Expression of type Conditional

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 Conditional, 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 = Conditional(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)), InSet(n, 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())

\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) \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 21 operand: 5
2	Operation	operator: 48 operands: 4
3	ExprTuple	5
4	ExprTuple	52, 6
5	Lambda	parameters: 7 body: 8
6	Literal
7	ExprTuple	9, 44, 40, 36
8	Conditional	value: 10 condition: 11
9	ExprRange	lambda_map: 12 start_index: 51 end_index: 52
10	Operation	operator: 13 operands: 14
11	Operation	operator: 15 operand: 19
12	Lambda	parameter: 58 body: 53
13	Literal
14	ExprTuple	17, 18
15	Literal
16	ExprTuple	19
17	Operation	operator: 48 operands: 20
18	Operation	operator: 21 operand: 25
19	Lambda	parameters: 45 body: 23
20	ExprTuple	36, 24
21	Literal
22	ExprTuple	25
23	Conditional	value: 42 condition: 26
24	Operation	operator: 27 operand: 31
25	Lambda	parameters: 45 body: 29
26	Operation	operator: 38 operands: 30
27	Literal
28	ExprTuple	31
29	Conditional	value: 32 condition: 35
30	ExprTuple	41
31	Lambda	parameters: 45 body: 33
32	Operation	operator: 48 operands: 34
33	Conditional	value: 37 condition: 35
34	ExprTuple	36, 37
35	Operation	operator: 38 operands: 39
36	Variable
37	Operation	operator: 40 operands: 45
38	Literal
39	ExprTuple	41, 42
40	Variable
41	ExprRange	lambda_map: 43 start_index: 51 end_index: 52
42	Operation	operator: 44 operands: 45
43	Lambda	parameter: 58 body: 46
44	Variable
45	ExprTuple	47
46	Operation	operator: 48 operands: 49
47	ExprRange	lambda_map: 50 start_index: 51 end_index: 52
48	Literal
49	ExprTuple	54, 53
50	Lambda	parameter: 58 body: 54
51	Literal
52	Variable
53	IndexedVar	variable: 55 index: 58
54	IndexedVar	variable: 56 index: 58
55	Variable
56	Variable
57	ExprTuple	58
58	Variable