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, 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(n, Conditional(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(), 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} \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} \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: 62 body: 2
1	ExprTuple	62
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 58 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	62, 9
8	Lambda	parameters: 27 body: 10
9	Literal
10	Conditional	value: 11 condition: 12
11	Operation	operator: 38 operands: 13
12	Operation	operator: 40 operands: 14
13	ExprTuple	15, 16
14	ExprTuple	17
15	Operation	operator: 58 operands: 18
16	Operation	operator: 19 operand: 24
17	ExprRange	lambda_map: 21 start_index: 61 end_index: 62
18	ExprTuple	22, 23
19	Literal
20	ExprTuple	24
21	Lambda	parameter: 76 body: 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 28 operand: 54
24	Lambda	parameters: 29 body: 30
25	Operation	operator: 58 operands: 31
26	Literal
27	ExprTuple	32
28	Literal
29	ExprTuple	33
30	Conditional	value: 34 condition: 35
31	ExprTuple	70, 54
32	ExprRange	lambda_map: 36 start_index: 61 end_index: 62
33	ExprRange	lambda_map: 37 start_index: 61 end_index: 62
34	Operation	operator: 38 operands: 39
35	Operation	operator: 40 operands: 41
36	Lambda	parameter: 76 body: 70
37	Lambda	parameter: 76 body: 71
38	Literal
39	ExprTuple	42, 43
40	Literal
41	ExprTuple	44, 45
42	Operation	operator: 46 operands: 47
43	Operation	operator: 48 operand: 54
44	ExprRange	lambda_map: 50 start_index: 61 end_index: 62
45	Operation	operator: 51 operands: 52
46	Literal
47	ExprTuple	53
48	Literal
49	ExprTuple	54
50	Lambda	parameter: 76 body: 55
51	Literal
52	ExprTuple	56
53	ExprRange	lambda_map: 57 start_index: 61 end_index: 62
54	Variable
55	Operation	operator: 58 operands: 59
56	ExprRange	lambda_map: 60 start_index: 61 end_index: 62
57	Lambda	parameter: 76 body: 63
58	Literal
59	ExprTuple	71, 64
60	Lambda	parameter: 76 body: 65
61	Literal
62	Variable
63	Operation	operator: 66 operands: 67
64	Variable
65	Operation	operator: 68 operands: 69
66	Literal
67	ExprTuple	71, 70
68	Literal
69	ExprTuple	71, 72
70	IndexedVar	variable: 73 index: 76
71	IndexedVar	variable: 74 index: 76
72	Literal
73	Variable
74	Variable
75	ExprTuple	76
76	Variable