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(v, Conditional(Exists(instance_param_or_params = [a_1_to_m], instance_expr = 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(), condition = InSet(Set(a_1_to_m), OrthoNormBases(A))).with_wrapping(), InSet(v, TensorProd(A, 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(v \mapsto \left\{\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{m}~|~\left\{a_{1}, a_{2}, \ldots, a_{m}\right\} \in \textrm{O.N.Bases}\left(A\right)}~\\
\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}\right]\end{array} \textrm{ if } v \in \left(A {\otimes} B\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	parameter: 51 body: 3
2	ExprTuple	51
3	Conditional	value: 4 condition: 5
4	Operation	operator: 26 operand: 8
5	Operation	operator: 68 operands: 7
6	ExprTuple	8
7	ExprTuple	51, 9
8	Lambda	parameters: 20 body: 10
9	Operation	operator: 76 operands: 11
10	Conditional	value: 12 condition: 13
11	ExprTuple	25, 44
12	Operation	operator: 26 operand: 16
13	Operation	operator: 68 operands: 15
14	ExprTuple	16
15	ExprTuple	17, 18
16	Lambda	parameters: 37 body: 19
17	Operation	operator: 36 operands: 20
18	Operation	operator: 38 operand: 25
19	Conditional	value: 22 condition: 23
20	ExprTuple	24
21	ExprTuple	25
22	Operation	operator: 26 operand: 30
23	Operation	operator: 68 operands: 28
24	ExprRange	lambda_map: 29 start_index: 82 end_index: 90
25	Variable
26	Literal
27	ExprTuple	30
28	ExprTuple	31, 32
29	Lambda	parameter: 74 body: 33
30	Lambda	parameters: 34 body: 35
31	Operation	operator: 36 operands: 37
32	Operation	operator: 38 operand: 44
33	IndexedVar	variable: 84 index: 74
34	ExprTuple	40
35	Conditional	value: 41 condition: 42
36	Literal
37	ExprTuple	43
38	Literal
39	ExprTuple	44
40	ExprRange	lambda_map: 45 start_index: 82 end_index: 83
41	Operation	operator: 46 operands: 47
42	Operation	operator: 48 operands: 49
43	ExprRange	lambda_map: 50 start_index: 82 end_index: 91
44	Variable
45	Lambda	parameter: 74 body: 64
46	Literal
47	ExprTuple	51, 52
48	Literal
49	ExprTuple	53
50	Lambda	parameter: 74 body: 54
51	Variable
52	Operation	operator: 55 operand: 58
53	ExprRange	lambda_map: 57 start_index: 82 end_index: 83
54	IndexedVar	variable: 85 index: 74
55	Literal
56	ExprTuple	58
57	Lambda	parameter: 74 body: 59
58	Lambda	parameter: 89 body: 60
59	Operation	operator: 68 operands: 61
60	Conditional	value: 62 condition: 63
61	ExprTuple	64, 65
62	Operation	operator: 66 operands: 67
63	Operation	operator: 68 operands: 69
64	IndexedVar	variable: 75 index: 74
65	Literal
66	Literal
67	ExprTuple	71, 72
68	Literal
69	ExprTuple	89, 73
70	ExprTuple	74
71	IndexedVar	variable: 75 index: 89
72	Operation	operator: 76 operands: 77
73	Operation	operator: 78 operands: 79
74	Variable
75	Variable
76	Literal
77	ExprTuple	80, 81
78	Literal
79	ExprTuple	82, 83
80	IndexedVar	variable: 84 index: 89
81	IndexedVar	variable: 85 index: 89
82	Literal
83	Operation	operator: 87 operands: 88
84	Variable
85	Variable
86	ExprTuple	89
87	Literal
88	ExprTuple	90, 91
89	Variable
90	Variable
91	Variable