Expression of type Conditional ¶

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, Conditional, 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 = Conditional(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(), InSet(v, TensorProd(A, B)))

# 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\{\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} \textrm{ if } v \in \left(A {\otimes} B\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: 23 operand: 5
2	Operation	operator: 65 operands: 4
3	ExprTuple	5
4	ExprTuple	48, 6
5	Lambda	parameters: 17 body: 7
6	Operation	operator: 73 operands: 8
7	Conditional	value: 9 condition: 10
8	ExprTuple	22, 41
9	Operation	operator: 23 operand: 13
10	Operation	operator: 65 operands: 12
11	ExprTuple	13
12	ExprTuple	14, 15
13	Lambda	parameters: 34 body: 16
14	Operation	operator: 33 operands: 17
15	Operation	operator: 35 operand: 22
16	Conditional	value: 19 condition: 20
17	ExprTuple	21
18	ExprTuple	22
19	Operation	operator: 23 operand: 27
20	Operation	operator: 65 operands: 25
21	ExprRange	lambda_map: 26 start_index: 79 end_index: 87
22	Variable
23	Literal
24	ExprTuple	27
25	ExprTuple	28, 29
26	Lambda	parameter: 71 body: 30
27	Lambda	parameters: 31 body: 32
28	Operation	operator: 33 operands: 34
29	Operation	operator: 35 operand: 41
30	IndexedVar	variable: 81 index: 71
31	ExprTuple	37
32	Conditional	value: 38 condition: 39
33	Literal
34	ExprTuple	40
35	Literal
36	ExprTuple	41
37	ExprRange	lambda_map: 42 start_index: 79 end_index: 80
38	Operation	operator: 43 operands: 44
39	Operation	operator: 45 operands: 46
40	ExprRange	lambda_map: 47 start_index: 79 end_index: 88
41	Variable
42	Lambda	parameter: 71 body: 61
43	Literal
44	ExprTuple	48, 49
45	Literal
46	ExprTuple	50
47	Lambda	parameter: 71 body: 51
48	Variable
49	Operation	operator: 52 operand: 55
50	ExprRange	lambda_map: 54 start_index: 79 end_index: 80
51	IndexedVar	variable: 82 index: 71
52	Literal
53	ExprTuple	55
54	Lambda	parameter: 71 body: 56
55	Lambda	parameter: 86 body: 57
56	Operation	operator: 65 operands: 58
57	Conditional	value: 59 condition: 60
58	ExprTuple	61, 62
59	Operation	operator: 63 operands: 64
60	Operation	operator: 65 operands: 66
61	IndexedVar	variable: 72 index: 71
62	Literal
63	Literal
64	ExprTuple	68, 69
65	Literal
66	ExprTuple	86, 70
67	ExprTuple	71
68	IndexedVar	variable: 72 index: 86
69	Operation	operator: 73 operands: 74
70	Operation	operator: 75 operands: 76
71	Variable
72	Variable
73	Literal
74	ExprTuple	77, 78
75	Literal
76	ExprTuple	79, 80
77	IndexedVar	variable: 81 index: 86
78	IndexedVar	variable: 82 index: 86
79	Literal
80	Operation	operator: 84 operands: 85
81	Variable
82	Variable
83	ExprTuple	86
84	Literal
85	ExprTuple	87, 88
86	Variable
87	Variable
88	Variable

Expression of type Conditional¶

from the theory of proveit.linear_algebra.inner_products¶

Expression of type Conditional ¶

from the theory of proveit.linear_algebra.inner_products ¶