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, NaturalPos, greater_eq, one, subtract, zero

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

\forall_{N \in \mathbb{N}^+}~\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\}} = \{0~\ldotp \ldotp~N - 1\}\right)\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 54 body: 4
3	ExprTuple	54
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 49 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	54, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	63, 64, 61
13	Conditional	value: 14 condition: 15
14	Operation	operator: 16 operands: 17
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	22, 23, 24, 25
20	Operation	operator: 26 operand: 34
21	Operation	operator: 59 operands: 28
22	Operation	operator: 49 operands: 29
23	Operation	operator: 49 operands: 30
24	Operation	operator: 49 operands: 31
25	Operation	operator: 32 operands: 33
26	Literal
27	ExprTuple	34
28	ExprTuple	35, 36
29	ExprTuple	63, 37
30	ExprTuple	64, 37
31	ExprTuple	61, 37
32	Literal
33	ExprTuple	54, 38
34	Lambda	parameter: 62 body: 40
35	Literal
36	Operation	operator: 57 operands: 41
37	Literal
38	Operation	operator: 42 operand: 55
39	ExprTuple	62
40	Conditional	value: 44 condition: 45
41	ExprTuple	54, 46
42	Literal
43	ExprTuple	55
44	Operation	operator: 47 operands: 48
45	Operation	operator: 49 operands: 50
46	Operation	operator: 51 operand: 56
47	Literal
48	ExprTuple	53, 54
49	Literal
50	ExprTuple	62, 55
51	Literal
52	ExprTuple	56
53	Operation	operator: 57 operands: 58
54	Variable
55	Operation	operator: 59 operands: 60
56	Literal
57	Literal
58	ExprTuple	61, 62
59	Literal
60	ExprTuple	63, 64
61	Variable
62	Variable
63	Variable
64	Variable