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 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, InSet, Set
from proveit.numbers import NaturalPos

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

K \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 18 body: 1
1	Operation	operator: 19 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 47 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 19 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	47, 11
10	Lambda	parameter: 53 body: 13
11	Operation	operator: 14 operand: 18
12	ExprTuple	53
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	18
16	Operation	operator: 19 operand: 22
17	Operation	operator: 44 operands: 21
18	Variable
19	Literal
20	ExprTuple	22
21	ExprTuple	53, 23
22	Lambda	parameters: 49 body: 24
23	Literal
24	Conditional	value: 25 condition: 26
25	Operation	operator: 34 operands: 27
26	Operation	operator: 28 operands: 29
27	ExprTuple	30, 31
28	Literal
29	ExprTuple	32
30	Operation	operator: 44 operands: 33
31	Operation	operator: 34 operands: 35
32	ExprRange	lambda_map: 36 start_index: 52 end_index: 53
33	ExprTuple	46, 37
34	Literal
35	ExprTuple	47, 38
36	Lambda	parameter: 57 body: 39
37	Operation	operator: 40 operand: 47
38	Operation	operator: 42 operand: 46
39	Operation	operator: 44 operands: 45
40	Literal
41	ExprTuple	47
42	Literal
43	ExprTuple	46
44	Literal
45	ExprTuple	54, 47
46	Operation	operator: 48 operands: 49
47	Variable
48	Literal
49	ExprTuple	50
50	ExprRange	lambda_map: 51 start_index: 52 end_index: 53
51	Lambda	parameter: 57 body: 54
52	Literal
53	Variable
54	IndexedVar	variable: 55 index: 57
55	Variable
56	ExprTuple	57
57	Variable