Expression of type Lambda

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, Lambda, 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 = Lambda([b_1_to_n], 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))))

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())

\left(b_{1}, b_{2}, \ldots, b_{n}\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

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