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 = [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_{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]

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: 46 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 18 operand: 9
5	Operation	operator: 7 operands: 8
6	ExprTuple	9
7	Literal
8	ExprTuple	46, 10
9	Lambda	parameter: 52 body: 12
10	Operation	operator: 13 operand: 17
11	ExprTuple	52
12	Conditional	value: 15 condition: 16
13	Literal
14	ExprTuple	17
15	Operation	operator: 18 operand: 21
16	Operation	operator: 43 operands: 20
17	Variable
18	Literal
19	ExprTuple	21
20	ExprTuple	52, 22
21	Lambda	parameters: 48 body: 23
22	Literal
23	Conditional	value: 24 condition: 25
24	Operation	operator: 33 operands: 26
25	Operation	operator: 27 operands: 28
26	ExprTuple	29, 30
27	Literal
28	ExprTuple	31
29	Operation	operator: 43 operands: 32
30	Operation	operator: 33 operands: 34
31	ExprRange	lambda_map: 35 start_index: 51 end_index: 52
32	ExprTuple	45, 36
33	Literal
34	ExprTuple	46, 37
35	Lambda	parameter: 56 body: 38
36	Operation	operator: 39 operand: 46
37	Operation	operator: 41 operand: 45
38	Operation	operator: 43 operands: 44
39	Literal
40	ExprTuple	46
41	Literal
42	ExprTuple	45
43	Literal
44	ExprTuple	53, 46
45	Operation	operator: 47 operands: 48
46	Variable
47	Literal
48	ExprTuple	49
49	ExprRange	lambda_map: 50 start_index: 51 end_index: 52
50	Lambda	parameter: 56 body: 53
51	Literal
52	Variable
53	IndexedVar	variable: 54 index: 56
54	Variable
55	ExprTuple	56
56	Variable