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, V, Variable, n, x
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import Bases, LinDepSets, SpanningSets
from proveit.logic import And, Equals, InSet, NotInSet, Set
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Set(x_1_to_n)
expr = Conditional(Equals(InSet(sub_expr2, Bases(V)), And(InSet(sub_expr2, SpanningSets(V)), NotInSet(sub_expr2, LinDepSets(V))).with_wrapping_at(2)).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{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} \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: 3 operands: 4
2	Operation	operator: 10 operands: 5
3	Literal
4	ExprTuple	6, 7
5	ExprTuple	8
6	Operation	operator: 21 operands: 9
7	Operation	operator: 10 operands: 11
8	ExprRange	lambda_map: 12 start_index: 34 end_index: 35
9	ExprTuple	24, 13
10	Literal
11	ExprTuple	14, 15
12	Lambda	parameter: 39 body: 16
13	Operation	operator: 17 operand: 32
14	Operation	operator: 21 operands: 18
15	Operation	operator: 19 operands: 20
16	Operation	operator: 21 operands: 22
17	Literal
18	ExprTuple	24, 23
19	Literal
20	ExprTuple	24, 25
21	Literal
22	ExprTuple	36, 32
23	Operation	operator: 26 operand: 32
24	Operation	operator: 27 operands: 28
25	Operation	operator: 29 operand: 32
26	Literal
27	Literal
28	ExprTuple	31
29	Literal
30	ExprTuple	32
31	ExprRange	lambda_map: 33 start_index: 34 end_index: 35
32	Variable
33	Lambda	parameter: 39 body: 36
34	Literal
35	Variable
36	IndexedVar	variable: 37 index: 39
37	Variable
38	ExprTuple	39
39	Variable