Expression of type ExprTuple

from the theory of proveit.physics.quantum.algebra

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 ExprTuple, M, i, n
from proveit.core_expr_types import a_1_to_n, a_i, v_1_to_n, v_i
from proveit.linear_algebra import Adj, Hspace, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Exists, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = Adj(M)
expr = ExprTuple(Equals(Qmult(sub_expr1, M), Qmult(M, sub_expr1)), Exists(instance_param_or_params = [v_1_to_n], instance_expr = Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(M, VecSum(index_or_indices = [i], summand = ScalarMult(a_i, Qmult(Ket(v_i), Bra(v_i))), domain = Interval(one, n))), domain = Complex).with_wrapping(), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(Hspace))).with_wrapping())

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(M^{\dagger} \thinspace M\right) = \left(M \thinspace M^{\dagger}\right), \begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)}~\\
\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 \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\end{array}\right]\end{array}\right)

stored_expr.style_options()

name	description	default	current value	related methods
wrap_positions	position(s) at which wrapping is to occur; 'n' is after the nth comma.	()	()	('with_wrapping_at',)
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	left	left	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1, 2
1	Operation	operator: 36 operands: 3
2	Operation	operator: 19 operand: 7
3	ExprTuple	5, 6
4	ExprTuple	7
5	Operation	operator: 71 operands: 8
6	Operation	operator: 71 operands: 9
7	Lambda	parameters: 10 body: 11
8	ExprTuple	12, 42
9	ExprTuple	42, 12
10	ExprTuple	13
11	Conditional	value: 14 condition: 15
12	Operation	operator: 16 operand: 42
13	ExprRange	lambda_map: 18 start_index: 77 end_index: 78
14	Operation	operator: 19 operand: 22
15	Operation	operator: 63 operands: 21
16	Literal
17	ExprTuple	42
18	Lambda	parameter: 69 body: 60
19	Literal
20	ExprTuple	22
21	ExprTuple	23, 24
22	Lambda	parameters: 25 body: 26
23	Operation	operator: 40 operand: 33
24	Operation	operator: 28 operand: 34
25	ExprTuple	30
26	Conditional	value: 31 condition: 32
27	ExprTuple	33
28	Literal
29	ExprTuple	34
30	ExprRange	lambda_map: 35 start_index: 77 end_index: 78
31	Operation	operator: 36 operands: 37
32	Operation	operator: 38 operands: 39
33	Operation	operator: 40 operands: 41
34	Variable
35	Lambda	parameter: 69 body: 58
36	Literal
37	ExprTuple	42, 43
38	Literal
39	ExprTuple	44
40	Literal
41	ExprTuple	45
42	Variable
43	Operation	operator: 46 operand: 50
44	ExprRange	lambda_map: 48 start_index: 77 end_index: 78
45	ExprRange	lambda_map: 49 start_index: 77 end_index: 78
46	Literal
47	ExprTuple	50
48	Lambda	parameter: 69 body: 51
49	Lambda	parameter: 69 body: 52
50	Lambda	parameter: 85 body: 53
51	Operation	operator: 63 operands: 54
52	Operation	operator: 79 operand: 60
53	Conditional	value: 56 condition: 57
54	ExprTuple	58, 59
55	ExprTuple	60
56	Operation	operator: 61 operands: 62
57	Operation	operator: 63 operands: 64
58	IndexedVar	variable: 70 index: 69
59	Literal
60	IndexedVar	variable: 83 index: 69
61	Literal
62	ExprTuple	66, 67
63	Literal
64	ExprTuple	85, 68
65	ExprTuple	69
66	IndexedVar	variable: 70 index: 85
67	Operation	operator: 71 operands: 72
68	Operation	operator: 73 operands: 74
69	Variable
70	Variable
71	Literal
72	ExprTuple	75, 76
73	Literal
74	ExprTuple	77, 78
75	Operation	operator: 79 operand: 82
76	Operation	operator: 80 operand: 82
77	Literal
78	Variable
79	Literal
80	Literal
81	ExprTuple	82
82	IndexedVar	variable: 83 index: 85
83	Variable
84	ExprTuple	85
85	Variable