Expression of type ExprTuple

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, ExprTuple, 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 = ExprTuple(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(\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..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameters: 11 body: 2
2	Conditional	value: 3 condition: 4
3	Operation	operator: 17 operand: 7
4	Operation	operator: 59 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 28 body: 10
8	Operation	operator: 27 operands: 11
9	Operation	operator: 29 operand: 16
10	Conditional	value: 13 condition: 14
11	ExprTuple	15
12	ExprTuple	16
13	Operation	operator: 17 operand: 21
14	Operation	operator: 59 operands: 19
15	ExprRange	lambda_map: 20 start_index: 73 end_index: 81
16	Variable
17	Literal
18	ExprTuple	21
19	ExprTuple	22, 23
20	Lambda	parameter: 65 body: 24
21	Lambda	parameters: 25 body: 26
22	Operation	operator: 27 operands: 28
23	Operation	operator: 29 operand: 35
24	IndexedVar	variable: 75 index: 65
25	ExprTuple	31
26	Conditional	value: 32 condition: 33
27	Literal
28	ExprTuple	34
29	Literal
30	ExprTuple	35
31	ExprRange	lambda_map: 36 start_index: 73 end_index: 74
32	Operation	operator: 37 operands: 38
33	Operation	operator: 39 operands: 40
34	ExprRange	lambda_map: 41 start_index: 73 end_index: 82
35	Variable
36	Lambda	parameter: 65 body: 55
37	Literal
38	ExprTuple	42, 43
39	Literal
40	ExprTuple	44
41	Lambda	parameter: 65 body: 45
42	Variable
43	Operation	operator: 46 operand: 49
44	ExprRange	lambda_map: 48 start_index: 73 end_index: 74
45	IndexedVar	variable: 76 index: 65
46	Literal
47	ExprTuple	49
48	Lambda	parameter: 65 body: 50
49	Lambda	parameter: 80 body: 51
50	Operation	operator: 59 operands: 52
51	Conditional	value: 53 condition: 54
52	ExprTuple	55, 56
53	Operation	operator: 57 operands: 58
54	Operation	operator: 59 operands: 60
55	IndexedVar	variable: 66 index: 65
56	Literal
57	Literal
58	ExprTuple	62, 63
59	Literal
60	ExprTuple	80, 64
61	ExprTuple	65
62	IndexedVar	variable: 66 index: 80
63	Operation	operator: 67 operands: 68
64	Operation	operator: 69 operands: 70
65	Variable
66	Variable
67	Literal
68	ExprTuple	71, 72
69	Literal
70	ExprTuple	73, 74
71	IndexedVar	variable: 75 index: 80
72	IndexedVar	variable: 76 index: 80
73	Literal
74	Operation	operator: 78 operands: 79
75	Variable
76	Variable
77	ExprTuple	80
78	Literal
79	ExprTuple	81, 82
80	Variable
81	Variable
82	Variable