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 A, B, Conditional, ExprRange, IndexedVar, Lambda, 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 = Lambda([a_1_to_m], 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))))

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(a_{1}, a_{2}, \ldots, a_{m}\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

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