Difference between revisions of "Notation"

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* A percent sign <code>%</code> preceded only by spaces starts a line comment <code>% [comment]</code>
 
* A percent sign <code>%</code> preceded only by spaces starts a line comment <code>% [comment]</code>
 
* We use <code>·</code> to denote that we do not care about the value some variable takes. For example, when quantifying over a set of tuples of the form <math>(x, y)</math> we may use a <math>(x, \cdot)</math> to express that we do not care about the value <math>y</math> takes in each of the tuples.
 
* We use <code>·</code> to denote that we do not care about the value some variable takes. For example, when quantifying over a set of tuples of the form <math>(x, y)</math> we may use a <math>(x, \cdot)</math> to express that we do not care about the value <math>y</math> takes in each of the tuples.
* Let <math>X</math> be a set or a sequence. We use <math>|X|</math> to denote the size/length of the set/sequence.
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* Let <code>X</code> be a set or a sequence. We use <code>|X|</code> to denote the size/length of the set/sequence.
 +
* Let <code>X</code> be a sequence. We use <code>X[i]</code> to access the element in <code>X</code> at index <code>i</code> and <code>X[u…​v]</code> to refer to a slice of the sequence starting at index <code>u</code> and ending at index <code>v</code>. For example, for a sequence <code>X := (a, b, c, d)</code>, the <code>slice X[2…​3] = (b, c)</code>.
 +
* We define the concatenation of two ordered sequences <math>a := (x_1, …​, x_m)</math> and <math>b := (y_1, …​, y_n)</math>
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* We often work with partial maps <math>M: X \mapsto Y</math>, which we write as <math>\{ (x_1 \mapsto y_1), \dots, (x_n \mapsto y_n) \}</math>, where <math>\forall~ i, 1 \leq i \leq n : x_i \in X \land y_i \in Y</math> and <math>\nexists~ i, j, 1 \leq i, j \leq n: x_i = x_j \land i \neq j</math>. We note that partial maps can be nested, i.e., the set <math>Y</math> could again be a partial map

Revision as of 09:14, 3 November 2022

Notation

  • v : τ means value v has type τ.
  • X = a means that a type or a value X is an alias for a.
  • X := [something] means that X is defined to be [something]
  • A percent sign % preceded only by spaces starts a line comment % [comment]
  • We use · to denote that we do not care about the value some variable takes. For example, when quantifying over a set of tuples of the form [math]\displaystyle{ (x, y) }[/math] we may use a [math]\displaystyle{ (x, \cdot) }[/math] to express that we do not care about the value [math]\displaystyle{ y }[/math] takes in each of the tuples.
  • Let X be a set or a sequence. We use |X| to denote the size/length of the set/sequence.
  • Let X be a sequence. We use X[i] to access the element in X at index i and X[u…​v] to refer to a slice of the sequence starting at index u and ending at index v. For example, for a sequence X := (a, b, c, d), the slice X[2…​3] = (b, c).
  • We define the concatenation of two ordered sequences [math]\displaystyle{ a := (x_1, …​, x_m) }[/math] and [math]\displaystyle{ b := (y_1, …​, y_n) }[/math]
  • We often work with partial maps [math]\displaystyle{ M: X \mapsto Y }[/math], which we write as [math]\displaystyle{ \{ (x_1 \mapsto y_1), \dots, (x_n \mapsto y_n) \} }[/math], where [math]\displaystyle{ \forall~ i, 1 \leq i \leq n : x_i \in X \land y_i \in Y }[/math] and [math]\displaystyle{ \nexists~ i, j, 1 \leq i, j \leq n: x_i = x_j \land i \neq j }[/math]. We note that partial maps can be nested, i.e., the set [math]\displaystyle{ Y }[/math] could again be a partial map