Theory Graph

Blocks

Cards (Nodes) in these graphs are :::theorem, :::definition and other blocks.

Directed edges are produced by labelled :::proof blocks like Add the forms, which create an edge pointing into node #thm-even-plus-odd from each of the nodes in uses.

A proof with several uses premises draws a line from each premise to the proof's label, and a single arrow from the label to the proves node.

Example

From Theory Graph Example:

:::definition[Even]{#def-even}
$n \in \mathbb{Z}$ is even if $n = 2j$ for some $j \in \mathbb{Z}$.
:::

:::definition[Odd]{#def-odd}
$n \in \mathbb{Z}$ is odd if $n = 2k + 1$ for some $k \in \mathbb{Z}$.
:::

:::theorem[Even plus odd]{#thm-even-plus-odd}
If $m$ is even and $n$ is odd, then $m + n$ is odd.
:::

:::proof[Add the forms]{proves="#thm-even-plus-odd" uses="#def-even #def-odd"}
$m + n = 2j + (2k + 1) = 2(j + k) + 1$.
:::

Renders as

One proof using two premises: both lines meet at the Add the forms label, and one arrow leaves it.

DefinitionEven

$n \in \mathbb{Z}$ is even if $n = 2j$ for some $j \in \mathbb{Z}$ .

DefinitionOdd

$n \in \mathbb{Z}$ is odd if $n = 2k + 1$ for some $k \in \mathbb{Z}$ .

TheoremEven plus odd

If $m$ is even and $n$ is odd, then $m + n$ is odd.

Add the forms: Even, Odd ⟹ Even plus odd

Interactive

Clicking a card expands its statement and opens its relationships.

PROVED BY in edges
USED IN out edges
HIGHLIGHT dims everything except direct neighbours, all upstream, or all downstream results

Graph Object

You can specify the pages or single blocks (e.g. 'theory.md#thm-IVT') to include. The graph below is produced by

<TheoremGraph include={['theory.md']} label="Companion to Analysis theory graph" />

DefinitionThe fundamental axiom of analysis

If,

$a_n \in \mathbb{R} \forall n \in \mathbb{N}$
$A \in R$
$a_{n} \leq a_{n+1} \forall n \in \mathbb{N}$
$a_{n} < A \forall n \in \mathbb{N}$

Then, $\exists a \in \mathbb{R} : a_{n} \to n \text{ as } n \to \infty$ .

DefinitionSupremum

Let $A \subseteq \mathbb{R}$ be non empty.

$\alpha$ is the least upper bound (supremum) of A if

$\alpha \geq a \ \forall a \in A$ .
$\beta \geq a \ \forall a \in A \implies \beta \geq \alpha$

Theorem1.35 Intermediate Value Theorem

If,

$f: [a,b] \to \mathbb{R}$ is continuous
$f(a) \geq 0 \geq f(b)$

Then, $\exists c \in [a,b] : f(c) = 0$

Theorem1.42 The mean value inequality

Let,

$U=\left( \alpha, \beta \right)$ be an open interval on $\mathbb{R}$ .
$a,b \in U$ and $a<b$
$f: U \to \mathbb{R}$ be differentiable
$f'(t) \leq K \ \forall t \in U$

Then, $f(b) - f(a) \leq K (b-a)$

TheoremTheorem 3.7 Supremum Principle

Let,

$A \subseteq \mathbb{R}$ be non empty.
$A$ is bounded above, $\exists K \in \mathbb{R} : a \leq K\ \forall a \in A$

Then, $A$ has a supremum.