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 x_1_to_n
from proveit.linear_algebra import Span, SpanningSets, VecSpaces
from proveit.logic import Equals, Forall, InSet, Set
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
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(InSet(sub_expr1, SpanningSets(V)), Equals(V, Span(sub_expr1))), 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(\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{SpanningSets}\left(V\right)\right) = \left(V = \textrm{Span}\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}\right)\right)\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: 20 body: 3
3	Operation	operator: 21 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 49 body: 6
6	Conditional	value: 7 condition: 8
7	Operation	operator: 21 operand: 12
8	Operation	operator: 10 operands: 11
9	ExprTuple	12
10	Literal
11	ExprTuple	49, 13
12	Lambda	parameter: 55 body: 15
13	Operation	operator: 16 operand: 20
14	ExprTuple	55
15	Conditional	value: 18 condition: 19
16	Literal
17	ExprTuple	20
18	Operation	operator: 21 operand: 24
19	Operation	operator: 46 operands: 23
20	Variable
21	Literal
22	ExprTuple	24
23	ExprTuple	55, 25
24	Lambda	parameters: 51 body: 26
25	Literal
26	Conditional	value: 27 condition: 28
27	Operation	operator: 36 operands: 29
28	Operation	operator: 30 operands: 31
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34
32	Operation	operator: 46 operands: 35
33	Operation	operator: 36 operands: 37
34	ExprRange	lambda_map: 38 start_index: 54 end_index: 55
35	ExprTuple	48, 39
36	Literal
37	ExprTuple	49, 40
38	Lambda	parameter: 59 body: 41
39	Operation	operator: 42 operand: 49
40	Operation	operator: 44 operand: 48
41	Operation	operator: 46 operands: 47
42	Literal
43	ExprTuple	49
44	Literal
45	ExprTuple	48
46	Literal
47	ExprTuple	56, 49
48	Operation	operator: 50 operands: 51
49	Variable
50	Literal
51	ExprTuple	52
52	ExprRange	lambda_map: 53 start_index: 54 end_index: 55
53	Lambda	parameter: 59 body: 56
54	Literal
55	Variable
56	IndexedVar	variable: 57 index: 59
57	Variable
58	ExprTuple	59
59	Variable