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, K, Lambda, V, n
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 Equals, Forall, InSet, Set, SetOfAll
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Lambda(n, Conditional(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(), 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(\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} \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: 48 body: 2
1	ExprTuple	48
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 53 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	48, 9
8	Lambda	parameters: 28 body: 10
9	Literal
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operands: 14
12	Operation	operator: 41 operands: 15
13	Literal
14	ExprTuple	16, 17
15	ExprTuple	18
16	Operation	operator: 19 operand: 24
17	Operation	operator: 21 operand: 25
18	ExprRange	lambda_map: 23 start_index: 47 end_index: 48
19	Literal
20	ExprTuple	24
21	Literal
22	ExprTuple	25
23	Lambda	parameter: 61 body: 26
24	Operation	operator: 27 operands: 28
25	Lambda	parameters: 29 body: 30
26	Operation	operator: 53 operands: 31
27	Literal
28	ExprTuple	32
29	ExprTuple	33
30	Conditional	value: 34 condition: 35
31	ExprTuple	55, 36
32	ExprRange	lambda_map: 37 start_index: 47 end_index: 48
33	ExprRange	lambda_map: 38 start_index: 47 end_index: 48
34	Operation	operator: 39 operands: 40
35	Operation	operator: 41 operands: 42
36	Variable
37	Lambda	parameter: 61 body: 55
38	Lambda	parameter: 61 body: 56
39	Literal
40	ExprTuple	43
41	Literal
42	ExprTuple	44
43	ExprRange	lambda_map: 45 start_index: 47 end_index: 48
44	ExprRange	lambda_map: 46 start_index: 47 end_index: 48
45	Lambda	parameter: 61 body: 49
46	Lambda	parameter: 61 body: 50
47	Literal
48	Variable
49	Operation	operator: 51 operands: 52
50	Operation	operator: 53 operands: 54
51	Literal
52	ExprTuple	56, 55
53	Literal
54	ExprTuple	56, 57
55	IndexedVar	variable: 58 index: 61
56	IndexedVar	variable: 59 index: 61
57	Variable
58	Variable
59	Variable
60	ExprTuple	61
61	Variable