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, 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 = Lambda([S_1_to_n, Q, R, x], 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(S_{1}, S_{2}, \ldots, S_{n}, Q, R, x\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

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