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, VecSpaces, lin_comb_axn
from proveit.logic import Equals, Forall, InClass, Set, SetOfAll
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [n], instance_expr = 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(), domain = NaturalPos), InClass(V, VecSpaces(K)))

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\{\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} \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}\right] \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\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: 14 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	45, 7
6	Lambda	parameter: 57 body: 9
7	Operation	operator: 10 operand: 66
8	ExprTuple	57
9	Conditional	value: 12 condition: 13
10	Literal
11	ExprTuple	66
12	Operation	operator: 14 operand: 17
13	Operation	operator: 62 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	57, 18
17	Lambda	parameters: 37 body: 19
18	Literal
19	Conditional	value: 20 condition: 21
20	Operation	operator: 22 operands: 23
21	Operation	operator: 50 operands: 24
22	Literal
23	ExprTuple	25, 26
24	ExprTuple	27
25	Operation	operator: 28 operand: 33
26	Operation	operator: 30 operand: 34
27	ExprRange	lambda_map: 32 start_index: 56 end_index: 57
28	Literal
29	ExprTuple	33
30	Literal
31	ExprTuple	34
32	Lambda	parameter: 70 body: 35
33	Operation	operator: 36 operands: 37
34	Lambda	parameters: 38 body: 39
35	Operation	operator: 62 operands: 40
36	Literal
37	ExprTuple	41
38	ExprTuple	42
39	Conditional	value: 43 condition: 44
40	ExprTuple	64, 45
41	ExprRange	lambda_map: 46 start_index: 56 end_index: 57
42	ExprRange	lambda_map: 47 start_index: 56 end_index: 57
43	Operation	operator: 48 operands: 49
44	Operation	operator: 50 operands: 51
45	Variable
46	Lambda	parameter: 70 body: 64
47	Lambda	parameter: 70 body: 65
48	Literal
49	ExprTuple	52
50	Literal
51	ExprTuple	53
52	ExprRange	lambda_map: 54 start_index: 56 end_index: 57
53	ExprRange	lambda_map: 55 start_index: 56 end_index: 57
54	Lambda	parameter: 70 body: 58
55	Lambda	parameter: 70 body: 59
56	Literal
57	Variable
58	Operation	operator: 60 operands: 61
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	Variable
67	Variable
68	Variable
69	ExprTuple	70
70	Variable