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
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

# build up the expression from sub-expressions
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()

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}{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}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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: 21 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 32 operands: 7
6	Operation	operator: 34 operands: 8
7	ExprTuple	9, 10
8	ExprTuple	11
9	Operation	operator: 52 operands: 12
10	Operation	operator: 13 operand: 18
11	ExprRange	lambda_map: 15 start_index: 55 end_index: 56
12	ExprTuple	16, 17
13	Literal
14	ExprTuple	18
15	Lambda	parameter: 70 body: 19
16	Operation	operator: 20 operands: 21
17	Operation	operator: 22 operand: 48
18	Lambda	parameters: 23 body: 24
19	Operation	operator: 52 operands: 25
20	Literal
21	ExprTuple	26
22	Literal
23	ExprTuple	27
24	Conditional	value: 28 condition: 29
25	ExprTuple	64, 48
26	ExprRange	lambda_map: 30 start_index: 55 end_index: 56
27	ExprRange	lambda_map: 31 start_index: 55 end_index: 56
28	Operation	operator: 32 operands: 33
29	Operation	operator: 34 operands: 35
30	Lambda	parameter: 70 body: 64
31	Lambda	parameter: 70 body: 65
32	Literal
33	ExprTuple	36, 37
34	Literal
35	ExprTuple	38, 39
36	Operation	operator: 40 operands: 41
37	Operation	operator: 42 operand: 48
38	ExprRange	lambda_map: 44 start_index: 55 end_index: 56
39	Operation	operator: 45 operands: 46
40	Literal
41	ExprTuple	47
42	Literal
43	ExprTuple	48
44	Lambda	parameter: 70 body: 49
45	Literal
46	ExprTuple	50
47	ExprRange	lambda_map: 51 start_index: 55 end_index: 56
48	Variable
49	Operation	operator: 52 operands: 53
50	ExprRange	lambda_map: 54 start_index: 55 end_index: 56
51	Lambda	parameter: 70 body: 57
52	Literal
53	ExprTuple	65, 58
54	Lambda	parameter: 70 body: 59
55	Literal
56	Variable
57	Operation	operator: 60 operands: 61
58	Variable
59	Operation	operator: 62 operands: 63
60	Literal
61	ExprTuple	65, 64
62	Literal
63	ExprTuple	65, 66
64	IndexedVar	variable: 67 index: 70
65	IndexedVar	variable: 68 index: 70
66	Literal
67	Variable
68	Variable
69	ExprTuple	70
70	Variable