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, n, 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
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_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)), domain = 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())

\forall_{n \in \mathbb{N}^+}~\left[\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)\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 49 body: 4
3	ExprTuple	49
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 45 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	49, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	14, 41, 37, 33
13	Operation	operator: 30 operands: 15
14	ExprRange	lambda_map: 16 start_index: 48 end_index: 49
15	ExprTuple	17, 18
16	Lambda	parameter: 55 body: 50
17	Operation	operator: 45 operands: 19
18	Operation	operator: 20 operand: 23
19	ExprTuple	33, 22
20	Literal
21	ExprTuple	23
22	Operation	operator: 24 operand: 27
23	Lambda	parameters: 42 body: 26
24	Literal
25	ExprTuple	27
26	Conditional	value: 28 condition: 32
27	Lambda	parameters: 42 body: 29
28	Operation	operator: 30 operands: 31
29	Conditional	value: 34 condition: 32
30	Literal
31	ExprTuple	33, 34
32	Operation	operator: 35 operands: 36
33	Variable
34	Operation	operator: 37 operands: 42
35	Literal
36	ExprTuple	38, 39
37	Variable
38	ExprRange	lambda_map: 40 start_index: 48 end_index: 49
39	Operation	operator: 41 operands: 42
40	Lambda	parameter: 55 body: 43
41	Variable
42	ExprTuple	44
43	Operation	operator: 45 operands: 46
44	ExprRange	lambda_map: 47 start_index: 48 end_index: 49
45	Literal
46	ExprTuple	51, 50
47	Lambda	parameter: 55 body: 51
48	Literal
49	Variable
50	IndexedVar	variable: 52 index: 55
51	IndexedVar	variable: 53 index: 55
52	Variable
53	Variable
54	ExprTuple	55
55	Variable