Expression of type Conditional

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 Conditional, 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 = Conditional(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(), InSet(n, 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())

\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} \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 6
2	Operation	operator: 56 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	60, 7
6	Lambda	parameters: 25 body: 8
7	Literal
8	Conditional	value: 9 condition: 10
9	Operation	operator: 36 operands: 11
10	Operation	operator: 38 operands: 12
11	ExprTuple	13, 14
12	ExprTuple	15
13	Operation	operator: 56 operands: 16
14	Operation	operator: 17 operand: 22
15	ExprRange	lambda_map: 19 start_index: 59 end_index: 60
16	ExprTuple	20, 21
17	Literal
18	ExprTuple	22
19	Lambda	parameter: 74 body: 23
20	Operation	operator: 24 operands: 25
21	Operation	operator: 26 operand: 52
22	Lambda	parameters: 27 body: 28
23	Operation	operator: 56 operands: 29
24	Literal
25	ExprTuple	30
26	Literal
27	ExprTuple	31
28	Conditional	value: 32 condition: 33
29	ExprTuple	68, 52
30	ExprRange	lambda_map: 34 start_index: 59 end_index: 60
31	ExprRange	lambda_map: 35 start_index: 59 end_index: 60
32	Operation	operator: 36 operands: 37
33	Operation	operator: 38 operands: 39
34	Lambda	parameter: 74 body: 68
35	Lambda	parameter: 74 body: 69
36	Literal
37	ExprTuple	40, 41
38	Literal
39	ExprTuple	42, 43
40	Operation	operator: 44 operands: 45
41	Operation	operator: 46 operand: 52
42	ExprRange	lambda_map: 48 start_index: 59 end_index: 60
43	Operation	operator: 49 operands: 50
44	Literal
45	ExprTuple	51
46	Literal
47	ExprTuple	52
48	Lambda	parameter: 74 body: 53
49	Literal
50	ExprTuple	54
51	ExprRange	lambda_map: 55 start_index: 59 end_index: 60
52	Variable
53	Operation	operator: 56 operands: 57
54	ExprRange	lambda_map: 58 start_index: 59 end_index: 60
55	Lambda	parameter: 74 body: 61
56	Literal
57	ExprTuple	69, 62
58	Lambda	parameter: 74 body: 63
59	Literal
60	Variable
61	Operation	operator: 64 operands: 65
62	Variable
63	Operation	operator: 66 operands: 67
64	Literal
65	ExprTuple	69, 68
66	Literal
67	ExprTuple	69, 70
68	IndexedVar	variable: 71 index: 74
69	IndexedVar	variable: 72 index: 74
70	Literal
71	Variable
72	Variable
73	ExprTuple	74
74	Variable