Expression of type Forall

from the theory of proveit.logic.sets.intersection

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 Q, R, n, x
from proveit.core_expr_types import Q__y_1_to_n, R__y_1_to_n, S_1_to_n, y_1_to_n
from proveit.logic import Equals, Exists, Forall, InSet
from proveit.logic.sets import general_intersectall_Ryn
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
sub_expr1 = [y_1_to_n]
sub_expr2 = [S_1_to_n]
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [S_1_to_n, Q, R, x], instance_expr = Equals(InSet(x, general_intersectall_Ryn), Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(x, R__y_1_to_n), domains = sub_expr2, condition = Q__y_1_to_n)).with_wrapping_at(1), condition = Exists(instance_param_or_params = sub_expr1, instance_expr = Q__y_1_to_n, domains = sub_expr2)), 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_{S_{1}, S_{2}, \ldots, S_{n}, Q, R, x~|~\exists_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)}~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(\begin{array}{c} \begin{array}{l} \left(x \in \left[\bigcap_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~R\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right]\right) \\  = \left[\forall_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(x \in R\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right)\right] \end{array} \end{array}\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: 25 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 56 body: 4
3	ExprTuple	56
4	Conditional	value: 5 condition: 6
5	Operation	operator: 25 operand: 9
6	Operation	operator: 52 operands: 8
7	ExprTuple	9
8	ExprTuple	56, 10
9	Lambda	parameters: 11 body: 12
10	Literal
11	ExprTuple	13, 48, 44, 40
12	Conditional	value: 14 condition: 15
13	ExprRange	lambda_map: 16 start_index: 55 end_index: 56
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operand: 23
16	Lambda	parameter: 62 body: 57
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23
21	Operation	operator: 52 operands: 24
22	Operation	operator: 25 operand: 29
23	Lambda	parameters: 49 body: 27
24	ExprTuple	40, 28
25	Literal
26	ExprTuple	29
27	Conditional	value: 46 condition: 30
28	Operation	operator: 31 operand: 35
29	Lambda	parameters: 49 body: 33
30	Operation	operator: 42 operands: 34
31	Literal
32	ExprTuple	35
33	Conditional	value: 36 condition: 39
34	ExprTuple	45
35	Lambda	parameters: 49 body: 37
36	Operation	operator: 52 operands: 38
37	Conditional	value: 41 condition: 39
38	ExprTuple	40, 41
39	Operation	operator: 42 operands: 43
40	Variable
41	Operation	operator: 44 operands: 49
42	Literal
43	ExprTuple	45, 46
44	Variable
45	ExprRange	lambda_map: 47 start_index: 55 end_index: 56
46	Operation	operator: 48 operands: 49
47	Lambda	parameter: 62 body: 50
48	Variable
49	ExprTuple	51
50	Operation	operator: 52 operands: 53
51	ExprRange	lambda_map: 54 start_index: 55 end_index: 56
52	Literal
53	ExprTuple	58, 57
54	Lambda	parameter: 62 body: 58
55	Literal
56	Variable
57	IndexedVar	variable: 59 index: 62
58	IndexedVar	variable: 60 index: 62
59	Variable
60	Variable
61	ExprTuple	62
62	Variable