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
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

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
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()

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())

\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}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 32 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 14 operands: 9
7	Literal
8	ExprTuple	10, 11
9	ExprTuple	12
10	Operation	operator: 25 operands: 13
11	Operation	operator: 14 operands: 15
12	ExprRange	lambda_map: 16 start_index: 38 end_index: 39
13	ExprTuple	28, 17
14	Literal
15	ExprTuple	18, 19
16	Lambda	parameter: 43 body: 20
17	Operation	operator: 21 operand: 36
18	Operation	operator: 25 operands: 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 25 operands: 26
21	Literal
22	ExprTuple	28, 27
23	Literal
24	ExprTuple	28, 29
25	Literal
26	ExprTuple	40, 36
27	Operation	operator: 30 operand: 36
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operand: 36
30	Literal
31	Literal
32	ExprTuple	35
33	Literal
34	ExprTuple	36
35	ExprRange	lambda_map: 37 start_index: 38 end_index: 39
36	Variable
37	Lambda	parameter: 43 body: 40
38	Literal
39	Variable
40	IndexedVar	variable: 41 index: 43
41	Variable
42	ExprTuple	43
43	Variable