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, 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

# 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 = [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))

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_{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)

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: 17 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	5, 40, 36, 32
4	Conditional	value: 6 condition: 7
5	ExprRange	lambda_map: 8 start_index: 47 end_index: 48
6	Operation	operator: 9 operands: 10
7	Operation	operator: 11 operand: 15
8	Lambda	parameter: 54 body: 49
9	Literal
10	ExprTuple	13, 14
11	Literal
12	ExprTuple	15
13	Operation	operator: 44 operands: 16
14	Operation	operator: 17 operand: 21
15	Lambda	parameters: 41 body: 19
16	ExprTuple	32, 20
17	Literal
18	ExprTuple	21
19	Conditional	value: 38 condition: 22
20	Operation	operator: 23 operand: 27
21	Lambda	parameters: 41 body: 25
22	Operation	operator: 34 operands: 26
23	Literal
24	ExprTuple	27
25	Conditional	value: 28 condition: 31
26	ExprTuple	37
27	Lambda	parameters: 41 body: 29
28	Operation	operator: 44 operands: 30
29	Conditional	value: 33 condition: 31
30	ExprTuple	32, 33
31	Operation	operator: 34 operands: 35
32	Variable
33	Operation	operator: 36 operands: 41
34	Literal
35	ExprTuple	37, 38
36	Variable
37	ExprRange	lambda_map: 39 start_index: 47 end_index: 48
38	Operation	operator: 40 operands: 41
39	Lambda	parameter: 54 body: 42
40	Variable
41	ExprTuple	43
42	Operation	operator: 44 operands: 45
43	ExprRange	lambda_map: 46 start_index: 47 end_index: 48
44	Literal
45	ExprTuple	50, 49
46	Lambda	parameter: 54 body: 50
47	Literal
48	Variable
49	IndexedVar	variable: 51 index: 54
50	IndexedVar	variable: 52 index: 54
51	Variable
52	Variable
53	ExprTuple	54
54	Variable