Expression of type Forall ¶

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, 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 = Forall(instance_param_or_params = [K], instance_expr = 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)))

# 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())

\forall_{K}~\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()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

	core type	sub-expressions
0	Operation	operator: 20 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 79 body: 3
3	Operation	operator: 20 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 69 body: 6
6	Conditional	value: 7 condition: 8
7	Operation	operator: 20 operand: 12
8	Operation	operator: 10 operands: 11
9	ExprTuple	12
10	Literal
11	ExprTuple	69, 13
12	Lambda	parameter: 77 body: 15
13	Operation	operator: 16 operand: 79
14	ExprTuple	77
15	Conditional	value: 18 condition: 19
16	Literal
17	ExprTuple	79
18	Operation	operator: 20 operand: 23
19	Operation	operator: 73 operands: 22
20	Literal
21	ExprTuple	23
22	ExprTuple	77, 24
23	Lambda	parameters: 42 body: 25
24	Literal
25	Conditional	value: 26 condition: 27
26	Operation	operator: 53 operands: 28
27	Operation	operator: 55 operands: 29
28	ExprTuple	30, 31
29	ExprTuple	32
30	Operation	operator: 73 operands: 33
31	Operation	operator: 34 operand: 39
32	ExprRange	lambda_map: 36 start_index: 76 end_index: 77
33	ExprTuple	37, 38
34	Literal
35	ExprTuple	39
36	Lambda	parameter: 91 body: 40
37	Operation	operator: 41 operands: 42
38	Operation	operator: 43 operand: 69
39	Lambda	parameters: 44 body: 45
40	Operation	operator: 73 operands: 46
41	Literal
42	ExprTuple	47
43	Literal
44	ExprTuple	48
45	Conditional	value: 49 condition: 50
46	ExprTuple	85, 69
47	ExprRange	lambda_map: 51 start_index: 76 end_index: 77
48	ExprRange	lambda_map: 52 start_index: 76 end_index: 77
49	Operation	operator: 53 operands: 54
50	Operation	operator: 55 operands: 56
51	Lambda	parameter: 91 body: 85
52	Lambda	parameter: 91 body: 86
53	Literal
54	ExprTuple	57, 58
55	Literal
56	ExprTuple	59, 60
57	Operation	operator: 61 operands: 62
58	Operation	operator: 63 operand: 69
59	ExprRange	lambda_map: 65 start_index: 76 end_index: 77
60	Operation	operator: 66 operands: 67
61	Literal
62	ExprTuple	68
63	Literal
64	ExprTuple	69
65	Lambda	parameter: 91 body: 70
66	Literal
67	ExprTuple	71
68	ExprRange	lambda_map: 72 start_index: 76 end_index: 77
69	Variable
70	Operation	operator: 73 operands: 74
71	ExprRange	lambda_map: 75 start_index: 76 end_index: 77
72	Lambda	parameter: 91 body: 78
73	Literal
74	ExprTuple	86, 79
75	Lambda	parameter: 91 body: 80
76	Literal
77	Variable
78	Operation	operator: 81 operands: 82
79	Variable
80	Operation	operator: 83 operands: 84
81	Literal
82	ExprTuple	86, 85
83	Literal
84	ExprTuple	86, 87
85	IndexedVar	variable: 88 index: 91
86	IndexedVar	variable: 89 index: 91
87	Literal
88	Variable
89	Variable
90	ExprTuple	91
91	Variable

Expression of type Forall¶

from the theory of proveit.linear_algebra.vector_sets¶

Expression of type Forall ¶

from the theory of proveit.linear_algebra.vector_sets ¶