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 = [K], instance_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_{K}~\left[\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]\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: 21 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 73 body: 3
3	Operation	operator: 21 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 52 body: 7
6	ExprTuple	52
7	Conditional	value: 8 condition: 9
8	Operation	operator: 21 operand: 13
9	Operation	operator: 11 operands: 12
10	ExprTuple	13
11	Literal
12	ExprTuple	52, 14
13	Lambda	parameter: 64 body: 16
14	Operation	operator: 17 operand: 73
15	ExprTuple	64
16	Conditional	value: 19 condition: 20
17	Literal
18	ExprTuple	73
19	Operation	operator: 21 operand: 24
20	Operation	operator: 69 operands: 23
21	Literal
22	ExprTuple	24
23	ExprTuple	64, 25
24	Lambda	parameters: 44 body: 26
25	Literal
26	Conditional	value: 27 condition: 28
27	Operation	operator: 29 operands: 30
28	Operation	operator: 57 operands: 31
29	Literal
30	ExprTuple	32, 33
31	ExprTuple	34
32	Operation	operator: 35 operand: 40
33	Operation	operator: 37 operand: 41
34	ExprRange	lambda_map: 39 start_index: 63 end_index: 64
35	Literal
36	ExprTuple	40
37	Literal
38	ExprTuple	41
39	Lambda	parameter: 77 body: 42
40	Operation	operator: 43 operands: 44
41	Lambda	parameters: 45 body: 46
42	Operation	operator: 69 operands: 47
43	Literal
44	ExprTuple	48
45	ExprTuple	49
46	Conditional	value: 50 condition: 51
47	ExprTuple	71, 52
48	ExprRange	lambda_map: 53 start_index: 63 end_index: 64
49	ExprRange	lambda_map: 54 start_index: 63 end_index: 64
50	Operation	operator: 55 operands: 56
51	Operation	operator: 57 operands: 58
52	Variable
53	Lambda	parameter: 77 body: 71
54	Lambda	parameter: 77 body: 72
55	Literal
56	ExprTuple	59
57	Literal
58	ExprTuple	60
59	ExprRange	lambda_map: 61 start_index: 63 end_index: 64
60	ExprRange	lambda_map: 62 start_index: 63 end_index: 64
61	Lambda	parameter: 77 body: 65
62	Lambda	parameter: 77 body: 66
63	Literal
64	Variable
65	Operation	operator: 67 operands: 68
66	Operation	operator: 69 operands: 70
67	Literal
68	ExprTuple	72, 71
69	Literal
70	ExprTuple	72, 73
71	IndexedVar	variable: 74 index: 77
72	IndexedVar	variable: 75 index: 77
73	Variable
74	Variable
75	Variable
76	ExprTuple	77
77	Variable