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 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
from proveit.numbers import NaturalPos

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