Expression of type Conditional ¶

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, 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 = 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)))

# 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())

\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()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Conditional	value: 1 condition: 2
1	Operation	operator: 14 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	63, 7
6	Lambda	parameter: 71 body: 9
7	Operation	operator: 10 operand: 73
8	ExprTuple	71
9	Conditional	value: 12 condition: 13
10	Literal
11	ExprTuple	73
12	Operation	operator: 14 operand: 17
13	Operation	operator: 67 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	71, 18
17	Lambda	parameters: 36 body: 19
18	Literal
19	Conditional	value: 20 condition: 21
20	Operation	operator: 47 operands: 22
21	Operation	operator: 49 operands: 23
22	ExprTuple	24, 25
23	ExprTuple	26
24	Operation	operator: 67 operands: 27
25	Operation	operator: 28 operand: 33
26	ExprRange	lambda_map: 30 start_index: 70 end_index: 71
27	ExprTuple	31, 32
28	Literal
29	ExprTuple	33
30	Lambda	parameter: 85 body: 34
31	Operation	operator: 35 operands: 36
32	Operation	operator: 37 operand: 63
33	Lambda	parameters: 38 body: 39
34	Operation	operator: 67 operands: 40
35	Literal
36	ExprTuple	41
37	Literal
38	ExprTuple	42
39	Conditional	value: 43 condition: 44
40	ExprTuple	79, 63
41	ExprRange	lambda_map: 45 start_index: 70 end_index: 71
42	ExprRange	lambda_map: 46 start_index: 70 end_index: 71
43	Operation	operator: 47 operands: 48
44	Operation	operator: 49 operands: 50
45	Lambda	parameter: 85 body: 79
46	Lambda	parameter: 85 body: 80
47	Literal
48	ExprTuple	51, 52
49	Literal
50	ExprTuple	53, 54
51	Operation	operator: 55 operands: 56
52	Operation	operator: 57 operand: 63
53	ExprRange	lambda_map: 59 start_index: 70 end_index: 71
54	Operation	operator: 60 operands: 61
55	Literal
56	ExprTuple	62
57	Literal
58	ExprTuple	63
59	Lambda	parameter: 85 body: 64
60	Literal
61	ExprTuple	65
62	ExprRange	lambda_map: 66 start_index: 70 end_index: 71
63	Variable
64	Operation	operator: 67 operands: 68
65	ExprRange	lambda_map: 69 start_index: 70 end_index: 71
66	Lambda	parameter: 85 body: 72
67	Literal
68	ExprTuple	80, 73
69	Lambda	parameter: 85 body: 74
70	Literal
71	Variable
72	Operation	operator: 75 operands: 76
73	Variable
74	Operation	operator: 77 operands: 78
75	Literal
76	ExprTuple	80, 79
77	Literal
78	ExprTuple	80, 81
79	IndexedVar	variable: 82 index: 85
80	IndexedVar	variable: 83 index: 85
81	Literal
82	Variable
83	Variable
84	ExprTuple	85
85	Variable

Expression of type Conditional¶

from the theory of proveit.linear_algebra.vector_sets¶

Expression of type Conditional ¶

from the theory of proveit.linear_algebra.vector_sets ¶