Expression of type Forall

from the theory of proveit.linear_algebra.vector_sets

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 K, V, n
from proveit.core_expr_types import a_1_to_n, x_1_to_n
from proveit.linear_algebra import LinDepSets, VecZero, lin_comb_axn, some_nonzero_a
from proveit.logic import Equals, Exists, Forall, InSet, Set
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 = [x_1_to_n], instance_expr = Equals(InSet(Set(x_1_to_n), LinDepSets(V)), Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(lin_comb_axn, VecZero(V)), domain = K, condition = some_nonzero_a).with_wrapping()).with_wrapping_at(2), domain = V).with_wrapping(), 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[\begin{array}{l}\forall_{x_{1}, x_{2}, \ldots, x_{n} \in V}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{LinDepSets}\left(V\right)\right) =  \\ \left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in K~|~\left(a_{1} \neq 0\right) \lor  \left(a_{2} \neq 0\right) \lor  \ldots \lor  \left(a_{n} \neq 0\right)}~\\
\left(\left(\left(a_{1} \cdot x_{1}\right) +  \left(a_{2} \cdot x_{2}\right) +  \ldots +  \left(a_{n} \cdot x_{n}\right)\right) = \vec{0}\left(V\right)\right)\end{array}\right] \end{array} \end{array}\right)\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 64 body: 4
3	ExprTuple	64
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 60 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	64, 11
10	Lambda	parameters: 29 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 40 operands: 15
14	Operation	operator: 42 operands: 16
15	ExprTuple	17, 18
16	ExprTuple	19
17	Operation	operator: 60 operands: 20
18	Operation	operator: 21 operand: 26
19	ExprRange	lambda_map: 23 start_index: 63 end_index: 64
20	ExprTuple	24, 25
21	Literal
22	ExprTuple	26
23	Lambda	parameter: 78 body: 27
24	Operation	operator: 28 operands: 29
25	Operation	operator: 30 operand: 56
26	Lambda	parameters: 31 body: 32
27	Operation	operator: 60 operands: 33
28	Literal
29	ExprTuple	34
30	Literal
31	ExprTuple	35
32	Conditional	value: 36 condition: 37
33	ExprTuple	72, 56
34	ExprRange	lambda_map: 38 start_index: 63 end_index: 64
35	ExprRange	lambda_map: 39 start_index: 63 end_index: 64
36	Operation	operator: 40 operands: 41
37	Operation	operator: 42 operands: 43
38	Lambda	parameter: 78 body: 72
39	Lambda	parameter: 78 body: 73
40	Literal
41	ExprTuple	44, 45
42	Literal
43	ExprTuple	46, 47
44	Operation	operator: 48 operands: 49
45	Operation	operator: 50 operand: 56
46	ExprRange	lambda_map: 52 start_index: 63 end_index: 64
47	Operation	operator: 53 operands: 54
48	Literal
49	ExprTuple	55
50	Literal
51	ExprTuple	56
52	Lambda	parameter: 78 body: 57
53	Literal
54	ExprTuple	58
55	ExprRange	lambda_map: 59 start_index: 63 end_index: 64
56	Variable
57	Operation	operator: 60 operands: 61
58	ExprRange	lambda_map: 62 start_index: 63 end_index: 64
59	Lambda	parameter: 78 body: 65
60	Literal
61	ExprTuple	73, 66
62	Lambda	parameter: 78 body: 67
63	Literal
64	Variable
65	Operation	operator: 68 operands: 69
66	Variable
67	Operation	operator: 70 operands: 71
68	Literal
69	ExprTuple	73, 72
70	Literal
71	ExprTuple	73, 74
72	IndexedVar	variable: 75 index: 78
73	IndexedVar	variable: 76 index: 78
74	Literal
75	Variable
76	Variable
77	ExprTuple	78
78	Variable