===== Abstract (algebraic) Data Types =====

==== Base constructors of a ADT ====

Base constructors express how values of a certain ADT are constructed **canonically** (not as a result of other ADT operations). Base constructors are defined via their **signature**, which expresses how values of a certain type are constructed.

Define **base constructors** for the following types:

1. Binary tree with keys as integers (''Tree'')

2. FIFO (First-In First Out) with elements of an arbitrary type ''E'' (''FIFO'')

3. Map (A map is a one-to-one association of "keys" and "values") with keys of an arbitrary type ''K'' and values of an arbitrary type ''V'' (''Map'')


==== ADT operators and their axioms ====

**Operators** are //functionalities// or //operations// available for the datatype at hand. Operators have:
  * a **signature** 
  * a set of **axioms** which specify how the operator **implementation** should behave. There is no fixed ''recipe'' for finding proper axioms, but a good set of axioms usually:
      * specifies behaviour in terms of **base constructors**
      * is general enough
      * allows **proving** sensible properties of the ADT

2.1. Define axioms for the following operators:
  * ''size : Tree -> Integer''
  * ''height : Tree -> Integer''
  * ''mirror : Tree -> Tree''
  * ''flatten : Tree -> List''

2.2. Define operators and their proper axioms for:
  * the concatenation of two ''FIFO''s
  * the operations ''top'' and ''dequeue'' used in a previous lecture (''top'' produces the //first-in// element of the ''FIFO'' while ''dequeue'' returns a ''FIFO'' without this very element)

==== ADT implementation ====

Implement the ''ADT'' ''Tree'' and ''FIFO'' in Python, together with thier operations. **Use two stacks to implement the ''FIFO''**.
For the implementation, you may use classes, following the example shown in the lecture. To what extent can you use the axioms to provide //**correct**// implementations for each operation?

==== The ADT ''Map'' ====

3.1. Define axioms for the following ''Map'' operations:
   * ''get'' which returns the value associated with a **given** key, if the **key exists**
   * ''update'' which modifies the value associated with a **given** key, if the **key exists**
   * ''is_defined'' which returns ''True'' if a certain key has a defined value
   * ''remove'' which deletes an entry based on it's key.

3.2. (Optional). Implement ''Map'' as a ''HashMap'' (ro: folosind tabele de dispersie).