Expression of type Conditional

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, 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 = 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())

\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()

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: 15 operand: 5
2	Operation	operator: 55 operands: 4
3	ExprTuple	5
4	ExprTuple	59, 6
5	Lambda	parameters: 7 body: 8
6	Literal
7	ExprTuple	36, 47, 44, 9
8	Operation	operator: 10 operands: 11
9	ExprRange	lambda_map: 12 start_index: 58 end_index: 59
10	Literal
11	ExprTuple	13, 14
12	Lambda	parameter: 67 body: 61
13	Operation	operator: 15 operand: 19
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19
17	Literal
18	ExprTuple	20, 21
19	Lambda	parameters: 45 body: 22
20	Operation	operator: 24 operand: 27
21	Operation	operator: 24 operand: 28
22	Conditional	value: 26 condition: 32
23	ExprTuple	27
24	Literal
25	ExprTuple	28
26	Operation	operator: 47 operands: 45
27	Lambda	parameters: 45 body: 29
28	Lambda	parameters: 48 body: 30
29	Conditional	value: 31 condition: 32
30	Conditional	value: 33 condition: 34
31	Operation	operator: 36 operands: 45
32	Operation	operator: 37 operands: 35
33	Operation	operator: 36 operands: 48
34	Operation	operator: 37 operands: 38
35	ExprTuple	39, 40
36	Variable
37	Literal
38	ExprTuple	41, 42
39	ExprRange	lambda_map: 43 start_index: 58 end_index: 59
40	Operation	operator: 44 operands: 45
41	ExprRange	lambda_map: 46 start_index: 58 end_index: 59
42	Operation	operator: 47 operands: 48
43	Lambda	parameter: 67 body: 49
44	Variable
45	ExprTuple	50
46	Lambda	parameter: 67 body: 51
47	Variable
48	ExprTuple	52
49	Operation	operator: 55 operands: 53
50	ExprRange	lambda_map: 54 start_index: 58 end_index: 59
51	Operation	operator: 55 operands: 56
52	ExprRange	lambda_map: 57 start_index: 58 end_index: 59
53	ExprTuple	60, 61
54	Lambda	parameter: 67 body: 60
55	Literal
56	ExprTuple	62, 61
57	Lambda	parameter: 67 body: 62
58	Literal
59	Variable
60	IndexedVar	variable: 63 index: 67
61	IndexedVar	variable: 64 index: 67
62	IndexedVar	variable: 65 index: 67
63	Variable
64	Variable
65	Variable
66	ExprTuple	67
67	Variable