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, x
from proveit.logic import Card, Equals, Forall, SetOfAll
from proveit.numbers import 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], instance_expr = Equals(SetOfAll(instance_param_or_params = [x], instance_element = Mod(x, 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 \in \mathbb{Z}~|~|\{a~\ldotp \ldotp~b\}| \geq N}~\left(\left\{x ~\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: 48 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11, 12
9	Literal
10	ExprTuple	13, 14, 15
11	Operation	operator: 16 operand: 23
12	Operation	operator: 47 operands: 18
13	Operation	operator: 39 operands: 19
14	Operation	operator: 39 operands: 20
15	Operation	operator: 21 operands: 22
16	Literal
17	ExprTuple	23
18	ExprTuple	24, 25
19	ExprTuple	49, 26
20	ExprTuple	50, 26
21	Literal
22	ExprTuple	43, 27
23	Lambda	parameter: 44 body: 29
24	Literal
25	Operation	operator: 30 operands: 31
26	Literal
27	Operation	operator: 32 operand: 45
28	ExprTuple	44
29	Conditional	value: 34 condition: 35
30	Literal
31	ExprTuple	43, 36
32	Literal
33	ExprTuple	45
34	Operation	operator: 37 operands: 38
35	Operation	operator: 39 operands: 40
36	Operation	operator: 41 operand: 46
37	Literal
38	ExprTuple	44, 43
39	Literal
40	ExprTuple	44, 45
41	Literal
42	ExprTuple	46
43	Variable
44	Variable
45	Operation	operator: 47 operands: 48
46	Literal
47	Literal
48	ExprTuple	49, 50
49	Variable
50	Variable