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 = [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(), InSet(Set(a_1_to_m), OrthoNormBases(A)))

# 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_{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} \textrm{ if } \left\{a_{1}, a_{2}, \ldots, a_{m}\right\} \in \textrm{O.N.Bases}\left(A\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: 5
2	Operation	operator: 57 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7
5	Lambda	parameters: 26 body: 8
6	Operation	operator: 25 operands: 9
7	Operation	operator: 27 operand: 14
8	Conditional	value: 11 condition: 12
9	ExprTuple	13
10	ExprTuple	14
11	Operation	operator: 15 operand: 19
12	Operation	operator: 57 operands: 17
13	ExprRange	lambda_map: 18 start_index: 71 end_index: 79
14	Variable
15	Literal
16	ExprTuple	19
17	ExprTuple	20, 21
18	Lambda	parameter: 63 body: 22
19	Lambda	parameters: 23 body: 24
20	Operation	operator: 25 operands: 26
21	Operation	operator: 27 operand: 33
22	IndexedVar	variable: 73 index: 63
23	ExprTuple	29
24	Conditional	value: 30 condition: 31
25	Literal
26	ExprTuple	32
27	Literal
28	ExprTuple	33
29	ExprRange	lambda_map: 34 start_index: 71 end_index: 72
30	Operation	operator: 35 operands: 36
31	Operation	operator: 37 operands: 38
32	ExprRange	lambda_map: 39 start_index: 71 end_index: 80
33	Variable
34	Lambda	parameter: 63 body: 53
35	Literal
36	ExprTuple	40, 41
37	Literal
38	ExprTuple	42
39	Lambda	parameter: 63 body: 43
40	Variable
41	Operation	operator: 44 operand: 47
42	ExprRange	lambda_map: 46 start_index: 71 end_index: 72
43	IndexedVar	variable: 74 index: 63
44	Literal
45	ExprTuple	47
46	Lambda	parameter: 63 body: 48
47	Lambda	parameter: 78 body: 49
48	Operation	operator: 57 operands: 50
49	Conditional	value: 51 condition: 52
50	ExprTuple	53, 54
51	Operation	operator: 55 operands: 56
52	Operation	operator: 57 operands: 58
53	IndexedVar	variable: 64 index: 63
54	Literal
55	Literal
56	ExprTuple	60, 61
57	Literal
58	ExprTuple	78, 62
59	ExprTuple	63
60	IndexedVar	variable: 64 index: 78
61	Operation	operator: 65 operands: 66
62	Operation	operator: 67 operands: 68
63	Variable
64	Variable
65	Literal
66	ExprTuple	69, 70
67	Literal
68	ExprTuple	71, 72
69	IndexedVar	variable: 73 index: 78
70	IndexedVar	variable: 74 index: 78
71	Literal
72	Operation	operator: 76 operands: 77
73	Variable
74	Variable
75	ExprTuple	78
76	Literal
77	ExprTuple	79, 80
78	Variable
79	Variable
80	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 ¶