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
from proveit.core_expr_types import a_1_to_n, x_1_to_n
from proveit.linear_algebra import Span, lin_comb_axn
from proveit.logic import Equals, Forall, Set, SetOfAll

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(Span(Set(x_1_to_n)), SetOfAll(instance_param_or_params = [a_1_to_n], instance_element = lin_comb_axn, domain = K)).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} \textrm{Span}\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}\right) =  \\ \left\{\left(a_{1} \cdot x_{1}\right) +  \left(a_{2} \cdot x_{2}\right) +  \ldots +  \left(a_{n} \cdot x_{n}\right)\right\}_{a_{1}, a_{2}, \ldots, a_{n} \in K} \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: 22 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 35 operands: 9
7	Literal
8	ExprTuple	10, 11
9	ExprTuple	12
10	Operation	operator: 13 operand: 18
11	Operation	operator: 15 operand: 19
12	ExprRange	lambda_map: 17 start_index: 41 end_index: 42
13	Literal
14	ExprTuple	18
15	Literal
16	ExprTuple	19
17	Lambda	parameter: 55 body: 20
18	Operation	operator: 21 operands: 22
19	Lambda	parameters: 23 body: 24
20	Operation	operator: 47 operands: 25
21	Literal
22	ExprTuple	26
23	ExprTuple	27
24	Conditional	value: 28 condition: 29
25	ExprTuple	49, 30
26	ExprRange	lambda_map: 31 start_index: 41 end_index: 42
27	ExprRange	lambda_map: 32 start_index: 41 end_index: 42
28	Operation	operator: 33 operands: 34
29	Operation	operator: 35 operands: 36
30	Variable
31	Lambda	parameter: 55 body: 49
32	Lambda	parameter: 55 body: 50
33	Literal
34	ExprTuple	37
35	Literal
36	ExprTuple	38
37	ExprRange	lambda_map: 39 start_index: 41 end_index: 42
38	ExprRange	lambda_map: 40 start_index: 41 end_index: 42
39	Lambda	parameter: 55 body: 43
40	Lambda	parameter: 55 body: 44
41	Literal
42	Variable
43	Operation	operator: 45 operands: 46
44	Operation	operator: 47 operands: 48
45	Literal
46	ExprTuple	50, 49
47	Literal
48	ExprTuple	50, 51
49	IndexedVar	variable: 52 index: 55
50	IndexedVar	variable: 53 index: 55
51	Variable
52	Variable
53	Variable
54	ExprTuple	55
55	Variable