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 Span, lin_comb_axn
from proveit.logic import Equals, Forall, InSet, Set, SetOfAll
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(Span(Set(x_1_to_n)), SetOfAll(instance_param_or_params = [a_1_to_n], instance_element = lin_comb_axn, domain = K)).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} \textrm{Span}\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}\right) =  \\ \left\{\left(a_{1} \cdot x_{1}\right) +  \left(a_{2} \cdot x_{2}\right) +  \ldots +  \left(a_{n} \cdot x_{n}\right)\right\}_{a_{1}, a_{2}, \ldots, a_{n} \in K} \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: 51 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	46, 7
6	Lambda	parameters: 26 body: 8
7	Literal
8	Conditional	value: 9 condition: 10
9	Operation	operator: 11 operands: 12
10	Operation	operator: 39 operands: 13
11	Literal
12	ExprTuple	14, 15
13	ExprTuple	16
14	Operation	operator: 17 operand: 22
15	Operation	operator: 19 operand: 23
16	ExprRange	lambda_map: 21 start_index: 45 end_index: 46
17	Literal
18	ExprTuple	22
19	Literal
20	ExprTuple	23
21	Lambda	parameter: 59 body: 24
22	Operation	operator: 25 operands: 26
23	Lambda	parameters: 27 body: 28
24	Operation	operator: 51 operands: 29
25	Literal
26	ExprTuple	30
27	ExprTuple	31
28	Conditional	value: 32 condition: 33
29	ExprTuple	53, 34
30	ExprRange	lambda_map: 35 start_index: 45 end_index: 46
31	ExprRange	lambda_map: 36 start_index: 45 end_index: 46
32	Operation	operator: 37 operands: 38
33	Operation	operator: 39 operands: 40
34	Variable
35	Lambda	parameter: 59 body: 53
36	Lambda	parameter: 59 body: 54
37	Literal
38	ExprTuple	41
39	Literal
40	ExprTuple	42
41	ExprRange	lambda_map: 43 start_index: 45 end_index: 46
42	ExprRange	lambda_map: 44 start_index: 45 end_index: 46
43	Lambda	parameter: 59 body: 47
44	Lambda	parameter: 59 body: 48
45	Literal
46	Variable
47	Operation	operator: 49 operands: 50
48	Operation	operator: 51 operands: 52
49	Literal
50	ExprTuple	54, 53
51	Literal
52	ExprTuple	54, 55
53	IndexedVar	variable: 56 index: 59
54	IndexedVar	variable: 57 index: 59
55	Variable
56	Variable
57	Variable
58	ExprTuple	59
59	Variable