Expression of type Lambda

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, Lambda, V, n
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import Span, SpanningSets
from proveit.logic import Equals, Forall, InSet, Set
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(InSet(sub_expr1, SpanningSets(V)), Equals(V, Span(sub_expr1))), 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())

n \mapsto \left\{\begin{array}{l}\forall_{x_{1}, x_{2}, \ldots, x_{n} \in V}~\\
\left(\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{SpanningSets}\left(V\right)\right) = \left(V = \textrm{Span}\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}\right)\right)\right)\end{array} \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 39 body: 2
1	ExprTuple	39
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 30 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	39, 9
8	Lambda	parameters: 35 body: 10
9	Literal
10	Conditional	value: 11 condition: 12
11	Operation	operator: 20 operands: 13
12	Operation	operator: 14 operands: 15
13	ExprTuple	16, 17
14	Literal
15	ExprTuple	18
16	Operation	operator: 30 operands: 19
17	Operation	operator: 20 operands: 21
18	ExprRange	lambda_map: 22 start_index: 38 end_index: 39
19	ExprTuple	32, 23
20	Literal
21	ExprTuple	33, 24
22	Lambda	parameter: 43 body: 25
23	Operation	operator: 26 operand: 33
24	Operation	operator: 28 operand: 32
25	Operation	operator: 30 operands: 31
26	Literal
27	ExprTuple	33
28	Literal
29	ExprTuple	32
30	Literal
31	ExprTuple	40, 33
32	Operation	operator: 34 operands: 35
33	Variable
34	Literal
35	ExprTuple	36
36	ExprRange	lambda_map: 37 start_index: 38 end_index: 39
37	Lambda	parameter: 43 body: 40
38	Literal
39	Variable
40	IndexedVar	variable: 41 index: 43
41	Variable
42	ExprTuple	43
43	Variable