Expression of type Lambda

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, Lambda, 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 = Lambda(n, 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())

n \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 54 body: 2
1	ExprTuple	54
2	Conditional	value: 3 condition: 4
3	Operation	operator: 23 operand: 7
4	Operation	operator: 50 operands: 6
5	ExprTuple	7
6	ExprTuple	54, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	11, 46, 42, 38
10	Conditional	value: 12 condition: 13
11	ExprRange	lambda_map: 14 start_index: 53 end_index: 54
12	Operation	operator: 15 operands: 16
13	Operation	operator: 17 operand: 21
14	Lambda	parameter: 60 body: 55
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21
19	Operation	operator: 50 operands: 22
20	Operation	operator: 23 operand: 27
21	Lambda	parameters: 47 body: 25
22	ExprTuple	38, 26
23	Literal
24	ExprTuple	27
25	Conditional	value: 44 condition: 28
26	Operation	operator: 29 operand: 33
27	Lambda	parameters: 47 body: 31
28	Operation	operator: 40 operands: 32
29	Literal
30	ExprTuple	33
31	Conditional	value: 34 condition: 37
32	ExprTuple	43
33	Lambda	parameters: 47 body: 35
34	Operation	operator: 50 operands: 36
35	Conditional	value: 39 condition: 37
36	ExprTuple	38, 39
37	Operation	operator: 40 operands: 41
38	Variable
39	Operation	operator: 42 operands: 47
40	Literal
41	ExprTuple	43, 44
42	Variable
43	ExprRange	lambda_map: 45 start_index: 53 end_index: 54
44	Operation	operator: 46 operands: 47
45	Lambda	parameter: 60 body: 48
46	Variable
47	ExprTuple	49
48	Operation	operator: 50 operands: 51
49	ExprRange	lambda_map: 52 start_index: 53 end_index: 54
50	Literal
51	ExprTuple	56, 55
52	Lambda	parameter: 60 body: 56
53	Literal
54	Variable
55	IndexedVar	variable: 57 index: 60
56	IndexedVar	variable: 58 index: 60
57	Variable
58	Variable
59	ExprTuple	60
60	Variable