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 LinDepSets, VecSpaces, VecZero, lin_comb_axn, some_nonzero_a
from proveit.logic import Equals, Exists, Forall, InClass, InSet, Set
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(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), 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} \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] \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: 64 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 15 operand: 7
3	Operation	operator: 5 operands: 6
4	ExprTuple	7
5	Literal
6	ExprTuple	64, 8
7	Lambda	parameter: 72 body: 10
8	Operation	operator: 11 operand: 74
9	ExprTuple	72
10	Conditional	value: 13 condition: 14
11	Literal
12	ExprTuple	74
13	Operation	operator: 15 operand: 18
14	Operation	operator: 68 operands: 17
15	Literal
16	ExprTuple	18
17	ExprTuple	72, 19
18	Lambda	parameters: 37 body: 20
19	Literal
20	Conditional	value: 21 condition: 22
21	Operation	operator: 48 operands: 23
22	Operation	operator: 50 operands: 24
23	ExprTuple	25, 26
24	ExprTuple	27
25	Operation	operator: 68 operands: 28
26	Operation	operator: 29 operand: 34
27	ExprRange	lambda_map: 31 start_index: 71 end_index: 72
28	ExprTuple	32, 33
29	Literal
30	ExprTuple	34
31	Lambda	parameter: 86 body: 35
32	Operation	operator: 36 operands: 37
33	Operation	operator: 38 operand: 64
34	Lambda	parameters: 39 body: 40
35	Operation	operator: 68 operands: 41
36	Literal
37	ExprTuple	42
38	Literal
39	ExprTuple	43
40	Conditional	value: 44 condition: 45
41	ExprTuple	80, 64
42	ExprRange	lambda_map: 46 start_index: 71 end_index: 72
43	ExprRange	lambda_map: 47 start_index: 71 end_index: 72
44	Operation	operator: 48 operands: 49
45	Operation	operator: 50 operands: 51
46	Lambda	parameter: 86 body: 80
47	Lambda	parameter: 86 body: 81
48	Literal
49	ExprTuple	52, 53
50	Literal
51	ExprTuple	54, 55
52	Operation	operator: 56 operands: 57
53	Operation	operator: 58 operand: 64
54	ExprRange	lambda_map: 60 start_index: 71 end_index: 72
55	Operation	operator: 61 operands: 62
56	Literal
57	ExprTuple	63
58	Literal
59	ExprTuple	64
60	Lambda	parameter: 86 body: 65
61	Literal
62	ExprTuple	66
63	ExprRange	lambda_map: 67 start_index: 71 end_index: 72
64	Variable
65	Operation	operator: 68 operands: 69
66	ExprRange	lambda_map: 70 start_index: 71 end_index: 72
67	Lambda	parameter: 86 body: 73
68	Literal
69	ExprTuple	81, 74
70	Lambda	parameter: 86 body: 75
71	Literal
72	Variable
73	Operation	operator: 76 operands: 77
74	Variable
75	Operation	operator: 78 operands: 79
76	Literal
77	ExprTuple	81, 80
78	Literal
79	ExprTuple	81, 82
80	IndexedVar	variable: 83 index: 86
81	IndexedVar	variable: 84 index: 86
82	Literal
83	Variable
84	Variable
85	ExprTuple	86
86	Variable