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, one, subtract, zero

# build up the expression from sub-expressions
sub_expr1 = Interval(a, b)
expr = Forall(instance_param_or_params = [a, b, c], instance_expr = Equals(SetOfAll(instance_param_or_params = [x], instance_element = Mod(Add(x, c), N), domain = sub_expr1), Interval(zero, subtract(N, one))), 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\}} = \{0~\ldotp \ldotp~N - 1\}\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	55, 56, 54
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15, 16, 17
12	Operation	operator: 18 operand: 26
13	Operation	operator: 51 operands: 20
14	Operation	operator: 41 operands: 21
15	Operation	operator: 41 operands: 22
16	Operation	operator: 41 operands: 23
17	Operation	operator: 24 operands: 25
18	Literal
19	ExprTuple	26
20	ExprTuple	27, 28
21	ExprTuple	55, 29
22	ExprTuple	56, 29
23	ExprTuple	54, 29
24	Literal
25	ExprTuple	46, 30
26	Lambda	parameter: 53 body: 32
27	Literal
28	Operation	operator: 49 operands: 33
29	Literal
30	Operation	operator: 34 operand: 47
31	ExprTuple	53
32	Conditional	value: 36 condition: 37
33	ExprTuple	46, 38
34	Literal
35	ExprTuple	47
36	Operation	operator: 39 operands: 40
37	Operation	operator: 41 operands: 42
38	Operation	operator: 43 operand: 48
39	Literal
40	ExprTuple	45, 46
41	Literal
42	ExprTuple	53, 47
43	Literal
44	ExprTuple	48
45	Operation	operator: 49 operands: 50
46	Variable
47	Operation	operator: 51 operands: 52
48	Literal
49	Literal
50	ExprTuple	53, 54
51	Literal
52	ExprTuple	55, 56
53	Variable
54	Variable
55	Variable
56	Variable