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 Span, lin_comb_axn
from proveit.logic import And, Equals, InSet, Set, SetOfAll
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Conditional(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), 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} \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} \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: 31 operands: 5
3	Literal
4	ExprTuple	6, 7
5	ExprTuple	8
6	Operation	operator: 9 operand: 14
7	Operation	operator: 11 operand: 15
8	ExprRange	lambda_map: 13 start_index: 37 end_index: 38
9	Literal
10	ExprTuple	14
11	Literal
12	ExprTuple	15
13	Lambda	parameter: 51 body: 16
14	Operation	operator: 17 operands: 18
15	Lambda	parameters: 19 body: 20
16	Operation	operator: 43 operands: 21
17	Literal
18	ExprTuple	22
19	ExprTuple	23
20	Conditional	value: 24 condition: 25
21	ExprTuple	45, 26
22	ExprRange	lambda_map: 27 start_index: 37 end_index: 38
23	ExprRange	lambda_map: 28 start_index: 37 end_index: 38
24	Operation	operator: 29 operands: 30
25	Operation	operator: 31 operands: 32
26	Variable
27	Lambda	parameter: 51 body: 45
28	Lambda	parameter: 51 body: 46
29	Literal
30	ExprTuple	33
31	Literal
32	ExprTuple	34
33	ExprRange	lambda_map: 35 start_index: 37 end_index: 38
34	ExprRange	lambda_map: 36 start_index: 37 end_index: 38
35	Lambda	parameter: 51 body: 39
36	Lambda	parameter: 51 body: 40
37	Literal
38	Variable
39	Operation	operator: 41 operands: 42
40	Operation	operator: 43 operands: 44
41	Literal
42	ExprTuple	46, 45
43	Literal
44	ExprTuple	46, 47
45	IndexedVar	variable: 48 index: 51
46	IndexedVar	variable: 49 index: 51
47	Variable
48	Variable
49	Variable
50	ExprTuple	51
51	Variable