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, c, x
from proveit.logic import Card, Equals, Forall, InSet, SetOfAll
from proveit.numbers import Add, Integer, Interval, Mod, NaturalPos, greater_eq

# build up the expression from sub-expressions
sub_expr1 = Interval(a, b)
expr = Conditional(Forall(instance_param_or_params = [a, b, c], instance_expr = Equals(Card(SetOfAll(instance_param_or_params = [x], instance_element = Mod(Add(c, x), N), domain = sub_expr1)), N), domain = Integer, condition = greater_eq(Card(sub_expr1), N)), InSet(N, NaturalPos))

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\{\forall_{a, b, c \in \mathbb{Z}~|~|\{a~\ldotp \ldotp~b\}| \geq N}~\left(|\left\{\left(c + x\right) ~\textup{mod}~ N\right\}_{x \in \{a~\ldotp \ldotp~b\}}| = N\right) \textrm{ if } N \in \mathbb{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 operand: 6
2	Operation	operator: 41 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	44, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	52, 53, 50
9	Conditional	value: 10 condition: 11
10	Operation	operator: 12 operands: 13
11	Operation	operator: 14 operands: 15
12	Literal
13	ExprTuple	16, 44
14	Literal
15	ExprTuple	17, 18, 19, 20
16	Operation	operator: 32 operand: 27
17	Operation	operator: 41 operands: 22
18	Operation	operator: 41 operands: 23
19	Operation	operator: 41 operands: 24
20	Operation	operator: 25 operands: 26
21	ExprTuple	27
22	ExprTuple	52, 28
23	ExprTuple	53, 28
24	ExprTuple	50, 28
25	Literal
26	ExprTuple	44, 29
27	Operation	operator: 30 operand: 34
28	Literal
29	Operation	operator: 32 operand: 45
30	Literal
31	ExprTuple	34
32	Literal
33	ExprTuple	45
34	Lambda	parameter: 51 body: 36
35	ExprTuple	51
36	Conditional	value: 37 condition: 38
37	Operation	operator: 39 operands: 40
38	Operation	operator: 41 operands: 42
39	Literal
40	ExprTuple	43, 44
41	Literal
42	ExprTuple	51, 45
43	Operation	operator: 46 operands: 47
44	Variable
45	Operation	operator: 48 operands: 49
46	Literal
47	ExprTuple	50, 51
48	Literal
49	ExprTuple	52, 53
50	Variable
51	Variable
52	Variable
53	Variable