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 A, B, ExprRange, IndexedVar, Variable, i, lambda_, m, n, v
from proveit.core_expr_types import a_1_to_m, 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 = [a_1_to_m], instance_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(), condition = InSet(Set(a_1_to_m), OrthoNormBases(A))).with_wrapping()

# 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_{a_{1}, a_{2}, \ldots, a_{m}~|~\left\{a_{1}, a_{2}, \ldots, a_{m}\right\} \in \textrm{O.N.Bases}\left(A\right)}~\\
\left[\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}\right]\end{array}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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',)

	core type	sub-expressions
0	Operation	operator: 18 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 12 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 18 operand: 8
5	Operation	operator: 60 operands: 7
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameters: 29 body: 11
9	Operation	operator: 28 operands: 12
10	Operation	operator: 30 operand: 17
11	Conditional	value: 14 condition: 15
12	ExprTuple	16
13	ExprTuple	17
14	Operation	operator: 18 operand: 22
15	Operation	operator: 60 operands: 20
16	ExprRange	lambda_map: 21 start_index: 74 end_index: 82
17	Variable
18	Literal
19	ExprTuple	22
20	ExprTuple	23, 24
21	Lambda	parameter: 66 body: 25
22	Lambda	parameters: 26 body: 27
23	Operation	operator: 28 operands: 29
24	Operation	operator: 30 operand: 36
25	IndexedVar	variable: 76 index: 66
26	ExprTuple	32
27	Conditional	value: 33 condition: 34
28	Literal
29	ExprTuple	35
30	Literal
31	ExprTuple	36
32	ExprRange	lambda_map: 37 start_index: 74 end_index: 75
33	Operation	operator: 38 operands: 39
34	Operation	operator: 40 operands: 41
35	ExprRange	lambda_map: 42 start_index: 74 end_index: 83
36	Variable
37	Lambda	parameter: 66 body: 56
38	Literal
39	ExprTuple	43, 44
40	Literal
41	ExprTuple	45
42	Lambda	parameter: 66 body: 46
43	Variable
44	Operation	operator: 47 operand: 50
45	ExprRange	lambda_map: 49 start_index: 74 end_index: 75
46	IndexedVar	variable: 77 index: 66
47	Literal
48	ExprTuple	50
49	Lambda	parameter: 66 body: 51
50	Lambda	parameter: 81 body: 52
51	Operation	operator: 60 operands: 53
52	Conditional	value: 54 condition: 55
53	ExprTuple	56, 57
54	Operation	operator: 58 operands: 59
55	Operation	operator: 60 operands: 61
56	IndexedVar	variable: 67 index: 66
57	Literal
58	Literal
59	ExprTuple	63, 64
60	Literal
61	ExprTuple	81, 65
62	ExprTuple	66
63	IndexedVar	variable: 67 index: 81
64	Operation	operator: 68 operands: 69
65	Operation	operator: 70 operands: 71
66	Variable
67	Variable
68	Literal
69	ExprTuple	72, 73
70	Literal
71	ExprTuple	74, 75
72	IndexedVar	variable: 76 index: 81
73	IndexedVar	variable: 77 index: 81
74	Literal
75	Operation	operator: 79 operands: 80
76	Variable
77	Variable
78	ExprTuple	81
79	Literal
80	ExprTuple	82, 83
81	Variable
82	Variable
83	Variable

Expression of type Exists¶

from the theory of proveit.linear_algebra.inner_products¶

Expression of type Exists ¶

from the theory of proveit.linear_algebra.inner_products ¶