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, f, x
from proveit.core_expr_types import Q__y_1_to_n, S_1_to_n, f__y_1_to_n, y_1_to_n
from proveit.logic import Equals, Exists, Forall, InSet
from proveit.logic.sets import general_comprehension_fyn

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [S_1_to_n, Q, f, x], instance_expr = Equals(InSet(x, general_comprehension_fyn), Exists(instance_param_or_params = [y_1_to_n], instance_expr = Equals(x, f__y_1_to_n), domains = [S_1_to_n], condition = Q__y_1_to_n)).with_wrapping_at(1))

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_{S_{1}, S_{2}, \ldots, S_{n}, Q, f, x}~\left(\begin{array}{c} \begin{array}{l} \left(x \in \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)}\right) \\  = \left[\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)}~\left(x = f\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right)\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	6, 33, 29, 25
5	Operation	operator: 22 operands: 7
6	ExprRange	lambda_map: 8 start_index: 40 end_index: 41
7	ExprTuple	9, 10
8	Lambda	parameter: 47 body: 42
9	Operation	operator: 37 operands: 11
10	Operation	operator: 12 operand: 15
11	ExprTuple	25, 14
12	Literal
13	ExprTuple	15
14	Operation	operator: 16 operand: 19
15	Lambda	parameters: 34 body: 18
16	Literal
17	ExprTuple	19
18	Conditional	value: 20 condition: 24
19	Lambda	parameters: 34 body: 21
20	Operation	operator: 22 operands: 23
21	Conditional	value: 26 condition: 24
22	Literal
23	ExprTuple	25, 26
24	Operation	operator: 27 operands: 28
25	Variable
26	Operation	operator: 29 operands: 34
27	Literal
28	ExprTuple	30, 31
29	Variable
30	ExprRange	lambda_map: 32 start_index: 40 end_index: 41
31	Operation	operator: 33 operands: 34
32	Lambda	parameter: 47 body: 35
33	Variable
34	ExprTuple	36
35	Operation	operator: 37 operands: 38
36	ExprRange	lambda_map: 39 start_index: 40 end_index: 41
37	Literal
38	ExprTuple	43, 42
39	Lambda	parameter: 47 body: 43
40	Literal
41	Variable
42	IndexedVar	variable: 44 index: 47
43	IndexedVar	variable: 45 index: 47
44	Variable
45	Variable
46	ExprTuple	47
47	Variable