Expression of type Conditional ¶

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, 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 = 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)))

# 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())

\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()

# display the expression information
stored_expr.expr_info()

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

Expression of type Conditional¶

from the theory of proveit.linear_algebra.vector_sets¶

Expression of type Conditional ¶

from the theory of proveit.linear_algebra.vector_sets ¶