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 x_1_to_n
from proveit.linear_algebra import Bases, LinDepSets, SpanningSets, VecSpaces
from proveit.logic import And, Equals, Forall, InSet, NotInSet, Set
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
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(sub_expr1, Bases(V)), And(InSet(sub_expr1, SpanningSets(V)), NotInSet(sub_expr1, LinDepSets(V))).with_wrapping_at(2)).with_wrapping_at(2), domain = V).with_wrapping(), domain = NaturalPos), domain = 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())

\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{Bases}\left(V\right)\right) =  \\ \left(\begin{array}{c} \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{SpanningSets}\left(V\right)\right) \land  \\ \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \notin \textrm{LinDepSets}\left(V\right)\right) \end{array}\right) \end{array} \end{array}\right)\end{array}\right]\right]

stored_expr.style_options()

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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 18 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 55 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 18 operand: 9
5	Operation	operator: 7 operands: 8
6	ExprTuple	9
7	Literal
8	ExprTuple	55, 10
9	Lambda	parameter: 58 body: 12
10	Operation	operator: 13 operand: 17
11	ExprTuple	58
12	Conditional	value: 15 condition: 16
13	Literal
14	ExprTuple	17
15	Operation	operator: 18 operand: 21
16	Operation	operator: 44 operands: 20
17	Variable
18	Literal
19	ExprTuple	21
20	ExprTuple	58, 22
21	Lambda	parameters: 51 body: 23
22	Literal
23	Conditional	value: 24 condition: 25
24	Operation	operator: 26 operands: 27
25	Operation	operator: 33 operands: 28
26	Literal
27	ExprTuple	29, 30
28	ExprTuple	31
29	Operation	operator: 44 operands: 32
30	Operation	operator: 33 operands: 34
31	ExprRange	lambda_map: 35 start_index: 57 end_index: 58
32	ExprTuple	47, 36
33	Literal
34	ExprTuple	37, 38
35	Lambda	parameter: 62 body: 39
36	Operation	operator: 40 operand: 55
37	Operation	operator: 44 operands: 41
38	Operation	operator: 42 operands: 43
39	Operation	operator: 44 operands: 45
40	Literal
41	ExprTuple	47, 46
42	Literal
43	ExprTuple	47, 48
44	Literal
45	ExprTuple	59, 55
46	Operation	operator: 49 operand: 55
47	Operation	operator: 50 operands: 51
48	Operation	operator: 52 operand: 55
49	Literal
50	Literal
51	ExprTuple	54
52	Literal
53	ExprTuple	55
54	ExprRange	lambda_map: 56 start_index: 57 end_index: 58
55	Variable
56	Lambda	parameter: 62 body: 59
57	Literal
58	Variable
59	IndexedVar	variable: 60 index: 62
60	Variable
61	ExprTuple	62
62	Variable