Expression of type Implies

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.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 = 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())

\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}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

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