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 LinDepSets, VecSpaces, VecZero, lin_comb_axn, some_nonzero_a
from proveit.logic import Equals, Exists, Forall, InSet, Set
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(InSet(Set(x_1_to_n), LinDepSets(V)), Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(lin_comb_axn, VecZero(V)), domain = K, condition = some_nonzero_a).with_wrapping()).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} \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{LinDepSets}\left(V\right)\right) =  \\ \left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in K~|~\left(a_{1} \neq 0\right) \lor  \left(a_{2} \neq 0\right) \lor  \ldots \lor  \left(a_{n} \neq 0\right)}~\\
\left(\left(\left(a_{1} \cdot x_{1}\right) +  \left(a_{2} \cdot x_{2}\right) +  \ldots +  \left(a_{n} \cdot x_{n}\right)\right) = \vec{0}\left(V\right)\right)\end{array}\right] \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: 77 body: 1
1	Operation	operator: 18 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 67 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 18 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	67, 11
10	Lambda	parameter: 75 body: 13
11	Operation	operator: 14 operand: 77
12	ExprTuple	75
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	77
16	Operation	operator: 18 operand: 21
17	Operation	operator: 71 operands: 20
18	Literal
19	ExprTuple	21
20	ExprTuple	75, 22
21	Lambda	parameters: 40 body: 23
22	Literal
23	Conditional	value: 24 condition: 25
24	Operation	operator: 51 operands: 26
25	Operation	operator: 53 operands: 27
26	ExprTuple	28, 29
27	ExprTuple	30
28	Operation	operator: 71 operands: 31
29	Operation	operator: 32 operand: 37
30	ExprRange	lambda_map: 34 start_index: 74 end_index: 75
31	ExprTuple	35, 36
32	Literal
33	ExprTuple	37
34	Lambda	parameter: 89 body: 38
35	Operation	operator: 39 operands: 40
36	Operation	operator: 41 operand: 67
37	Lambda	parameters: 42 body: 43
38	Operation	operator: 71 operands: 44
39	Literal
40	ExprTuple	45
41	Literal
42	ExprTuple	46
43	Conditional	value: 47 condition: 48
44	ExprTuple	83, 67
45	ExprRange	lambda_map: 49 start_index: 74 end_index: 75
46	ExprRange	lambda_map: 50 start_index: 74 end_index: 75
47	Operation	operator: 51 operands: 52
48	Operation	operator: 53 operands: 54
49	Lambda	parameter: 89 body: 83
50	Lambda	parameter: 89 body: 84
51	Literal
52	ExprTuple	55, 56
53	Literal
54	ExprTuple	57, 58
55	Operation	operator: 59 operands: 60
56	Operation	operator: 61 operand: 67
57	ExprRange	lambda_map: 63 start_index: 74 end_index: 75
58	Operation	operator: 64 operands: 65
59	Literal
60	ExprTuple	66
61	Literal
62	ExprTuple	67
63	Lambda	parameter: 89 body: 68
64	Literal
65	ExprTuple	69
66	ExprRange	lambda_map: 70 start_index: 74 end_index: 75
67	Variable
68	Operation	operator: 71 operands: 72
69	ExprRange	lambda_map: 73 start_index: 74 end_index: 75
70	Lambda	parameter: 89 body: 76
71	Literal
72	ExprTuple	84, 77
73	Lambda	parameter: 89 body: 78
74	Literal
75	Variable
76	Operation	operator: 79 operands: 80
77	Variable
78	Operation	operator: 81 operands: 82
79	Literal
80	ExprTuple	84, 83
81	Literal
82	ExprTuple	84, 85
83	IndexedVar	variable: 86 index: 89
84	IndexedVar	variable: 87 index: 89
85	Literal
86	Variable
87	Variable
88	ExprTuple	89
89	Variable