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, VecSpaces, 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 = [V], instance_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), 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_{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} \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]\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: 18 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 49 body: 4
3	ExprTuple	49
4	Conditional	value: 5 condition: 6
5	Operation	operator: 18 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	49, 11
10	Lambda	parameter: 61 body: 13
11	Operation	operator: 14 operand: 70
12	ExprTuple	61
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	70
16	Operation	operator: 18 operand: 21
17	Operation	operator: 66 operands: 20
18	Literal
19	ExprTuple	21
20	ExprTuple	61, 22
21	Lambda	parameters: 41 body: 23
22	Literal
23	Conditional	value: 24 condition: 25
24	Operation	operator: 26 operands: 27
25	Operation	operator: 54 operands: 28
26	Literal
27	ExprTuple	29, 30
28	ExprTuple	31
29	Operation	operator: 32 operand: 37
30	Operation	operator: 34 operand: 38
31	ExprRange	lambda_map: 36 start_index: 60 end_index: 61
32	Literal
33	ExprTuple	37
34	Literal
35	ExprTuple	38
36	Lambda	parameter: 74 body: 39
37	Operation	operator: 40 operands: 41
38	Lambda	parameters: 42 body: 43
39	Operation	operator: 66 operands: 44
40	Literal
41	ExprTuple	45
42	ExprTuple	46
43	Conditional	value: 47 condition: 48
44	ExprTuple	68, 49
45	ExprRange	lambda_map: 50 start_index: 60 end_index: 61
46	ExprRange	lambda_map: 51 start_index: 60 end_index: 61
47	Operation	operator: 52 operands: 53
48	Operation	operator: 54 operands: 55
49	Variable
50	Lambda	parameter: 74 body: 68
51	Lambda	parameter: 74 body: 69
52	Literal
53	ExprTuple	56
54	Literal
55	ExprTuple	57
56	ExprRange	lambda_map: 58 start_index: 60 end_index: 61
57	ExprRange	lambda_map: 59 start_index: 60 end_index: 61
58	Lambda	parameter: 74 body: 62
59	Lambda	parameter: 74 body: 63
60	Literal
61	Variable
62	Operation	operator: 64 operands: 65
63	Operation	operator: 66 operands: 67
64	Literal
65	ExprTuple	69, 68
66	Literal
67	ExprTuple	69, 70
68	IndexedVar	variable: 71 index: 74
69	IndexedVar	variable: 72 index: 74
70	Variable
71	Variable
72	Variable
73	ExprTuple	74
74	Variable