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, Implies, 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 = Implies(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) \\  \Rightarrow \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, 34, 30, 26
5	Operation	operator: 7 operands: 8
6	ExprRange	lambda_map: 9 start_index: 41 end_index: 42
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 48 body: 43
10	Operation	operator: 38 operands: 12
11	Operation	operator: 13 operand: 16
12	ExprTuple	26, 15
13	Literal
14	ExprTuple	16
15	Operation	operator: 17 operand: 20
16	Lambda	parameters: 35 body: 19
17	Literal
18	ExprTuple	20
19	Conditional	value: 21 condition: 25
20	Lambda	parameters: 35 body: 22
21	Operation	operator: 23 operands: 24
22	Conditional	value: 27 condition: 25
23	Literal
24	ExprTuple	26, 27
25	Operation	operator: 28 operands: 29
26	Variable
27	Operation	operator: 30 operands: 35
28	Literal
29	ExprTuple	31, 32
30	Variable
31	ExprRange	lambda_map: 33 start_index: 41 end_index: 42
32	Operation	operator: 34 operands: 35
33	Lambda	parameter: 48 body: 36
34	Variable
35	ExprTuple	37
36	Operation	operator: 38 operands: 39
37	ExprRange	lambda_map: 40 start_index: 41 end_index: 42
38	Literal
39	ExprTuple	44, 43
40	Lambda	parameter: 48 body: 44
41	Literal
42	Variable
43	IndexedVar	variable: 45 index: 48
44	IndexedVar	variable: 46 index: 48
45	Variable
46	Variable
47	ExprTuple	48
48	Variable