Expression of type Conditional

from the theory of proveit.numbers.modular

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, N, a, b, x
from proveit.logic import And, Card, Equals, InSet, SetOfAll
from proveit.numbers import Integer, Interval, LessEq, Mod

# build up the expression from sub-expressions
sub_expr1 = Interval(a, b)
sub_expr2 = Card(sub_expr1)
expr = Conditional(Equals(Card(SetOfAll(instance_param_or_params = [x], instance_element = Mod(x, N), domain = sub_expr1)), sub_expr2), And(InSet(a, Integer), InSet(b, Integer), LessEq(sub_expr2, N)))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{|\left\{x ~\textup{mod}~ N\right\}_{x \in \{a~\ldotp \ldotp~b\}}| = |\{a~\ldotp \ldotp~b\}| \textrm{ if } a \in \mathbb{Z} ,  b \in \mathbb{Z} ,  |\{a~\ldotp \ldotp~b\}| \leq N\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operands: 4
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7, 18
5	Literal
6	ExprTuple	8, 9, 10
7	Operation	operator: 21 operand: 16
8	Operation	operator: 30 operands: 12
9	Operation	operator: 30 operands: 13
10	Operation	operator: 14 operands: 15
11	ExprTuple	16
12	ExprTuple	37, 17
13	ExprTuple	38, 17
14	Literal
15	ExprTuple	18, 32
16	Operation	operator: 19 operand: 23
17	Literal
18	Operation	operator: 21 operand: 34
19	Literal
20	ExprTuple	23
21	Literal
22	ExprTuple	34
23	Lambda	parameter: 33 body: 25
24	ExprTuple	33
25	Conditional	value: 26 condition: 27
26	Operation	operator: 28 operands: 29
27	Operation	operator: 30 operands: 31
28	Literal
29	ExprTuple	33, 32
30	Literal
31	ExprTuple	33, 34
32	Variable
33	Variable
34	Operation	operator: 35 operands: 36
35	Literal
36	ExprTuple	37, 38
37	Variable
38	Variable