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, ExprRange, IndexedVar, K, V, Variable, n, x
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 And, Equals, Exists, InSet, Set
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Conditional(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), And(ExprRange(sub_expr1, InSet(IndexedVar(x, sub_expr1), V), one, n)))

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}{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} \textrm{ if } \left(x_{1} \in V\right) \land  \left(x_{2} \in V\right) \land  \ldots \land  \left(x_{n} \in V\right)\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: 28 operands: 3
2	Operation	operator: 30 operands: 4
3	ExprTuple	5, 6
4	ExprTuple	7
5	Operation	operator: 48 operands: 8
6	Operation	operator: 9 operand: 14
7	ExprRange	lambda_map: 11 start_index: 51 end_index: 52
8	ExprTuple	12, 13
9	Literal
10	ExprTuple	14
11	Lambda	parameter: 66 body: 15
12	Operation	operator: 16 operands: 17
13	Operation	operator: 18 operand: 44
14	Lambda	parameters: 19 body: 20
15	Operation	operator: 48 operands: 21
16	Literal
17	ExprTuple	22
18	Literal
19	ExprTuple	23
20	Conditional	value: 24 condition: 25
21	ExprTuple	60, 44
22	ExprRange	lambda_map: 26 start_index: 51 end_index: 52
23	ExprRange	lambda_map: 27 start_index: 51 end_index: 52
24	Operation	operator: 28 operands: 29
25	Operation	operator: 30 operands: 31
26	Lambda	parameter: 66 body: 60
27	Lambda	parameter: 66 body: 61
28	Literal
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 36 operands: 37
33	Operation	operator: 38 operand: 44
34	ExprRange	lambda_map: 40 start_index: 51 end_index: 52
35	Operation	operator: 41 operands: 42
36	Literal
37	ExprTuple	43
38	Literal
39	ExprTuple	44
40	Lambda	parameter: 66 body: 45
41	Literal
42	ExprTuple	46
43	ExprRange	lambda_map: 47 start_index: 51 end_index: 52
44	Variable
45	Operation	operator: 48 operands: 49
46	ExprRange	lambda_map: 50 start_index: 51 end_index: 52
47	Lambda	parameter: 66 body: 53
48	Literal
49	ExprTuple	61, 54
50	Lambda	parameter: 66 body: 55
51	Literal
52	Variable
53	Operation	operator: 56 operands: 57
54	Variable
55	Operation	operator: 58 operands: 59
56	Literal
57	ExprTuple	61, 60
58	Literal
59	ExprTuple	61, 62
60	IndexedVar	variable: 63 index: 66
61	IndexedVar	variable: 64 index: 66
62	Literal
63	Variable
64	Variable
65	ExprTuple	66
66	Variable