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 x_1_to_n
from proveit.linear_algebra import Bases, LinDepSets, SpanningSets, VecSpaces
from proveit.logic import And, Equals, Forall, InClass, InSet, NotInSet, Set
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
expr = Conditional(Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(InSet(sub_expr1, Bases(V)), And(InSet(sub_expr1, SpanningSets(V)), NotInSet(sub_expr1, LinDepSets(V))).with_wrapping_at(2)).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} \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{Bases}\left(V\right)\right) =  \\ \left(\begin{array}{c} \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{SpanningSets}\left(V\right)\right) \land  \\ \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \notin \textrm{LinDepSets}\left(V\right)\right) \end{array}\right) \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: 15 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	52, 7
6	Lambda	parameter: 55 body: 9
7	Operation	operator: 10 operand: 14
8	ExprTuple	55
9	Conditional	value: 12 condition: 13
10	Literal
11	ExprTuple	14
12	Operation	operator: 15 operand: 18
13	Operation	operator: 41 operands: 17
14	Variable
15	Literal
16	ExprTuple	18
17	ExprTuple	55, 19
18	Lambda	parameters: 48 body: 20
19	Literal
20	Conditional	value: 21 condition: 22
21	Operation	operator: 23 operands: 24
22	Operation	operator: 30 operands: 25
23	Literal
24	ExprTuple	26, 27
25	ExprTuple	28
26	Operation	operator: 41 operands: 29
27	Operation	operator: 30 operands: 31
28	ExprRange	lambda_map: 32 start_index: 54 end_index: 55
29	ExprTuple	44, 33
30	Literal
31	ExprTuple	34, 35
32	Lambda	parameter: 59 body: 36
33	Operation	operator: 37 operand: 52
34	Operation	operator: 41 operands: 38
35	Operation	operator: 39 operands: 40
36	Operation	operator: 41 operands: 42
37	Literal
38	ExprTuple	44, 43
39	Literal
40	ExprTuple	44, 45
41	Literal
42	ExprTuple	56, 52
43	Operation	operator: 46 operand: 52
44	Operation	operator: 47 operands: 48
45	Operation	operator: 49 operand: 52
46	Literal
47	Literal
48	ExprTuple	51
49	Literal
50	ExprTuple	52
51	ExprRange	lambda_map: 53 start_index: 54 end_index: 55
52	Variable
53	Lambda	parameter: 59 body: 56
54	Literal
55	Variable
56	IndexedVar	variable: 57 index: 59
57	Variable
58	ExprTuple	59
59	Variable