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 = [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_{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]

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