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, 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 = Conditional(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), 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())

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