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