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 Conditional, ExprTuple, Lambda, M, i, n
from proveit.core_expr_types import a_1_to_n, a_i, x_1_to_n, x_i
from proveit.linear_algebra import Hspace, OrthoNormBases, OrthoProj, ScalarMult, Span, VecSum
from proveit.logic import Equals, Exists, InSet, Set
from proveit.numbers import Complex, Interval, one

# build up the expression from sub-expressions
expr = ExprTuple(Lambda([x_1_to_n], Conditional(Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(M, VecSum(index_or_indices = [i], summand = ScalarMult(a_i, OrthoProj(Hspace, Span(Set(x_i)))), domain = Interval(one, n))), domain = Complex).with_wrapping(), InSet(Set(x_1_to_n), OrthoNormBases(Hspace)))))

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(x_{1}, x_{2}, \ldots, x_{n}\right) \mapsto \left\{\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\\
\left(M = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \textrm{OrthoProj}\left(\mathcal{H}, \textrm{Span}\left(\left\{x_{i}\right\}\right)\right)\right)\right)\right)\end{array} \textrm{ if } \left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\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: 13 body: 2
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 43 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameters: 11 body: 12
9	Operation	operator: 62 operands: 13
10	Operation	operator: 14 operand: 55
11	ExprTuple	16
12	Conditional	value: 17 condition: 18
13	ExprTuple	19
14	Literal
15	ExprTuple	55
16	ExprRange	lambda_map: 20 start_index: 57 end_index: 58
17	Operation	operator: 21 operands: 22
18	Operation	operator: 23 operands: 24
19	ExprRange	lambda_map: 25 start_index: 57 end_index: 58
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: 65 index: 49
30	Literal
31	ExprTuple	33
32	Lambda	parameter: 49 body: 34
33	Lambda	parameter: 67 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	67, 48
45	ExprTuple	49
46	IndexedVar	variable: 50 index: 67
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	Variable
56	Operation	operator: 59 operand: 61
57	Literal
58	Variable
59	Literal
60	ExprTuple	61
61	Operation	operator: 62 operand: 64
62	Literal
63	ExprTuple	64
64	IndexedVar	variable: 65 index: 67
65	Variable
66	ExprTuple	67
67	Variable