Expression of type Lambda

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 Conditional, K, Lambda, 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, InClass, InSet, Set
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
sub_expr1 = Set(x_1_to_n)
expr = Lambda(V, Conditional(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), InClass(V, 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())

V \mapsto \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] \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 44 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 16 operand: 7
3	Operation	operator: 5 operands: 6
4	ExprTuple	7
5	Literal
6	ExprTuple	44, 8
7	Lambda	parameter: 50 body: 10
8	Operation	operator: 11 operand: 15
9	ExprTuple	50
10	Conditional	value: 13 condition: 14
11	Literal
12	ExprTuple	15
13	Operation	operator: 16 operand: 19
14	Operation	operator: 41 operands: 18
15	Variable
16	Literal
17	ExprTuple	19
18	ExprTuple	50, 20
19	Lambda	parameters: 46 body: 21
20	Literal
21	Conditional	value: 22 condition: 23
22	Operation	operator: 31 operands: 24
23	Operation	operator: 25 operands: 26
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	29
27	Operation	operator: 41 operands: 30
28	Operation	operator: 31 operands: 32
29	ExprRange	lambda_map: 33 start_index: 49 end_index: 50
30	ExprTuple	43, 34
31	Literal
32	ExprTuple	44, 35
33	Lambda	parameter: 54 body: 36
34	Operation	operator: 37 operand: 44
35	Operation	operator: 39 operand: 43
36	Operation	operator: 41 operands: 42
37	Literal
38	ExprTuple	44
39	Literal
40	ExprTuple	43
41	Literal
42	ExprTuple	51, 44
43	Operation	operator: 45 operands: 46
44	Variable
45	Literal
46	ExprTuple	47
47	ExprRange	lambda_map: 48 start_index: 49 end_index: 50
48	Lambda	parameter: 54 body: 51
49	Literal
50	Variable
51	IndexedVar	variable: 52 index: 54
52	Variable
53	ExprTuple	54
54	Variable