is the queen of sciences, and number theory is the queen of mathematics.
  — Gauss
]

#emph-block[
  Chapter Summary:
  - We introduced basic number theory concepts
  - Proved several important theorems
  - Demonstrated different types of mathematical environments
]

== Restated Theorems

// 1. Restate all theorems
#theorion-restate(
  filter: it => it.outlined and it.identifier == "theorem",
  render: it => it.render,
)
// 2. Restate all theorems with custom render function
// #theorion-restate(
//   filter: it => it.outlined and it.identifier == "theorem",
//   render: it => (prefix: none, title: "", full-title: auto, body) => block[#strong[#full-title.]#sym.space#emph(body)],
// )
// 3. Restate a specific theorem
// #theorion-restate(filter: it => it.label == <thm:euclid>)
// or we can use
// #theorion-restate(filter: <thm:euclid>)