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, VecSpaces, lin_comb_axn
from proveit.logic import Equals, Forall, InClass, Set, SetOfAll
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Lambda(V, Conditional(Forall(instance_param_or_params = [n], instance_expr = 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(), 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())

V \mapsto \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} \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}\right] \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 47 body: 2
1	ExprTuple	47
2	Conditional	value: 3 condition: 4
3	Operation	operator: 16 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	47, 9
8	Lambda	parameter: 59 body: 11
9	Operation	operator: 12 operand: 68
10	ExprTuple	59
11	Conditional	value: 14 condition: 15
12	Literal
13	ExprTuple	68
14	Operation	operator: 16 operand: 19
15	Operation	operator: 64 operands: 18
16	Literal
17	ExprTuple	19
18	ExprTuple	59, 20
19	Lambda	parameters: 39 body: 21
20	Literal
21	Conditional	value: 22 condition: 23
22	Operation	operator: 24 operands: 25
23	Operation	operator: 52 operands: 26
24	Literal
25	ExprTuple	27, 28
26	ExprTuple	29
27	Operation	operator: 30 operand: 35
28	Operation	operator: 32 operand: 36
29	ExprRange	lambda_map: 34 start_index: 58 end_index: 59
30	Literal
31	ExprTuple	35
32	Literal
33	ExprTuple	36
34	Lambda	parameter: 72 body: 37
35	Operation	operator: 38 operands: 39
36	Lambda	parameters: 40 body: 41
37	Operation	operator: 64 operands: 42
38	Literal
39	ExprTuple	43
40	ExprTuple	44
41	Conditional	value: 45 condition: 46
42	ExprTuple	66, 47
43	ExprRange	lambda_map: 48 start_index: 58 end_index: 59
44	ExprRange	lambda_map: 49 start_index: 58 end_index: 59
45	Operation	operator: 50 operands: 51
46	Operation	operator: 52 operands: 53
47	Variable
48	Lambda	parameter: 72 body: 66
49	Lambda	parameter: 72 body: 67
50	Literal
51	ExprTuple	54
52	Literal
53	ExprTuple	55
54	ExprRange	lambda_map: 56 start_index: 58 end_index: 59
55	ExprRange	lambda_map: 57 start_index: 58 end_index: 59
56	Lambda	parameter: 72 body: 60
57	Lambda	parameter: 72 body: 61
58	Literal
59	Variable
60	Operation	operator: 62 operands: 63
61	Operation	operator: 64 operands: 65
62	Literal
63	ExprTuple	67, 66
64	Literal
65	ExprTuple	67, 68
66	IndexedVar	variable: 69 index: 72
67	IndexedVar	variable: 70 index: 72
68	Variable
69	Variable
70	Variable
71	ExprTuple	72
72	Variable