Expression of type Exists

from the theory of proveit.linear_algebra.inner_products

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 B, ExprRange, IndexedVar, Variable, i, lambda_, m, n, v
from proveit.core_expr_types import a_i, b_1_to_n, b_i, lambda_i
from proveit.linear_algebra import OrthoNormBases, ScalarMult, TensorProd, VecSum
from proveit.logic import Equals, Exists, InSet, Set
from proveit.numbers import Interval, Min, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Min(m, n)
expr = Exists(instance_param_or_params = [b_1_to_n], instance_expr = Exists(instance_param_or_params = [ExprRange(sub_expr1, IndexedVar(lambda_, sub_expr1), one, sub_expr2)], instance_expr = Equals(v, VecSum(index_or_indices = [i], summand = ScalarMult(lambda_i, TensorProd(a_i, b_i)), domain = Interval(one, sub_expr2))), domain = RealNonNeg).with_wrapping(), condition = InSet(Set(b_1_to_n), OrthoNormBases(B))).with_wrapping()

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())

\begin{array}{l}\exists_{b_{1}, b_{2}, \ldots, b_{n}~|~\left\{b_{1}, b_{2}, \ldots, b_{n}\right\} \in \textrm{O.N.Bases}\left(B\right)}~\\
\left[\begin{array}{l}\exists_{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{{\rm Min}\left(m, n\right)} \in \mathbb{R}^{\ge 0}}~\\
\left(v = \left(\sum_{i=1}^{{\rm Min}\left(m, n\right)} \left(\lambda_{i} \cdot \left(a_{i} {\otimes} b_{i}\right)\right)\right)\right)\end{array}\right]\end{array}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 15 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 9
5	Operation	operator: 46 operands: 8
6	Literal
7	ExprTuple	9
8	ExprTuple	10, 11
9	Lambda	parameters: 12 body: 13
10	Operation	operator: 14 operands: 15
11	Operation	operator: 16 operand: 22
12	ExprTuple	18
13	Conditional	value: 19 condition: 20
14	Literal
15	ExprTuple	21
16	Literal
17	ExprTuple	22
18	ExprRange	lambda_map: 23 start_index: 60 end_index: 61
19	Operation	operator: 24 operands: 25
20	Operation	operator: 26 operands: 27
21	ExprRange	lambda_map: 28 start_index: 60 end_index: 69
22	Variable
23	Lambda	parameter: 52 body: 42
24	Literal
25	ExprTuple	29, 30
26	Literal
27	ExprTuple	31
28	Lambda	parameter: 52 body: 32
29	Variable
30	Operation	operator: 33 operand: 36
31	ExprRange	lambda_map: 35 start_index: 60 end_index: 61
32	IndexedVar	variable: 63 index: 52
33	Literal
34	ExprTuple	36
35	Lambda	parameter: 52 body: 37
36	Lambda	parameter: 67 body: 38
37	Operation	operator: 46 operands: 39
38	Conditional	value: 40 condition: 41
39	ExprTuple	42, 43
40	Operation	operator: 44 operands: 45
41	Operation	operator: 46 operands: 47
42	IndexedVar	variable: 53 index: 52
43	Literal
44	Literal
45	ExprTuple	49, 50
46	Literal
47	ExprTuple	67, 51
48	ExprTuple	52
49	IndexedVar	variable: 53 index: 67
50	Operation	operator: 54 operands: 55
51	Operation	operator: 56 operands: 57
52	Variable
53	Variable
54	Literal
55	ExprTuple	58, 59
56	Literal
57	ExprTuple	60, 61
58	IndexedVar	variable: 62 index: 67
59	IndexedVar	variable: 63 index: 67
60	Literal
61	Operation	operator: 65 operands: 66
62	Variable
63	Variable
64	ExprTuple	67
65	Literal
66	ExprTuple	68, 69
67	Variable
68	Variable
69	Variable