Expression of type Lambda

from the theory of proveit.logic.sets.comprehension

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, f, n
from proveit.core_expr_types import Q__x_1_to_n, R__x_1_to_n, S_1_to_n, f__x_1_to_n, x_1_to_n
from proveit.logic import Forall, Implies, InSet, SetOfAll, SubsetEq
from proveit.logic.sets import general_comprehension_fyn
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
sub_expr2 = [S_1_to_n]
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [f, Q, R, S_1_to_n], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Q__x_1_to_n, domains = sub_expr2, condition = R__x_1_to_n), SubsetEq(SetOfAll(instance_param_or_params = sub_expr1, instance_element = f__x_1_to_n, domains = sub_expr2, condition = R__x_1_to_n), general_comprehension_fyn).with_wrapping_at(2)).with_wrapping_at(2)), 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_{f, Q, R, S_{1}, S_{2}, \ldots, S_{n}}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{\left(x_{1} \in S_{1}\right), \left(x_{2} \in S_{2}\right), \ldots, \left(x_{n} \in S_{n}\right)~|~R\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left\{f\left(x_{1}, x_{2}, \ldots, x_{n}\right)~|~R\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right\}_{\left(x_{1} \in S_{1}\right), \left(x_{2} \in S_{2}\right), \ldots, \left(x_{n} \in S_{n}\right)} \subseteq  \\ \left\{f\left(y_{1}, y_{2}, \ldots, y_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right\}_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)} \end{array} \end{array}\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: 61 body: 2
1	ExprTuple	61
2	Conditional	value: 3 condition: 4
3	Operation	operator: 17 operand: 7
4	Operation	operator: 57 operands: 6
5	ExprTuple	7
6	ExprTuple	61, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	38, 49, 46, 11
10	Operation	operator: 12 operands: 13
11	ExprRange	lambda_map: 14 start_index: 60 end_index: 61
12	Literal
13	ExprTuple	15, 16
14	Lambda	parameter: 69 body: 63
15	Operation	operator: 17 operand: 21
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21
19	Literal
20	ExprTuple	22, 23
21	Lambda	parameters: 47 body: 24
22	Operation	operator: 26 operand: 29
23	Operation	operator: 26 operand: 30
24	Conditional	value: 28 condition: 34
25	ExprTuple	29
26	Literal
27	ExprTuple	30
28	Operation	operator: 49 operands: 47
29	Lambda	parameters: 47 body: 31
30	Lambda	parameters: 50 body: 32
31	Conditional	value: 33 condition: 34
32	Conditional	value: 35 condition: 36
33	Operation	operator: 38 operands: 47
34	Operation	operator: 39 operands: 37
35	Operation	operator: 38 operands: 50
36	Operation	operator: 39 operands: 40
37	ExprTuple	41, 42
38	Variable
39	Literal
40	ExprTuple	43, 44
41	ExprRange	lambda_map: 45 start_index: 60 end_index: 61
42	Operation	operator: 46 operands: 47
43	ExprRange	lambda_map: 48 start_index: 60 end_index: 61
44	Operation	operator: 49 operands: 50
45	Lambda	parameter: 69 body: 51
46	Variable
47	ExprTuple	52
48	Lambda	parameter: 69 body: 53
49	Variable
50	ExprTuple	54
51	Operation	operator: 57 operands: 55
52	ExprRange	lambda_map: 56 start_index: 60 end_index: 61
53	Operation	operator: 57 operands: 58
54	ExprRange	lambda_map: 59 start_index: 60 end_index: 61
55	ExprTuple	62, 63
56	Lambda	parameter: 69 body: 62
57	Literal
58	ExprTuple	64, 63
59	Lambda	parameter: 69 body: 64
60	Literal
61	Variable
62	IndexedVar	variable: 65 index: 69
63	IndexedVar	variable: 66 index: 69
64	IndexedVar	variable: 67 index: 69
65	Variable
66	Variable
67	Variable
68	ExprTuple	69
69	Variable