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 V, n
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import Bases, LinDepSets, SpanningSets
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 = [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)

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_{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]

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