Expression of type Forall

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 Q, R, f
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, SetOfAll, SubsetEq
from proveit.logic.sets import general_comprehension_fyn

# build up the expression from sub-expressions
sub_expr1 = [x_1_to_n]
sub_expr2 = [S_1_to_n]
expr = 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))

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())

\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)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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