Expression of type Forall

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 N, a, b, c, x
from proveit.logic import Card, Equals, Forall, SetOfAll
from proveit.numbers import Add, Integer, Interval, Mod, greater_eq

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

\forall_{a, b, c \in \mathbb{Z}~|~|\{a~\ldotp \ldotp~b\}| \geq N}~\left(|\left\{\left(x + c\right) ~\textup{mod}~ N\right\}_{x \in \{a~\ldotp \ldotp~b\}}| = N\right)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	48, 49, 47
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 40
10	Literal
11	ExprTuple	13, 14, 15, 16
12	Operation	operator: 28 operand: 23
13	Operation	operator: 37 operands: 18
14	Operation	operator: 37 operands: 19
15	Operation	operator: 37 operands: 20
16	Operation	operator: 21 operands: 22
17	ExprTuple	23
18	ExprTuple	48, 24
19	ExprTuple	49, 24
20	ExprTuple	47, 24
21	Literal
22	ExprTuple	40, 25
23	Operation	operator: 26 operand: 30
24	Literal
25	Operation	operator: 28 operand: 41
26	Literal
27	ExprTuple	30
28	Literal
29	ExprTuple	41
30	Lambda	parameter: 46 body: 32
31	ExprTuple	46
32	Conditional	value: 33 condition: 34
33	Operation	operator: 35 operands: 36
34	Operation	operator: 37 operands: 38
35	Literal
36	ExprTuple	39, 40
37	Literal
38	ExprTuple	46, 41
39	Operation	operator: 42 operands: 43
40	Variable
41	Operation	operator: 44 operands: 45
42	Literal
43	ExprTuple	46, 47
44	Literal
45	ExprTuple	48, 49
46	Variable
47	Variable
48	Variable
49	Variable