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 B, Conditional, 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 = Conditional(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(), InSet(Set(b_1_to_n), OrthoNormBases(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_{\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} \textrm{ if } \left\{b_{1}, b_{2}, \ldots, b_{n}\right\} \in \textrm{O.N.Bases}\left(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: 3 operand: 6
2	Operation	operator: 43 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7, 8
6	Lambda	parameters: 9 body: 10
7	Operation	operator: 11 operands: 12
8	Operation	operator: 13 operand: 19
9	ExprTuple	15
10	Conditional	value: 16 condition: 17
11	Literal
12	ExprTuple	18
13	Literal
14	ExprTuple	19
15	ExprRange	lambda_map: 20 start_index: 57 end_index: 58
16	Operation	operator: 21 operands: 22
17	Operation	operator: 23 operands: 24
18	ExprRange	lambda_map: 25 start_index: 57 end_index: 66
19	Variable
20	Lambda	parameter: 49 body: 39
21	Literal
22	ExprTuple	26, 27
23	Literal
24	ExprTuple	28
25	Lambda	parameter: 49 body: 29
26	Variable
27	Operation	operator: 30 operand: 33
28	ExprRange	lambda_map: 32 start_index: 57 end_index: 58
29	IndexedVar	variable: 60 index: 49
30	Literal
31	ExprTuple	33
32	Lambda	parameter: 49 body: 34
33	Lambda	parameter: 64 body: 35
34	Operation	operator: 43 operands: 36
35	Conditional	value: 37 condition: 38
36	ExprTuple	39, 40
37	Operation	operator: 41 operands: 42
38	Operation	operator: 43 operands: 44
39	IndexedVar	variable: 50 index: 49
40	Literal
41	Literal
42	ExprTuple	46, 47
43	Literal
44	ExprTuple	64, 48
45	ExprTuple	49
46	IndexedVar	variable: 50 index: 64
47	Operation	operator: 51 operands: 52
48	Operation	operator: 53 operands: 54
49	Variable
50	Variable
51	Literal
52	ExprTuple	55, 56
53	Literal
54	ExprTuple	57, 58
55	IndexedVar	variable: 59 index: 64
56	IndexedVar	variable: 60 index: 64
57	Literal
58	Operation	operator: 62 operands: 63
59	Variable
60	Variable
61	ExprTuple	64
62	Literal
63	ExprTuple	65, 66
64	Variable
65	Variable
66	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 ¶