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 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, Set, SetOfAll
from proveit.numbers import NaturalPos

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

K \mapsto \left[\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\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]\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: 71 body: 1
1	Operation	operator: 19 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 50 body: 5
4	ExprTuple	50
5	Conditional	value: 6 condition: 7
6	Operation	operator: 19 operand: 11
7	Operation	operator: 9 operands: 10
8	ExprTuple	11
9	Literal
10	ExprTuple	50, 12
11	Lambda	parameter: 62 body: 14
12	Operation	operator: 15 operand: 71
13	ExprTuple	62
14	Conditional	value: 17 condition: 18
15	Literal
16	ExprTuple	71
17	Operation	operator: 19 operand: 22
18	Operation	operator: 67 operands: 21
19	Literal
20	ExprTuple	22
21	ExprTuple	62, 23
22	Lambda	parameters: 42 body: 24
23	Literal
24	Conditional	value: 25 condition: 26
25	Operation	operator: 27 operands: 28
26	Operation	operator: 55 operands: 29
27	Literal
28	ExprTuple	30, 31
29	ExprTuple	32
30	Operation	operator: 33 operand: 38
31	Operation	operator: 35 operand: 39
32	ExprRange	lambda_map: 37 start_index: 61 end_index: 62
33	Literal
34	ExprTuple	38
35	Literal
36	ExprTuple	39
37	Lambda	parameter: 75 body: 40
38	Operation	operator: 41 operands: 42
39	Lambda	parameters: 43 body: 44
40	Operation	operator: 67 operands: 45
41	Literal
42	ExprTuple	46
43	ExprTuple	47
44	Conditional	value: 48 condition: 49
45	ExprTuple	69, 50
46	ExprRange	lambda_map: 51 start_index: 61 end_index: 62
47	ExprRange	lambda_map: 52 start_index: 61 end_index: 62
48	Operation	operator: 53 operands: 54
49	Operation	operator: 55 operands: 56
50	Variable
51	Lambda	parameter: 75 body: 69
52	Lambda	parameter: 75 body: 70
53	Literal
54	ExprTuple	57
55	Literal
56	ExprTuple	58
57	ExprRange	lambda_map: 59 start_index: 61 end_index: 62
58	ExprRange	lambda_map: 60 start_index: 61 end_index: 62
59	Lambda	parameter: 75 body: 63
60	Lambda	parameter: 75 body: 64
61	Literal
62	Variable
63	Operation	operator: 65 operands: 66
64	Operation	operator: 67 operands: 68
65	Literal
66	ExprTuple	70, 69
67	Literal
68	ExprTuple	70, 71
69	IndexedVar	variable: 72 index: 75
70	IndexedVar	variable: 73 index: 75
71	Variable
72	Variable
73	Variable
74	ExprTuple	75
75	Variable